Cooperative Base Station Coloring for Pair-wise Multi-Cell Coordination
Jeonghun~Park
Namyoon~Lee
and~Robert W. Heath Jr. %\IEEEauthorblockA{Department of Electrical and Computer Eng.
%The University of Texas at Austin
%Austin
TX 78712 USA
%Email: \{jeonghun\
%A semi-static base station (BS) coordination strategy exploits predefined multiple BS cluster patterns, i.e., a set of BS clusters that use the same time-frequency resource, for cooperative transmissions to improve the cell-edge user throughput. %To obtain high edge user throughput, CM requires efficient BS cluster patterns, where edge users are guaranteed to be protected from out-of-cluster interference. To design such BS cluster patterns in cellular networks whose BS locations are irregular, however, is difficult in general. %In implementing dynamic clustering in a network, a base station (BS) selection conflict problem, which occurs when two different users select the same BS, is a main bottleneck. Conventionally, a global scheduler is needed to solve this problem, which brings huge complexity. %To resolve the BS selection conflict problem with a systematic way, This paper proposes a method for designing base station (BS) clusters and cluster patterns for pair-wise BS coordination. %where edge users are guaranteed to be protected from dominant out-of-cluster interference in an irregular cellular network. %The proposed BS cluster pattern is designed to use two core ideas: 1) the 2nd-order Voronoi regions of deployed BSs' locations and 2) the edge-coloring for the graph whose each vertex is a BS location and each edge is the Delaunay triangulation. The key idea is that each BS cluster is formed by using the 2nd-order Voronoi region, and the BS clusters are assigned to a specific cluster pattern by using edge-coloring for a graph drawn by Delaunay triangulation. %The proposed BS cluster pattern is designed to use two core features: 1) make each BS cluster by using the notion of the 2nd-order Voronoi region and 2) the edge-coloring for the graph comprising a set of vertices for BS locations and a set of edges induced by the Delaunay triangulation. The main advantage of the proposed method is that the BS selection conflict problem is prevented, while users are guaranteed to communicate with their two closest BSs in any irregular BS topology. %that users are guaranteed to communicate with their two closest BSs set without the BS selection conflict problemin any irregular BS topology. %irrespective of a BS deployment scenario. % via coordinated beamforming. With the proposed coordination method, analytical expressions for the rate distribution and the ergodic spectral efficiency are derived as a function of relevant system parameters in a fixed irregular network model. In a random network model with a homogeneous Poisson point process, a lower bound on the ergodic spectral efficiency is characterized. %for a random network where locations of BSs and users are modeled by using a homogeneous Poisson point process. %Through simulations, the derived analytical expressions are verified. Through system level simulations, the performance of the proposed method is compared with that of conventional coordination methods: dynamic clustering and static clustering. Our major finding is that, when users are dense enough in a network, the proposed method provides the same level of coordination benefit with dynamic clustering to edge users. %achieves the same performance of dynamic clustering with much less complexity. %provides considerable performance gains in both the rate coverage probability and the ergodic spectral efficiency for cell-edge users compared to conventional strategies. % so that the relatively fair performance is able to be provided. One noticeable point is the CM is a network centric method, where the clustering sets are made independently to users' location, thereby low overheads are entailed.
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\begin{document}

\title{Cooperative Base Station Coloring for Pair-wise Multi-Cell Coordination}

\author{Jeonghun~Park,
 Namyoon~Lee,
 and~Robert W. Heath Jr.
%\IEEEauthorblockA{Department of Electrical and Computer Eng.\\
%The University of Texas at Austin\\
%Austin, TX 78712 USA\\
%Email: \{jeonghun\}@utexas.edu}}
\thanks{J. Park and R. W. Heath Jr. are with the Wireless Networking and Communication Group (WNCG), Department of Electrical and Computer Engineering, 
The University of Texas at Austin, TX 78701, USA. (E-mail: $\left\{\right.$jeonghun, rheath$\left\}\right.$@utexas.edu) 

N. Lee is with Intel Labs, 2200 Mission College Blvd, Santa Clara, CA 95054, USA (Email:namyoon.lee@intel.com)

This research was supported by a gift from Huawei Technologies Co. Ltd.}}

 
\maketitle \setcounter{page}{1} 

%\markboth{\emph{Transaction on Wireless Comm.},~2014}
%{Park \emph{et al}: Rate Coverage and Ergodic Spectral Efficiency of Cluster Multiplexing with Pairwise Multi-Cell Coordination}

\begin{abstract}
%A semi-static base station (BS) coordination strategy exploits predefined multiple BS cluster patterns, i.e., a set of BS clusters that use the same time-frequency resource, for cooperative transmissions to improve the cell-edge user throughput. 
%To obtain high edge user throughput, CM requires efficient BS cluster patterns, where edge users are guaranteed to be protected from out-of-cluster interference. To design such BS cluster patterns in cellular networks whose BS locations are irregular, however, is difficult in general. 
%In implementing dynamic clustering in a network, a base station (BS) selection conflict problem, which occurs when two different users select the same BS, is a main bottleneck. Conventionally, a global scheduler is needed to solve this problem, which brings huge complexity.
%To resolve the BS selection conflict problem with a systematic way, 
This paper proposes a method for designing base station (BS) clusters and cluster patterns for pair-wise BS coordination.
%where edge users are guaranteed to be protected from dominant out-of-cluster interference in an irregular cellular network.
%The proposed BS cluster pattern is designed to use two core ideas: 1) the 2nd-order Voronoi regions of deployed BSs' locations and 2) the edge-coloring for the graph whose each vertex is a BS location and each edge is the Delaunay triangulation. 
The key idea is that each BS cluster is formed by using the 2nd-order Voronoi region, and the BS clusters are assigned to a specific cluster pattern by using edge-coloring for a graph drawn by Delaunay triangulation.
%The proposed BS cluster pattern is designed to use two core features: 1) make each BS cluster by using the notion of the 2nd-order Voronoi region and 2) the edge-coloring for the graph comprising a set of vertices for BS locations and a set of edges induced by the Delaunay triangulation. 
The main advantage of the proposed method is that the BS selection conflict problem is prevented, while users are guaranteed to communicate with their two closest BSs in any irregular BS topology.
%that users are guaranteed to communicate with their two closest BSs set without the BS selection conflict problemin any irregular BS topology.
%irrespective of a BS deployment scenario.
% via coordinated beamforming. 
With the proposed coordination method, analytical expressions for the rate distribution and the ergodic spectral efficiency are derived as a function of relevant system parameters in a fixed irregular network model. 
In a random network model with a homogeneous Poisson point process,
a lower bound on the ergodic spectral efficiency is characterized. 
%for a random network where locations of BSs and users are modeled by using a homogeneous Poisson point process. 
%Through simulations, the derived analytical expressions are verified.
Through system level simulations, the performance of the proposed method is compared with that of conventional coordination methods: dynamic clustering and static clustering. Our major finding is that, when users are dense enough in a network, the proposed method provides the same level of coordination benefit with dynamic clustering to edge users.
%achieves the same performance of dynamic clustering with much less complexity.
%provides considerable performance gains in both the rate coverage probability and the ergodic spectral efficiency for cell-edge users compared to conventional strategies.
% so that the relatively fair performance is able to be provided. One noticeable point is the CM is a network centric method, where the clustering sets are made independently to users' location, thereby low overheads are entailed.
\end{abstract}

\section{Introduction}
\subsection{Motivation}

%Multicell coordination is a technique for mitigating inter-cell interference where the transmission strategy are coordinated (e.g. beamforming, scheduling, and power control) among base stations (BSs) in cellular systems. 

Coordination among base stations (BSs) is a powerful approach for mitigating inter-cell interference in cellular systems \cite{lee:comp2012}. It has been shown that the sum spectral efficiency scales linearly with the signal-to-noise ratio (SNR) when all the BSs are coordinated \cite{somekh:co_multicell_zf}. 
In practice, however, coordination with a large number of BSs may not be feasible due to excessive overhead associated with the coordination, e.g., complexity, channel estimation and channel feedback \cite{lozano:2013:limitcoop}. 
One practical solution for implementing multicell coordination is to form a BS cluster so that a limited number of BSs are coordinated to control intra-cluster interference with a reasonable amount of overhead \cite{lozano:2013:limitcoop, jun:09:network}. Unfortunately, when clustering-based BSs coordination is applied in a static way \cite{Huang:static, ges10},  or equivalently in a random way \cite{6410048} independent to users' conditions, 
the performance is mainly limited by unmanageable out-of-cluster interference \cite{lozano:2013:limitcoop, Huang:static, ges10}.
%which gives us an intuition of how to make a cluster.

%Motivated by the limitation of static (or random) clustering, 
Dynamic clustering, where a BS cluster is dynamically made by users, was proposed in \cite{Papadogiannis:dclus} to overcome the performance limitation of static clustering. While dynamic clustering improves the signal-to-interference and noise ratio (SINR) of a user significantly, there are several challenges that make it difficult to realize. 
%the main bottleneck is that dynamic clustering brings network-wise issues when implementing it in a network. 
In particular, different users who share the same spectrum can select the same BS at the same time. We refer to this as ``the BS selection conflict problem."
%which is called as ``a BS selection conflict problem" in this paper.
To resolve this, users in a network need to send feedback regarding desirable BS clusters and a global scheduler is required to schedule BSs and users based on this feedback information.
%a global scheduler is needed to schedule multiple BSs and users based on feedback information regarding cooperative BS sets chosen by every user in a network. 
For this reason, performance chiefly depends on the choice of this scheduler. 
The scheduling itself, however, requires a lot of feedback overhead from the users and a substantial amount of coordination of all the BSs and the users, which causes huge system overhead.
%and the associated feedback mechanism to implement such dynamic clustering. 
%This approach, however, increases system overheads since the feedback should be collected to a centralized scheduler and also the amount of computation of the global scheduler increases as the density of users increases.
%. The scheduling itself requires a lot of feedback overhead from the users and a substantial amount of coordination of all the users. 

To solve the BS selection conflict problem without requiring global coordination, semi-static clustering was proposed in \cite{periodic:frequency, Purmehdi:2013icc}. In this method, multiple predefined BS clusters are used with different time-frequency resources. A set of BS clusters that use the same time-frequency resource is referred as ``a BS cluster pattern.'' For instance, in a square grid BS topology, four BS cluster patterns are required to cover the whole region without the BS selection conflict problem. 
Using the semi-static clustering, users can communicate with their favorable BS set. The main limitation of the existing work \cite{periodic:frequency, Purmehdi:2013icc} is that the BS clustering methods are applicable for a regular BS topology where each BS is located regularly on a grid. In practice, however, the BS topology of cellular networks is irregular \cite{Andrew:10:stocg}. In such case, BS clustering pattern design is challenging. In this work, by leveraging geometry properties, we make progress in designing BS clusters and BS cluster patterns in a semi-static way, which is applicable for any wireless networks whose topology is irregular. 

%To protect a user effectively from out-of-cluster interference, a semi-static multicell coordination strategy was proposed in \cite{periodic:frequency, Purmehdi:2013icc}. The key idea is to use different predefined BS cluster patterns, i.e., a set of BS cluster that uses the same time-frequency resource.
% for cooperative transmissions. 
%From a reason that the used BS cluster patterns are predefined, overheads and complexity caused by BS coordination is reduced by this method. 
%This approach provides an opportunity for users to communicate with their favorable set of BSs in a particular phase, improving the cluster edge-user throughput \cite{periodic:frequency, Purmehdi:2013icc}. The performance of the semi-static multicell coordination strategy depends on how the BS cluster patterns are determined. The problem of designing the BS cluster pattern is therefore important, yet it is difficult in general cellular networks with irregular BS locations. The reason is that it becomes unclear how to form multiple BS cluster patterns so that every user obtains equivalent opportunities to communicate with their favorable set of BSs. In this paper, we provide a solution to the cluster pattern design problem for pair-wise BS coordination to improve the edge-user performance in cellular downlink networks with irregular BSs' locations.


\subsection{Related Work}
%The most efficient way to protect a user from out-of-cluster interference is dynamic BS clustering, where a BS cluster is formed in a user-centric way depending on the users' conditions, e.g., distances to interfering BSs and channel quality. 
Cooperative strategies for dynamic clustering have been extensively studied in the literature \cite{Papadogiannis:dclus, pa10, wd11, gkb12, 6848065, ry07, ry09, nh10, song:vcellcoop, nrm12, NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420, 6909064, 6879305}. In \cite{Papadogiannis:dclus, pa10, wd11, gkb12}, an algorithm for making a BS cluster in dynamic clustering was proposed. For instance, given the users' conditions such as distances or channel conditions, an algorithm selects BSs to include into a BS cluster so that the joint achievable throughput of the formed BS cluster is maximized.
In \cite{6848065, ry07, ry09}, a resource allocation method to enhance sum spectral efficiency was proposed for dynamic clustering. By jointly allocating resources to each BS cluster depending on their channel qualities, the sum spectral efficiency in a BS cluster can be improved.
%In \cite{mf07, gsz11}, a user scheduling method for improving user throughput was proposed. 
%a resource allocation or a user scheduling method to enhance sum spectral efficiency was proposed also for a dynamic BS clustering approach.
In \cite{nh10, song:vcellcoop, nrm12}, a beamforming algorithm 
was proposed in dynamic clustering aiming complexity reduction or sum spectral efficiency improvement.
In this work \cite{Papadogiannis:dclus, pa10, wd11, gkb12, 6848065, ry07, ry09, nh10, song:vcellcoop, nrm12}, a toy network, where only limited number of BSs and users are 
assumed, was considered.
%Specifically, in \cite{??}, -- was proposed for --.
%In \cite{nh10}, a linear interference nulling technique called soft interference nulling is proposed in a dynamic BS clustering method with overlapping coordination clusters. Similar to \cite{nh10}, in \cite{song:vcellcoop}, a beamformer optimization problem was tackled in overlapped BS clusters, formed by a user-centric way. 
In \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420}, the performance of dynamic clustering was characterized in a random network where BSs are distributed according to a homogeneous Poisson point process (PPP).
In \cite{6909064}, pair-wise dynamic clustering with rate splitting method was proposed assuming a Poisson network.
In \cite{6879305}, dynamic clustering in two-tier network was proposed and analyzed also in a Poisson network.
%By applying the stochastic geometry approach as in \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420, 6909064, 6879305 }, while taking a different performance metric, the average error symbol probability were characterized when using a frequency reuse interference management method in \cite{7151123}. Unlike \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420, 6909064, 6879305}, in \cite{7151123}, no BS clustering was considered.
The main limitation of the prior work considering BS clustering in a random network \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420, 6909064, 6879305} is that they did not provide a solution for resolving the BS selection conflict problem, rather they focused on the performance characterization (e.g., the coverage probability) under the assumption that a global scheduler allows such BS clustering to a typical user without the BS selection conflict problem. Global scheduling has higher complexity as overhead and scheduling are required between all possible BS clusters. 
%As mentioned in the previous section, this approach causes huge system overheads. 
The main difference between this paper and the prior work in \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6928420, 6909064, 6879305} is that we focus on providing a systematic solution for the BS selection conflict problem without using a global scheduler, so that the proposed method is able to be implemented in a network with reasonable overhead.

%work including [R1] and [R2], however, did not resolve this BS selection conflict problem, rather they focused on the performance characterization (e.g. the coverage probability) when a genie scheduler allows such dynamic BS clustering without any BS selection conflict.

%In these approaches, each user dynamically selects a set of cooperative BSs based on user's location or user's channel quality. Therefore, a user always can be located near the cluster-center, thereby reducing the out-of-cluster interference effectively.

%In general, the main limitations of dynamic clustering are the overheads in dynamically forming BS clusters and the high complexity of the user scheduling \cite{rc09}. 
%Basically, to form a BS cluster dynamically, every user's channel condition and distance to each BS are required to use 
%are reported to a centralized unit. Even with cloud radio access network \cite{cran_whitep} is employed, Furthermore, 
%For example, in \cite{nh10, song:vcellcoop}, the transmitted data is sent to other BSs through backhaul within the same BS cluster, but the BS cluster changes by users' conditions. A centralized unit \cite{cran_whitep}, therefore, should control the backhaul traffic whenever the instantaneous channel conditions of each user change, which causes high complexity. In \cite{NY:dynamic, junzhang:dynamic}, each user selects a serving BS set, in which users should be carefully scheduled to prevent conflict, i.e., the situation where multiple users want the same BS simultaneously.

%Motivated by overcoming the drawbacks of dynamic clustering, in \cite{vtc_usercentric, yuhuan:noedge14} 
%semi-dynamic clustering was proposed, where dynamic BS clustering is applied under a limited number of candidate cooperative BSs. 
%Although semi-dynamic clustering reduces complexity from selecting a BS cluster 
%compared to fully dynamic ways, overheads from complex user scheduling are still required. 
%Addition to that, a network operator
%These methods, however, still need high complexity signaling overheads compared to the static BS clustering methods since a network operator should exploit users' channel quality in determining cooperative sets. 
%To reduce complexity even than the semi-dynamic clustering, 
%To refine this,
%in \cite{periodic:frequency, Purmehdi:2013icc}, semi-static BS clustering was proposed. The main idea of semi-static clustering is to use multiple predefined cluster patterns  so that edge users have a chance to be protected from dominant out-of-cluster interference. 
%Since used BS cluster patterns are predefined, required complexity is decreased compared to dynamic or semi-dynamic clustering.
%The existing semi-static BS clustering approaches \cite{periodic:frequency, Purmehdi:2013icc}, however, mainly focused on network topologies where the BSs' locations are regularly placed on a grid, and in those network models, it is trivial to design BS cluster patterns.
% even if the network size is big. 
%In a more practical network model where BSs' locations are irregular, it is not clear how to form a BS cluster and map the formed BS cluster into multiple cluster patterns. 


%There is no general rule for forming a BS cluster and constructing BS cluster patterns in a irregular network topology with which every user on a plane is guaranteed to be protected from the dominant interference, i.e., the interference from the closest interfering BS. 
%it is not trivial that which BS cluster patterns guarantee every user to be protected from dominant interference, i.e., the interference from the closest interfering BS.
%When employing a semi-static BS coordination without appropriate cluster pattern in irregular network topologies, significant performance improvement cannot be achieved since it is possible that an edge-user is exposed to the dominant interference.
%It is not clear, however, how to form a BS cluster in a network whose BSs' locations are irregular.
%Furthermore, they relied on simulation results to evaluate the performance. 

%In this paper, we first propose how to form a BS cluster for the CM approach to assure that users is always in a cluster-center region, in which no dominant interference significantly degrades the performance. Then, we 
%find analytical expressions of the performance in a closed form to provide intuition that which system parameters has a significant effect to the performance.

%analyze the rate coverage and the ergodic spectral efficiency to show that the CM with the proposed cluster patterns leads significant performance improvement especially for edge users.
%The main idea in \cite{periodic: frequency} is to use multiple frequency bands based on predetermined patterns so that a cluster-edge user is able to be protected from other cell interference. In \cite{periodic: frequency}, however, the transmission method is restricted to a single-user scenario, and a cluster-edge user can suffer from other cell interference, which cannot lead to enough performance improvements for cluster-edge users. 
% to dominant interference in some phases, which degrades the performance.  

%In contrast, a multi-user transmission method using multiple transmit antennas and an universal frequency reuse are considered in our work, and other cell interference is removed by using the coordinated beamforming (CBF) in conjunction with the proposed CM. We also characterize the ergodic spectral efficiency and the rate coverage probability of the proposed CM. To the best of our knowledge, none of the existing work studying semi-static BS clustering approaches 
%including \cite{periodic: frequency} 
%have tackled this.

%Stochastic geometry approach are 
%In prior work \cite{Huang:static, NY:dynamic, junzhang:dynamic}, the performance of BS clustering was analyzed by using tools from stochastic geometry due to its tractability in analyzing systems with many interferers. 
%While stochastic geometry makes the analysis of dynamic BS clustering tractable, in a static BS clustering case, an analytical expression of ergodic spectral efficiency is difficult to obtain since behaviors of static BS clusters cannot be captured in a random network modeled by using stochastic geometry. Therefore, \cite{Huang:static} derived only scaling behaviors of outage probability. 
%Another drawback of modeling BSs' location using stochastic geometry is that there is non-zero probability that two BSs' locations can be arbitrarily close to each other. In practice, however, locations of BSs are determined by cell planning, which keeps certain distance among BSs' locations. 
%In contrast to the stochastic geometry approach, the deterministic grid model is able to reflect this feature of certain distance among BSs' locations, and also capture behaviors of static BS clusters as well. 
%In contrast to the stochastic geometry approach, however, the performance analysis using deterministic grid models relied on simulations and they were seldom analyzed theoretically. 
%For these reasons, we analyze the performance of the CM considering the deterministic grid network model.
%\begin{remark}
%A grid model is, however, worth to be tackled since its regular geometry explains well a network-centric way of cooperation, like the proposed CM method. In this paper, we adopt a two-dimensional deterministic square grid model and analyze the performance as a function of distances between a user and BSs in the network.
%\end{remark}

%scenario is considered, and by using the proposed CM, a user is free from dominant interference, achieving more performance improvement for a edge user.

%Even in the semi-dynamical approaches \cite{vtc_usercentric, yuhuan:noedge14}, however, the overheads required for BS clustering incresases as the number of users increases. This is because a BS clustering set is dependent on a user, which leads us to necessity for developing new semi-static way of clustering.
%
%
%The frequency reuse strategy where the cell center and the cell edge are served with different frequency can improve the performance more, to focus on the proposed CM concept, that is now considered in this paper.
%
%
%The most distinguished feature of our approach is that we use the clustering patterns designed from a network-centric way, which is totally independent to users. The advantage of this network-centric approach is that much lower overhead is required compared to the user centric (dynamic \cite{ltw04, bh07b, ssl05, ven07, mf07a, wa07, ki09, ht09, jun:09:network, ges10, co11, zhn12, chns12} or semi-dynamic BS clustering \cite{vtc_usercentric, yuhuan:noedge14}) approach, which implies the performance improvement to cluster-edge users can be provided with low complexity.
%This work, therefore, proposes a way to take advantages from two conventional clustering methods while avoiding their weakness.
 
%Since a BS is implemented with a regular distance in a real cellular system, a deterministic grid model can capture a feature of the real cellular model. In addition to this, the regularity of deterministic grid models is fitted to explain a network centric way of cooperation like the proposed CM method.
%intuition of the proposed CM method is well explained in the deterministic grid model since network-centric
%For these reasons, in this paper, we consider a deterministic grid model to analyze the performance.
%Since it is impossible to obtain a closed-form expression in a deterministic grid model, an approximated model is exploited and we find that well approximated model including major interference terms is able to reflect an original model within sufficient exactness. (??)
%additional overhead than the static BS clustering case cannot be avoided since a BS cluster set is dependent on a user.
%Even \cite{vtc_usercentric, yuhuan:noedge14} uses the limited set of BSs, they still employed the user-centric clustering, where the cluster set is determined by a user. 

%\cite{vtc_usercentric} assumed that there are predetermined candidate BSs sets, termed by cooperation areas (CAs). The most appropriate CA is assigned to a user group (UG), then three BSs that have the largest cooperation gain are selected in the assigned CA .
%a BSs clustering method consists of two phase. Assuming that there are various predetermined candidate BSs set, termed by cooperation area (CA), the most appropriate CA is selected depending on a user group at the first phase. Then, three BSs that have the largest cooperation gain are chosen in the selected CA. 
%On the other hand, \cite{yuhuan:noedge14} assumed that a set of BSs possible to be clustered is 2-decomposable, for efficiently computing a BS clustering set. 
%More details can be found in \cite{vtc_usercentric, yuhuan:noedge14}. 
%It still, however, causes overhead since the assigned CA should be changed whenever the UG changes its location. Selecting three BSs in the assigned CA is also burdensome.
%employed a user-centric BS clustering method, which implies additional overhead is inevitable. For example, in \cite{vtc_usercentric},  assigning the most appropriate CA to a UG is burdensome since the assigned CA should be changed whenever the UG changes its location. Selecting three BSs in the CA also causes overhead.
%each UG would have to be associated to a different CA. 


\subsection{Contributions} 
In this paper, we propose a method for designing BS clusters and BS cluster patterns for pair-wise multicell coordination in a cellular downlink network whose BSs' locations are irregular. The goal of the proposed method is to ensure that selected users communicate with their two closest BSs avoiding the BS selection conflict problem. To do this, we 
leverage the concept of 2nd-order Voronoi region. Suppose two BSs are located at ${\bf d}_i \in\mathbb{R}^2$ and ${\bf d}_j \in\mathbb{R}^2$ in a two dimensional space. Then, 
the set of closer points to ${\bf{d}}_{i}$ and ${\bf{d}}_{j}$ than any other points is defined by the 2nd-order Voronoi region as
%\begin{align} \label{voronoi_defn}
%&\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right) \nonumber \\
%&= \left\{\left. {\bf{d}}\in \mathbb{R}^2 \right| 
%\left\{ \left\|{\bf{d}} - {\bf{d}}_{i} \right\| \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\|  \right\} \cap \left\{ \left\|{\bf{d}} - {\bf{d}}_{j} \right\|  \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\| \right\},\; \forall k \in \mathbb{Z}^+,\; k\ne i,j   \right\}.
%\end{align}
\begin{align} \label{voronoi_defn}
&\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right) \nonumber \\
&= \left\{ {\bf{d}}\in \mathbb{R}^2 \left| 
\begin{array}{lc}
\left\{ \left\|{\bf{d}} - {\bf{d}}_{i} \right\| \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\|  \right\}  \\
\cap \left\{ \left\|{\bf{d}} - {\bf{d}}_{j} \right\|  \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\| \right\},
\\\; \forall k \in \mathbb{Z}^+,\; k\ne i,j  \end{array} \right. \right\},
\end{align}
The core feature of the 2nd-order Voronoi region is that 
%The 2nd-order Voronoi region $\mathcal{V}_{2}\left({\bf{d}}_{i}, {\bf{d}}_{j} \right)$, which can be defined by an arbitrary point pair located at $\{{\bf{d}}_{i}, {\bf{d}}_{j}\}$ in any irregular point process, 
it characterizes an area where any point in that area has the minimum distances to $\{{\bf{d}}_{i}, {\bf{d}}_{j}\}$. If a BS pair located at $\{{\bf{d}}_{i}, {\bf{d}}_{j} \}$
forms a BS cluster and serves users in $\mathcal{V}_{2}\left({\bf{d}}_{i}, {\bf{d}}_{j} \right)$, those users are guaranteed to communicate with their two closest BSs. 
%To apply this strategy to the whole network, 
Applying this to the whole network,
the first step of the method is tessellating a network plane into the 2nd-order Voronoi regions of every feasible pair of BS locations and making a BS cluster according to 2nd-order Voronoi regions.
% and serve users in each region by the two closest BSs.
%using a BS cluster formed by the corresponding 2nd-order Voronoi region. 
Next step is designing BS cluster patterns by allocating time-frequency resources to each BS cluster for preventing the BS selection conflict problem. 
To do this, we adopt the concept of Delaunay triangulation and edge-coloring in graph theory. At the best of authors' knowledge, it is the first work to solve the BS selection conflict problem in irregular BS topologies. By solving this, users are able to obtain BS coordination benefit by communicating with their two closest BSs.
%This strategy, however, can cause BS conflicts, i.e., more than two users from different regions want to communicate with the same BS simultaneously. 
%To prevent this, BS clusters are assigned into multiple cluster patterns and each cluster pattern uses different time-frequency resource.
% are allocated to each cluster pattern.
%we also propose a cluster pattern construction method by solving the edge-coloring problem of the graph whose each vertex is a BS location and each edge is the Delaunay triangulation of the network.
%allocating 
%different time-frequency resource to each BS cluster. 
%by using edge-coloring from graph theory. 
We explain the proposed idea more rigorously in Section II. 
%Further, we propose an additional algorithm called as ``edge cutting and restore algorithm'' 
%for improving the more in Section III.
%With the proposed strategy, we are able to make an efficient clustering pattern for semi-static BS coordination regardless of BS deployment scenario. 
%At the best of authors' knowledge, this is 

With the proposed method, we characterize the performance of the proposed BS coordination strategy in a fixed irregular network model. Under the premise that multi-user coordinated beamforming (CBF) \cite{chae12} is applied to mitigate intra-cluster interference, we derive analytical expressions for the rate coverage probability and the ergodic spectral efficiency as a function of relevant system parameters: 1) distances between BSs and users, 2) the number of antennas per BS, 3) the number of selected users, 4) signal-to-noise ratio (SNR), and 5) the pathloss exponent. 
Next, considering a random network model where a location of each BS is modeled by using a homogeneous PPP, 
%Similarly to the case of a deterministic network model, a user is guaranteed to communicate with the most favorable BS set by alternatively exploiting the proposed cluster patterns. 
we derive a lower bound on the ergodic spectral efficiency as a function of the relevant system parameters.
% 1) the number of antennas per BS, 2) the number of selected users, and 3) the pathloss exponent.
% under the interference-limited assumption.
%The intra-cluster interference is mitigated by a CBF method as in the fixed network model case.

To demonstrate efficiency of the proposed method, we use a system level simulation to compare the edge users sum throughput of the proposed method and that of other conventional coordination methods: dynamic clustering and static clustering. 
One main finding is that when users are dense enough, the proposed method achieves the same edge user performance as dynamic clustering in which a global scheduler is assumed.
%in some favorable BS topologies. Even with other cases, the proposed algorithm has performance improvement compared to static clustering. 
%the rate coverage probability and the ergodic spectral efficiency are compared with those obtained from other ways of coordination strategies: single cell operation, fractional frequency reuse and a random BS clustering method. One main finding is that the proposed cluster pattern substantially improves the rate outage probability and the ergodic spectral efficiency of cell-edge users compared to those of the existing approaches.
%For clear understanding, a cell-edge user is defined as a user whose SINR is less than 1 with high probability, so that a rate is less than 1 bps/Hz.
This gain comes from the fact that the proposed method ensures that selected users communicate with their two closest BSs with high probability as the user density increases.
%, therefore users are effectively protected from out-of-cluster interference.

%The main contribution of this paper is on proposing a solution for a BS selection conflict problem by using a semi-static way, i.e., predefined multiple BS clusters and BS cluster patterns. At the best of authors' knowledge, it is the first work to solve this in irregular BS topologies. By solving this, users are able to obtain BS coordination benefit by communicating with their two closest BSs set.


%Since the proposed algorithm is non-dynamic, system complexity is 

%By solving a BS selection conflict problem, coordination benefit can be obtained with reasonable complexity.
%Since the proposed algorithm is a BS topology, and also is able to be applied in any irregular BS topology. 

%protected from dominant interference by the definition of the 2nd-order Voronoi region of $\{{\bf{d}}_1, {\bf{d}}_2\}$ is that a set of points closer than any other points 
%BS clustering method is to alternatively use multiple fixed BS clusters so that any user in the network is ensured to be a cluster-center user. 
%\begin{figure}[t]
%\centerline{\resizebox{1\columnwidth}{!}{\includegraphics{1_D_example.pdf}}}   
%\caption{An illustration of the proposed BS clustering pattern in one dimensional gird network model. Employing two different clustering patterns for each phase, every user in the network is able to be in the cluster-center region.}  
%   \label{1d_example} 
%\end{figure}
%We illustrate the proposed idea by an example depicted in Fig. \ref{1d_example}.
%Let us consider a one-dimensional grid network model where locations of BSs are placed on grid points and two cell-edge users exist per cell as depicted in Fig.~\ref{1d_example}. Each BS has two antennas and the users have a single antenna. 
%One should note that in this network model, the 2nd-order Voronoi region of BS $i$ and BS $i+1$ for $i \in \mathbb{Z}$ is corresponding to the line segment between BS $i$ and BS $i+1$. Using the proposed cluster pattern, BS 0 has two cluster sets each of which is with BS -1 and BS 1, therefore it alternatively uses two clusters. In phase one, BS 0 and BS -1 form a BS cluster and cooperatively serve a set of user (-1,1) and user (0,0) by applying coordinated beamforming (CBF) transmission that eliminates intra-cluster interference. In this phase, user (-1,1) and user (0,0) are protected from interference of BS 0 and BS -1, each of which is the closest interferer to each user. In the next phase, BS 0 and BS 1 make a different cluster and serve user (0,1) and user (1,0) by using the CBF method. 
%As shown in this example, by alternatively exploiting the two clustering patterns designed by the 2nd-order Voronoi region, every user is guaranteed to be in the cluster-center region, where the dominant interference from the closest BS is removed.
%
%


%, if we use two clustering patterns, called BS cluster 1 and BS cluster 2, let user 1 is in a cluster-center region for BS cluster 1 and is in a cluster-edge region for BS cluster 2, while user 2 is in a cluster-edge region for BS cluster 1 and is in a cluster-center region for BS cluster 2. When using only BS cluster 1 or 2 during two phases, user 1 or 2 is in a cluster-edge region, which suffers from out-of-cluster interference. When alternatively using BS cluster 1 and 2 one for each phase, however, the performance of user 1 or 2 has averaged resulting that none of user 1 or 2 has severely degraded performance, i.e., the edge user performance is improved. This intuition will be explained in more detail in the next section.
%Specifically, comparing to a conventional static cooperation, a user in a cluster-edge region cannot  improve its performance with a static clustering method since the used clustering set is not changed.
%By exploiting the proposed CM, however, the edge user is able to have a chance to be a cluster-center user since multiple different clustering sets is applied for the user, and this improves the edge-user performance. 
%By using multiple clusters, the performance obtained by different BS clusters are averaged, so that a user that was in a cluster-edge with one clustering set would be in a cluster-center with other clustering set.
%We call the proposed method as clustering multiplexing (CM).
%For ease of understanding, let compare the proposed CM to a static clustering method. In a static BS clustering case, the users in the cluster-edge region have the degraded performance because of inter-cluster interference. Using the CM, however, these users are able to be in the cluster-center region for some duration by multiplexing the clustering set, which improves the performance. 
%Through this, any user can be served by the uniform rate obtaining cooperation gain regardless of its location. 
%Specifically, since a user that is in a cluster-edge region with a clustering pattern, will be in a cluster-center region with another clustering pattern, users are able to be in a cluster-center region at least one time by multiplexing all the clustering patterns. This improves the edge-user performance.
% assume that a user is in a cluster-edge region with clustering pattern 1, and the user is in a cluster-center region with clustering pattern 2. Then, by multiplexing those clustering patterns, 
%the user will be a cluster-center region with other clustering pattern. 


%%Only low overhead is required in the proposed BS clustering method since fixed BS clustering patterns are used, where a BSs clustering set is determined by a system-centric way, that is totally independent to users. The main differences between our work and prior works, especially \cite{vtc_usercentric, yuhuan:noedge14}, are organized as the followings.
%\begin{itemize}
%\item In our method, a BSs clustering set is determined by a system-centric way, \emph{i.e.}, a BSs clustering set is determined independently to users. In prior works, a user-centric way of selecting a BSs clustering set is used, \emph{i.e.}, a BSs clustering set is determined depending on a user, or user group.
%\item Our target is to obtain a cooperation gain achieving the uniform user rate, while prior works aimed to obtain the largest cooperation gain.
%\end{itemize}
%The main idea of the proposed BS clustering method is to create two fixed  BS (BS) cooperation sets so that any user in the network can have an opportunity to be a cluster-center user. Specifically, as illustrated in Fig.~\ref{pairwise_cluster}, in phase one, user 0 and user -1 communicate with a  BS cluster $\left\{B_{-1}, B_{0}\right\}$. In this phase, while user 0 is well protected from dominant out-of-cluster interference, user -1 severely suffers from it as user -1 is in the cluster-edge. In the next phase, the set of BS cooperation is changed so that user -1 becomes the cluster-center user, while the user 0 is in the cluster-edge as depicted in Fig.~\ref{pairwise_cluster}. 


%There are two main points of novelty in the proposed clustering method: generality and uniformity. The generality means that the proposed clustering is able to be applied in any BS deployment scenario.
%The first and the most powerful one is the generality. The proposed clustering is applied in any BS deployment scenario.
%We propose a general set of guidelines for making an efficient BS cluster patterns for semi-static BS coordination applied in any BS deployment scenario.
%while the other strategy 
%has no limitation of application, which means it is more powerful. 
%When a network becomes densified by adopting small cells, a BS topology is irregular rather than a conventional hexagonal grid. In this environment, designing a BS coordination pattern can be more complicated. The proposed clustering strategy gives a solution for such problem. 
%The uniformity means that with the proposed clustering, all the users in the network are guaranteed to communicate with their two closest BSs, with the result that the edge-user performance is enhanced by protecting from the dominant interference. 
%To the best of authors' knowledge, there has been no work that achieved the two points simultaneously.

%In contrary to the existing dynamic clustering strategy \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014, 6879305}, where a clustering method was described only from a user-centric view, the proposed strategy is able to provide a network-wise view of cooperative regions and time-frequency resource allocation to each BS cluster. Thanks to this, the used BS cluster set and the allocated time-frequency resources are predefined, therefore the proposed strategy is operated with a semi-static way. 
%, which causes less complexity to dynamic clustering.
%which makes it possible to implement the proposed strategy in a semi-static way. 
%In additional to that, comparing to the conventional static clustering \cite{Huang:static, salam:multicell}, where some users can be located in a cluster-edge region, in the proposed strategy, every user communicates with two nearest BSs, so that there is no cluster-edge user. This guarantees higher cooperation benefit than static clustering. Summarizing, the proposed strategy achieves the performance of dynamic clustering, with the implement complexity of static clustering, i.e., better performance with less complexity.

% the proposed strategy has a strong point in implementation complexity. In contrary to 

%there are two main differences. In that we propose a strategy for a semi-static BS clustering approach, i.e., CM, our work is fundamentally different from the work that used a dynamic BS clustering approach \cite{NY:dynamic, junzhang:dynamic, Tanbourgi:2014}. From a point that the proposed strategy is able to be applied in any BS deployment scenario, our work is more general compared to the work that assumed a grid BS deployment scenario \cite{periodic:frequency, Purmehdi:2013icc}.
%In that we propose a strategy for a semi-static BS clustering approach, our paper , and in that we propose a strategy that can be applied in any BS deployment scenario, it also 




The remainder of the paper is organized as follows. In Section II, 
the system model including the proposed cluster pattern is explained. 
In Section III, the rate coverage probability is characterized and the ergodic spectral efficiency is obtained in Section IV. In Section V, 
%we show that the proposed cluster pattern is able to be applied in a random network, where locations of BSs are modeled by using a homogeneous PPP and analyze a lower bound on the ergodic spectral efficiency. 
the ergodic spectral efficiency in a random network is analyzed, followed by Section VI which provides comparisons with simulations. Section VII concludes the paper.


 
\section{System Model} 

In this section, we introduce the system model used in this paper. We first describe the network model, the pair-wise coordination model, and user association assumptions. Then the proposed BS cluster and BS cluster pattern are explained by using a toy example, and extend them to a general network model in the next subsection. Further, we introduce an algorithm for enhancing the proposed method, called as ``edge cutting algorithm."
Signal model and performance metrics are specified in the following subsection.
%The proposed cluster pattern is also explained in this section. 




\subsection{Network Model}
\begin{figure}[t]
\centering
$\begin{array}{cc}
{\resizebox{0.45\columnwidth}{!}
{\includegraphics{net_model_actual_eps.eps}}}  &
{\resizebox{0.45\columnwidth}{!}
{\includegraphics{net_model_actual_delaunay_eps.eps}}}  \\ 
\mbox{(a)} &
\mbox{(b)}
\end{array}$
\caption{(a) An example of the considered network model. 
This example is generated by a repulsive point process where a distance between any two BSs is larger than $1.4 \rm km$. 
%Generally, however, we do not assume a specific network model.
% based on a snapshot of PPP. Specifically, each point is not allowed to be located within range of $\sqrt{2}$ to other points. 
(b) A graph induced by the Delaunay triangulation corresponding to (a).} 
   \label{2d_net_model}
\end{figure} 

We consider a fixed downlink cellular network where each BS has $N$ antennas.
%Denoting the number of BS antennas as $N$, $K = N/2$ users 
As illustrated in Fig.~\ref{2d_net_model} (a), BSs are located at irregular points on a two dimensional plane. We denote each location of a BS as ${\bf{d}}_i \in \mathbb{R}^2$ for $i \in \mathbb{Z}^+$, so that $\mathcal{N} = \left\{ \left. {\bf{d}}_{i} \right| i \in \mathbb{Z}^+\right\}$ includes all the locations of the BS on the plane. 
A BS located at ${\bf{d}}_i $ for $i \in \mathbb{Z}^+$ has its own 1st-order Voronoi region ${\mathcal{V}}_1\left({\bf{d}}_i \right)$, and each 1st-order Voronoi region tessellates the network plane into $\left| \mathcal{N} \right|$ regions.

%Each Voronoi cell is sectorized into four sectors, denoted by $S_1$, $S_2$, $S_3$, and $S_4$, where $K$ users are included in each sector. As described in Fig.~\ref{2d_net_model}, by splitting the total time duration into four same durations, denoted by $T_1$, $T_2$, $T_3$, and $T_4$, a BS communicates to users in $S_{\ell}$ at $T_{\ell}$ for $\ell = 1,2,3,4$.

\subsection{Pair-wise Multicell Coordination}
%We consider that two adjacent BSs are coordinated, i.e., pair-wise BS coordination. 
In this paper, we focus on the case where only two adjacent BSs form a cooperative cluster.
The rationale behind focusing on pair-wise BS coordination is that it is highly representative for characterizing coordination gain without causing too much overhead. For example, in \cite{NY:dynamic}, it was shown that a BS cluster including more than two BSs decreased the ergodic spectral efficiency when considering signaling overhead. This is because including more BSs into a BS cluster requires more overhead for estimating channel coefficients of the BSs in the cluster, which decrease the ratio of the transmitted data in a packet. This degrades the spectral efficiency.
% pre-log term of the ergodic spectral efficiency.
For pair-wise multicell coordination, a BS located at ${\bf{d}}_i$ is able to coordinate with a BS located at ${\bf{d}}_j$ if $\lambda\left(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)  \right) \ne 0$, where $\lambda\left(\cdot\right)$ is the Lebesgue measure in two dimensional space. 

\subsection{User Association}
We assume that $2K$ single antenna users are selected in each 2nd-order Voronoi region irrespective of the size of the region. Among them, $K$ users are associated with the BS composing the corresponding 2nd-order Voronoi region while the other $K$ users are associated with the other BS. For instance, if $2K$ users are in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$, $K$ users in $\mathcal{V}_1\left({\bf{d}}_i\right) \cap \mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ are associated with the BS located at ${\bf{d}}_i$ and the other $K$ users in $\mathcal{V}_1\left({\bf{d}}_j\right) \cap \mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ are associated with the BS located at ${\bf{d}}_j$.
%, where $K$ users are in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ and other other $K$ users are in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$.
%to be associated with a BS in each cooperative area, 
%resulting in that there are total $2K$ users to be served in each 2nd-order Voronoi region irrespective of the size of the region. For instance, $2K$ users are in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$, where $K$ users are associated with the BS located at ${\bf{d}}_i$, and the other $K$ users are associated with the BS located at ${\bf{d}}_j$.
This underlying assumption is justified by assuming that the number of users in a network is far larger than the number of deployed BSs, so that there are at least $2K$ users in each 2nd-order Voronoi region satisfying our assumption. More specific user scheduling algorithms are beyond the scope of this paper. 

%${\bf{d}}_i$ and ${\bf{d}}_j$ are connected by the Delaunay triangulation defined in $\mathcal{N}$.

%such that no point in $\mathcal{N}$ is inside the circumcircle of any triangle.

%Due to its simplicity, pairwise cooperation is friendly to analysis, and also gives an insight of fundamentals of BSs cooperation as the smallest unit of BSs cluster. Additionally, from a practical perspective, pairwise cooperation is attractive for its low complexity. 
%For these reasons, pairwise cooperation has been studied in \cite{baccelli:pairwise}. 
%Specifically, for pairwise coordination, $B_{\left(i,j\right)}$ is able to coordinate with four adjacent BSs, i.e.,
%$B_{\left(i,j-1\right)}$, $B_{\left(i-1,j\right)}$, $B_{\left(i,j+1\right)}$, or $B_{\left(i+1,j\right)}$ by forming a coordination set; thereby a BS cluster includes two BSs.
%Exchanging CSI and data information within the same cluster is allowed by using an infinite capacity backhaul.
%Clustering sets including three BSs are not considered since it does not satisfy symmetrical clustering condition, which we will discuss in the later section.

%BS at $\left(0,0\right)$, indicated by $B_{\left(0,0\right)}$, serves four symmetrically located users, indicated by ${\rm{User}}_{\left\{i, \left(0, 0\right) \right\}}$, $1\le i \le 4$. Each BS is at distance $R$, and each user is located at distance $d$ for the serving BS. Therefore, the location of the users served by $B_{\left(i,j \right)}$

\subsection{The Proposed Clustering Model - Toy Example}

\begin{figure}[t]
\centerline{\resizebox{0.8\columnwidth}{!}{\includegraphics{tri_example_eps.eps}}}    
\caption{A motivational example where all BSs are assumed to be on a finite triangular plane $\mathcal{A}$, each of which is located at ${\bf{d}}_i$ for $i \in \{0,1,2, 3\}$.
The whole plane can be divided by the 2nd-order Voronoi region of $\{{\bf{d}}_i, {\bf{d}}_j \}$ for $i,j \in \{0,1,2,3\}$ and $i \ne j$. 
%A user in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ is served by $\{ {\bf{d}}_i, {\bf{d}}_j \}$, so that each user is guaranteed to communicate with two closest BSs.
The areas which has the same shade pattern are assigned to the same cluster pattern, resulting in that three cluster patterns, i.e., $\mathcal{P}_1, \mathcal{P}_2,$ and $\mathcal{P}_3$, are defined. 
%With the defined cluster patterns, there is no same BS in the same cluster pattern. By allocating different time-frequency resource to each cluster pattern, BS conflicts are avoided.
}  
   \label{tri_example} 
\end{figure}

%The proposed clustering strategy leverages the notion of the 2nd-order Voronoi region, defined as
%\begin{align} \label{voronoi_defn}
%&\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right) \nonumber \\
%&= \left\{\left. {\bf{d}}\in \mathbb{R}^2 \right| 
%\left\{ \left\|{\bf{d}} - {\bf{d}}_{i} \right\| \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\|  \right\} \cap \left\{ \left\|{\bf{d}} - {\bf{d}}_{j} \right\|  \le \left\|{\bf{d}} - {\bf{d}}_{k} \right\| \right\},\; \forall k \in \mathbb{Z}^+,\; k\ne i,j   \right\},
%\end{align}
%i.e., the set of closer points to ${\bf{d}}_{i}$ and ${\bf{d}}_{j}$ than any other points.
%The core feature of the 2nd-order Voronoi region is that it characterizes two closest BSs for each user. 
%From this intuition, 
%from users inherently by the definition.
%For instance, if a BS coordination pair $\{{\bf{d}}_i, {\bf{d}}_j\}$ is formed to serve a user in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$, the user is guaranteed to communicate with the two closest BSs without dominant out-of-cluster interference. 
%Specifically, we create a pair-wise BS cooperative region by tessellating the plane into the 2nd-order Voronoi region around the two neighbor BSs. 
In this subsection, we explain the proposed clustering model with a motivational example. To this end, suppose that BS $i$ is located at ${\bf{d}}_i$ in a finite area $ \mathcal{A} \subset{\mathbb{R}^2}$ for $i\in\{0,1,2, 3\}$, as illustrated in Fig.~\ref{tri_example}.
By exploiting the notion of the 2nd-order Voronoi region \eqref{voronoi_defn}, 
it is possible to create $\binom{4}{2} = 6$ different pair-wise cooperative areas $\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)$ for $i\neq j$ and $i,j\in\{0,1,2, 3\}$ such that all points within the given region $\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)$ have the same two nearest BSs, i.e., BS $i$ and BS $j$. 
In the proposed algorithm, users located in $\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)$ are served by a BS pair located at $\{{\bf{d}}_i, {\bf{d}}_j \}$, so that they are guaranteed to communicate with the two closest BSs.
Next, once each of 2nd-order Voronoi region is defined, multiple cluster patterns $\mathcal{P}_{\ell}$ for $\ell = \{1,...,L\}$ are created, each of which contains different 2nd-order Voronoi regions and $\bigcup_{\ell = 1}^{L} \mathcal{P}_{\ell} = \mathcal{A}$. Each cluster pattern uses different time-frequency resources, so that there is no BS conflict between two different cluster patterns.
%while it has the non-overlapping BSs. 
%If there are same BSs in the same cluster pattern, 
%a cluster pattern includes the same BSs, 
%e.g., $\mathcal{V}_2 \left( {\bf{d}}_0, {\bf{d}}_1\right) \in \mathcal{P}_{1}$ and $\mathcal{V}_2 \left( {\bf{d}}_0, {\bf{d}}_2\right) \in \mathcal{P}_{2}$,
%In the same cluster pattern, however, BS conflicts still can occur.
An example of the BS selection conflict problem occurs when $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1 \right)\in  \mathcal{P}_1$ and also $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_2 \right)\in  \mathcal{P}_1$. In this case, BS 0 is conflicted, and it cannot serve all the associated users. 
To avoid this, one possible example is to create six cluster patterns such that $\mathcal{P}_1 = \{ \mathcal{V}_2({\bf{d}}_0, {\bf{d}}_1)\} $, $\mathcal{P}_2 = \{ \mathcal{V}_2({\bf{d}}_0, {\bf{d}}_2)\} $, $\mathcal{P}_3 = \{ \mathcal{V}_2({\bf{d}}_0, {\bf{d}}_3)\} $, $\mathcal{P}_4 = \{ \mathcal{V}_2({\bf{d}}_1, {\bf{d}}_2)\} $, $\mathcal{P}_5 = \{ \mathcal{V}_2({\bf{d}}_1, {\bf{d}}_3)\} $, and $\mathcal{P}_6 = \{ \mathcal{V}_2({\bf{d}}_2, {\bf{d}}_3)\} $. 
%Since $\bigcup _{i=1}^{6} \mathcal{P}_i = \mathcal{A}$, all users in the network are able to communicate the two closest BSs.
%Each cluster pattern uses different time-frequency resource, so that a BS conflict problem is avoided. 
This example, however, requires too many time-frequency resources, i.e., 6. It is even larger than the case where each BS uses different time-frequency resources without BS coordination, i.e., 4.
A better example is to create three BS cluster patterns such that 
%$\mathcal{P}_1=\{\mathcal{V}_2({\bf{d}}_0, {\bf{d}}_1)\}$, $\mathcal{P}_2=\{\mathcal{V}_2({\bf{d}}_0, {\bf{d}}_2)\}$, and $\mathcal{P}_3=\{\mathcal{V}_2({\bf{d}}_1, {\bf{d}}_2)\}$. 
$\mathcal{P}_1=\{\mathcal{V}_2({\bf{d}}_0, {\bf{d}}_1),\mathcal{V}_2({\bf{d}}_2, {\bf{d}}_3)\}$, $\mathcal{P}_2=\{\mathcal{V}_2({\bf{d}}_0, {\bf{d}}_2),\mathcal{V}_2({\bf{d}}_1, {\bf{d}}_3)\}$, and $\mathcal{P}_3=\{\mathcal{V}_2({\bf{d}}_0, {\bf{d}}_3),\mathcal{V}_2({\bf{d}}_1, {\bf{d}}_2)\}$. 
%Since there is no overlapping BS, no BS is conflicted with these cluster patterns.
%the union of the three cluster patterns $\bigcup_{\ell =1}^{3}\mathcal{P}_{\ell}$ covers the whole plane $\mathcal{A}$, all the users in the network are able to be served through the proposed clustering pattern where each cluster pattern uses a different time-frequency resource. 
With this example, BS conflicts are prevented and the required number of time-frequency resources is reduced to 3. 
To figure out how much performance gain is achieved by using this cluster pattern, assume that a user is in cell-edge region where the interference chiefly comes from the closest interfering BS. When no BS coordination is applied, the performance of the user might be severely degraded by the dominant interference which has almost same power with the desired signal. When each BS has its own separate resource, the dominant interference can be mitigated but it requires 4 time-frequency resources. By applying the proposed pattern, the dominant interference is mitigated with 3 time-frequency resources, which can lead to the performance improvement.
%Under the assumption that the interference chiefly comes from the closest interfering BS, using this pattern can improve the performance than a case where each BS has their own resources.
%, which is reasonable than that of the previous approach. 




\subsection{The Proposed Clustering Model - General Network}

%\begin{figure}[t]
%\centerline{\resizebox{0.9\columnwidth}{!}{\includegraphics{net_model_irregular.pdf}}}    
%\caption{An example of a pair-wise coordination method using the proposed cluster pattern. 
%By alternatively using the BS cluster pattern in the different time duration, the users are able to be protected from the dominant interference by a property the 2nd-order Voronoi region. 
%The considered network model is divided by the $2$nd order Voronoi region for the proposed cluster pattern. The equal number of users are selected in one region, thereby $K/4$ users associated to a BS are included in $S_{\ell}$ for $\ell = 1,2,3,4$. The users in $S_{\ell}$ are communicated to the corresponding BS in time duration $T_{\ell}$.
%}  
%\label{2d_net_model_cm}
%\end{figure} 
\begin{figure}[t]
\centering
{\resizebox{0.8\columnwidth}{!}
{\includegraphics{general_net_example_eps.eps}}} 
\caption{An illustration of cooperative areas characterized by the 2nd-order Voronoi region (denoted as dotted red line) focusing on the BS located at ${\bf{d}}_0$. 
Each cooperative area marked by different shade patterns is assigned to a different cluster pattern to avoid BS conflicts. 
%By employing the proposed clustering strategy, users in each cooperative area are protected from the dominant interference.
} 
   \label{gen_net_example}
\end{figure} 

We extend the proposed clustering concept 
%explained in the previous section 
to a general network. At first, we illustrate in Fig.~\ref{gen_net_example} how the proposed BS clusters are formed in a general network. Focusing on a BS located at ${\bf{d}}_0$ and its adjacent BSs at ${\bf{d}}_j$ for $j \in \{1,2,...,5 \}$,
%As in the motivational example Fig.~\ref{tri_example}, 
users in $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_j \right)$ for $j \in \{1,2,...,5\}$ are served through a BS pair $\{{\bf{d}}_0, {\bf{d}}_j \}$ by using the proposed clustering method.
%in a general network whose BSs' locations are irregular as in Fig.~\ref{gen_net_example}, each 2nd-order Voronoi region 
After forming BS clusters, we design cluster patterns $\mathcal{P}_{\ell}$ for $\ell \in \{1,2,...,L\}$ where $\mathcal{P}_{\ell} = \{ \left. \mathcal{V}_2\left( {\bf{d}}_i, {\bf{d}}_j \right)\right| i,j \in \mathbb{Z}^+ \}$.
%and $\bigcup_{\ell = 1}^{L} \mathcal{P}_{\ell} = \mathbb{R}^2$.
%Each clustering pattern uses different time-frequency resource. 
%As in the toy example case, 
%The important condition of designing clustering patterns is to avoid BS conflicts, %i.e., one BS is included in two BS clusters that can be used in the same time-frequency resource. 
%For instance, in Fig.~\ref{gen_net_example}, if $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1 \right)\in  \mathcal{P}_1$ and also $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_2 \right)\in  \mathcal{P}_1$, conflict of the BS at ${\bf{d}}_0$ occurs, and the BS at ${\bf{d}}_0$ cannot serve all users. However, from the intuition in the previous section, if we design $L$ is large, it may cause excessive use of time-frequency resource.
From the intuition of the toy example, 
we summarize the conditions for a desirable cluster pattern.
% as in the followings.
\begin{enumerate}[1.]
\item $\bigcup_{\ell = 1}^{L} \mathcal{P}_{\ell} = \mathbb{R}^2$: (all users in the network are covered).
%: it is for covering all the users in the network.
\item Any two $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right) \in \mathcal{P}_{\ell}, \mathcal{V}_2\left({\bf{d}}_j, {\bf{d}}_{w} \right)\in \mathcal{P}_{\ell}$ for $\ell \in \{1,...,L\}$, $i \ne j \ne v \ne w$: (to avoid BS conflicts).
%: it is for avoiding BS conflicts.
\item Small $L$ satisfying 1) and 2): (to minimize the use of unnecessary resources).
% should be minimized.
%: it is for minimizing use of unnecessary resources.
\end{enumerate}
%The first condition is for covering all the users in the network, the second one is for avoiding a BS conflict problem, and the third one is for reducing unnecessary resource use. 
In a network whose BSs' locations are irregular, it is challenging to figure out how to map each of the BS clusters into appropriate BS cluster patterns satisfying the above conditions. To solve this, we draw a graph ${G}\left(\mathcal{N}\right)$ whose a vertex is ${\bf{d}}_i \in \mathcal{N}$ and an edge is made by Delaunay triangulation defined in $\mathcal{N}$. Delaunay triangulation for $\mathcal{N}$ is a triangulation of a plane, where each vertex of a triangle is ${\bf{d}}_i \in \mathcal{N}$. One important condition of the Delaunay triangulation is that when drawing the circumcircle of a triangle, there should be no point of $\mathcal{N}$ inside that circumcircle. Note that Delaunay triangulation always can be defined if there are more than three points, i.e., $\left| \mathcal{N} \right| \ge 3$.
The example of $G\left( \mathcal{N} \right)$ is illustrated in Fig.~\ref{2d_net_model} (b), when $\mathcal{N}$ is corresponding to Fig.~\ref{2d_net_model} (a).
%${G}\left(\mathcal{N}\right)$ is a simple, planar graph. 
%\begin{itemize}
%\item $G\left(\mathcal{N} \right)$ is a simple, planar graph.
%\item $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j\right)$ is non-empty if and only if ${\bf{d}}_i$ and ${\bf{d}}_j$ are connected by an edge of the graph $G\left(\mathcal{N}\right)$, denoted by $E\left({\bf{d}}_i, {\bf{d}}_j \right) = 1$.
%\item 
%\end{itemize}
A key relationship between 2nd-order Voronoi region and Delaunay triangulation is that
%In $G\left( \mathcal{N} \right)$, 
if there is an edge between ${\bf{d}}_i$ and ${\bf{d}}_j$, i.e., $E\left({\bf{d}}_i, {\bf{d}}_j \right)=1$ in $G\left( \mathcal{N} \right)$, the corresponding 2nd-order Voronoi region is non-empty, i.e., $\lambda\left(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j\right)\right) \ne 0$ \cite{higer_voro}.
%, denoted by $E\left({\bf{d}}_i, {\bf{d}}_j \right) = 1$ 
%Since the proposed BS cluster is determined by using the 2nd-order Voronoi region, and the 2nd-order Voronoi region ${\mathcal{V}}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ is not empty if and only if ${\bf{d}}_i$ and ${\bf{d}}_j$ are connected via the Delaunay triangulation , 
%Since each edge of $G\left(\mathcal{N}\right)$ is corresponding to the 2nd-order Voronoi region, 
%This implies that 
For this reason, designing BS cluster patterns, i.e., mapping $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j\right)$ into a cluster pattern, is equivalent to mapping $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ into a cluster pattern.
%to assign $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j\right)$ in $\mathcal{P}_{\ell}$ is equivalent to assign $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ in $\mathcal{P}_{\ell}$. 
%To satisfy the condition $2)$, any two edges sharing the same vertex should not be assigned to the same cluster pattern.
%Therefore, if create cluster patterns $\mathcal{P}_{\ell} = \{\left. \mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)\right| i,j \in \mathbb{Z}^+, i \ne j\}$ is equivalent to allocate 
When considering the cluster patterns as ``colors,"
the cluster pattern design problem is equivalent to the edge-coloring in graph theory \cite{intro_graph}.
The goal of edge-coloring is to assign colors to the edges of the graph so that any two edges sharing the same vertex have different colors, while minimizing the required number of colors to cover the whole graph. 
%A solution of the edge-coloring problem satisfies all the conditions for a desirable cluster pattern. At first, since the solution covers the whole graph, it satisfies 1). Next, a solution of
%Since a solution of the edge-coloring problem satisfies the 
By using edge-coloring for $G\left(\mathcal{N} \right)$, we are able to create cluster patterns $\mathcal{P}_{\ell}$, $\ell \in \{ 1,2,...,L\}$ that satisfies the above conditions.

\begin{figure}[t]
\centering
$\begin{array}{cc}
{\resizebox{0.4\columnwidth}{!}
{\includegraphics{example1_sym_eps.eps}}}  &
{\resizebox{0.4\columnwidth}{!}
{\includegraphics{example2_asym_eps.eps}}}  \\  
\mbox{(a)} &  
\mbox{(b)}
\end{array}$
\caption{
Illustration of two different Delaunay triangulations. The membership of the cluster for its cluster pattern is indicated by the edge label.
(a) An example of the Delaunay triangulation of a symmetric network, i.e., the maximum degree is equal to the minimum degree.
(b) An example of the Delaunay triangulation of an asymmetric network, i.e., the maximum degree is strictly larger than the minimum degree. 
%In each example, it is observed that a cluster pattern is mapped to each edge so that there is no BS conflict.
} 
   \label{net_example_sym_asym}
\end{figure} 

Now we study the relationship between the cluster pattern design and the edge-coloring specifically. At first, we characterize $L$, which is the required number of different time-frequency resources to avoid the BS selection conflict problem. Thanks to the equivalence between designing cluster patterns and the edge-coloring, $L$ is equal to the minimum required number of colors for the edges. The minimum required number of colors is characterized in the following Theorem.
\begin{theorem}[Vizing's theorem, \cite{vizing}] \label{th_vizing}
A simple planar graph $G$ of maximum degree $\Delta(G)$ has chromatic index $\Delta(G)$ or $\Delta(G) + 1$ in general.
\end{theorem}
\begin{proof}
See the reference \cite{vizing}.
\end{proof}
The chromatic index is known as the required minimum number of colors and the degree means the number of edges connected to the corresponding vertex. In general, it is NP-complete to find the chromatic index given an arbitrary graph, but we can observe that a graph $G$ made by Delaunay triangulation has chromatic index $\Delta(G)$ in many cases.
%It has been also proven that if $\Delta > 6$, the chromatic index of the given graph is $\Delta$ \cite{Sanders2001201}. Therefore, if $G\left(\mathcal{N}\right)$ has $\Delta \le 6$, 
%$L = \Delta + 1$ is enough to avoid BS conflicts, while if $\Delta > 6$, $L = \Delta$ is enough. 
%Since the planar graph has an average degree strictly less than $6$, and 
%the graph induced by the Delaunay triangulation,  $\Delta < 6$ rarely occurs.
Now we provide two examples to show how to create the cluster patterns for a given graph. The first example, described in Fig.~\ref{net_example_sym_asym} (a), is a symmetric network where every vertex of $G\left(\mathcal{N}\right)$ has the same degree, 6. This example is fitted to a scenario where each BS is deployed with a guard distance for preventing that two BSs are located very close together.
%The degree is the number of edges connected to the vertex.
Specifically, each BS is randomly located within a certain circle, and a center of each circle is located at a grid. %This scenario will be explained in Section VI.
%we have BS locations as illustrated in Fig.~\ref{net_example_sym_asym} (a).
For this example, $L = 6$. According to the description in Fig.~\ref{net_example_sym_asym} (a), the cluster pattern $\mathcal{P}_{\ell}$ for $\ell \in \{1,2,...,6\}$ can be designed as

\begin{align}
&\mathcal{P}_1 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_1 \right), \mathcal{V}_2 \left({\bf{d}}_2, {\bf{d}}_{10} \right), \mathcal{V}_2 \left({\bf{d}}_3, {\bf{d}}_4 \right), \cdots \}, \nonumber \\
&\mathcal{P}_2 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{2} \right), \mathcal{V}_2 \left({\bf{d}}_3, {\bf{d}}_{10} \right), \mathcal{V}_2 \left({\bf{d}}_4, {\bf{d}}_{11} \right), \cdots \}, \nonumber \\
&\mathcal{P}_3 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{3} \right), \mathcal{V}_2 \left({\bf{d}}_1, {\bf{d}}_{2} \right), \mathcal{V}_2 \left({\bf{d}}_4, {\bf{d}}_5 \right), \cdots \}, \nonumber \\
&\mathcal{P}_4 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{4} \right), \mathcal{V}_2 \left({\bf{d}}_2, {\bf{d}}_{3} \right), \mathcal{V}_2 \left({\bf{d}}_5, {\bf{d}}_6 \right), \cdots \}, \nonumber \\
&\mathcal{P}_5 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{5} \right), \mathcal{V}_2 \left({\bf{d}}_1, {\bf{d}}_{6} \right), \mathcal{V}_2 \left({\bf{d}}_7, {\bf{d}}_8 \right), \cdots \}, \nonumber \\
&\mathcal{P}_6 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{6} \right), \mathcal{V}_2 \left({\bf{d}}_4, {\bf{d}}_{8} \right), \mathcal{V}_2 \left({\bf{d}}_3, {\bf{d}}_{11} \right), 
 \cdots \},
\end{align}
by using edge-coloring.
%and $\bigcup_{\ell=1}^{6}\mathcal{P}_{\ell} = \mathbb{R}^2$. 
By allocating different time-frequency resources to each cluster pattern $\mathcal{P}_{\ell}$ for $\ell = \{1,2,...,6\}$, no BS conflict occurs and users are guaranteed to communicate with their two closest BSs set.
%Since the union of the six cluster patterns covers the whole plane $\mathbb{R}^2$, all the users in the network can be served through the proposed cluster pattern, and are guaranteed to communicate without the dominant interference. 

Now we look an example in Fig.~\ref{net_example_sym_asym} (b), which describes an asymmetric network, i.e., each vertex of $G\left(\mathcal{N}\right)$ can have different numbers of degrees. 
This example is fitted to a case where each BS is deployed with a more random manner than the symmetric network case. 
In this case, $L = 7$ is enough. Similar to the previous example, the cluster pattern $\mathcal{P}_{\ell}$ for $\{1,2,...,7\}$ can be 
\begin{align} \label{ex_clus_pattern_asym}
&\mathcal{P}_1 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_2 \right), \mathcal{V}_2 \left({\bf{d}}_5, {\bf{d}}_{8} \right), \mathcal{V}_2 \left({\bf{d}}_6, {\bf{d}}_{7} \right), \cdots \}, \nonumber \\
&\mathcal{P}_2 = \{ \mathcal{V}_2 \left({\bf{d}}_2, {\bf{d}}_{3} \right), \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{4} \right), \mathcal{V}_2 \left({\bf{d}}_6, {\bf{d}}_{8} \right), \cdots \}, \nonumber \\
&\mathcal{P}_3 = \{ \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{5} \right), \cdots \}, \nonumber \\
&\mathcal{P}_4 = \{ \mathcal{V}_2 \left({\bf{d}}_4, {\bf{d}}_{5} \right), \mathcal{V}_2 \left({\bf{d}}_0, {\bf{d}}_{8} \right), \cdots \}, \nonumber \\
&\mathcal{P}_5 = \{ \mathcal{V}_2 \left({\bf{d}}_7, {\bf{d}}_{8} \right), \cdots \}, \nonumber \\
&\mathcal{P}_6 = \{ \mathcal{V}_2 \left({\bf{d}}_2, {\bf{d}}_{4} \right),  \cdots \}, \nonumber \\
&\mathcal{P}_7 = \{ \mathcal{V}_2 \left({\bf{d}}_3, {\bf{d}}_{4} \right),  \cdots \}, %\nonumber \\
%&\mathcal{P}_8 = \{ ,  \cdots \}, %\nonumber \\
%&\mathcal{P}_9 = \{ ,  \cdots \}, 
\end{align}
also by using edge-coloring.
%and $\bigcup_{\ell=1}^{7}\mathcal{P}_{\ell} = \mathbb{R}^2$. 
No BS conflict occurs as long as a different time-frequency resource is assigned to a different cluster pattern. In this paper, we do not specify an algorithm for solving edge-coloring, rather we assume that edge-coloring is solved perfectly. Some useful algorithms for edge-coloring are found in \cite{gabow_simple, edgecutalgo1}. In \cite{gabow_simple}, an edge-coloring algorithm for simple graphs was proposed and it was proven that the algorithm complexity is $\mathcal{O}(\Delta(G)\left|\mathcal{N}\right| \sqrt{\left| \mathcal{N}\right|\log(\left|\mathcal{N}\right|)})$. In \cite{edgecutalgo1}, an edge-coloring algorithm for planar graphs whose $\Delta(G) \ge 9$ was proposed and the algorithm complexity was proven to be $\mathcal{O}(\left|\mathcal{N}\right| \sqrt{\left| \mathcal{N}\right|\log(\left|\mathcal{N}\right|)})$.


%by assigning different time-frequency resources to different cluster patterns. 
Observe that although the general edge-coloring problem is NP-complete, 
complexity is not a bottleneck since the proposed method (making BS clusters and BS cluster patterns) does not have to be performed frequently. 
% are for semi-static BS coordination. 
Once BS clusters and cluster patterns are made, they are preserved until the geometry of the BSs changes.
%Unless the geometry of the BSs changes, the cluster patterns are preserved. 
Conventionally, the BS geometry would not be changed on the order of months or years.


\subsection{Edge Cutting }
In some BS topologies, particularly in an asymmetric network case, it is possible that unnecessary time-frequency resources are wasted. For instance, in Fig.~\ref{net_example_sym_asym} (b), we found that $L=7$ time-frequency resources are required for the corresponding edge-coloring. Focusing on BS located at ${\bf{d}}_0$ in Fig.~\ref{net_example_sym_asym} (b), however, it only uses $5$ resources therefore $2$ resources are wasted. 
%For instance, we numerically obtain that the required number of time-frequency resources is $\approx 11$ when considering a network where each BS's location is distributed according to a homogeneous PPP with $\lambda = 3 \times 10^{-7}$. Therefore, the ergodic spectral efficiency is decreased by $\approx 0.09$. 
%Therefore, to apply the proposed clustering in the whole network plane without BS conflicts, $11$ time-frequency resources are needed, which is excessive. %Since the average degree of vertex is $6$, we can conjecture that some of the 2nd-order Voronoi regions are small, which means only 
Due to these wasted resources, the network sum throughput even can be degraded.
%even though the proposed clustering improves the rate coverage probability performance, the ergodic spectral efficiency can be degraded due to the excessive time-frequency use. 
To relieve this, the edge cutting algorithm is proposed. The key idea of this algorithm is decreasing the maximum degrees $\Delta(G)$ to $\Delta_{\rm EC}(G) < \Delta(G)$ by cutting some edges in the original graph $G(\mathcal{N})$.
%the edge-cutting algorithm is cutting edges of $G\left(\mathcal{N} \right)$ so that the maximum number of degrees of $G\left(\mathcal{N} \right)$ is forced to be $\Delta_{\rm EC} < \Delta$. 
Specifically, in $G\left(\mathcal{N}\right)$, if a vertex ${\bf{d}}_i \in \mathcal{N}$ has degree $D\left({\bf{d}}_i \right) > \Delta_{\rm EC}(G)$,
$D\left({\bf{d}}_i \right) - \Delta_{\rm EC}(G)$ edges are selected and cut.
Repeating this to every vertex in a graph, we construct a cut-graph $G^{\rm cut}\left( \mathcal{N}\right)$ 
whose the maximum degree is $\Delta_{\rm EC}(G)$.
%, and this decreases the required number of time-frequency resources to $\Delta_{\rm EC}(G)$ (See Theorem \ref{th_vizing}). 
%The cut-graph $G^{\rm cut}\left( \mathcal{N}\right)$ is a subgraph of $G\left(\mathcal{N} \right)$.
Note that the constructed cut-graph $G^{\rm cut}\left( \mathcal{N}\right)$ is a subgraph of $G\left(\mathcal{N} \right)$.
Assuming that an edge $E\left({\bf{d}}_i , {\bf{d}}_j\right)$ is cut in $G^{\rm cut}\left( \mathcal{N}\right)$, users in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ are 
not served.
By doing this, the required time-frequency resources are $\Delta_{\rm EC}(G)$.
%not served while exploiting $G^{\rm cut}\left( \mathcal{N}\right)$. By alternatively using different cut-graphs $G^{{\rm cut},1}\left(\mathcal{N} \right)$, $G^{{\rm cut},2}\left(\mathcal{N} \right), \cdots$, all the regions can be fairly covered. 
By applying the edge-cutting algorithm, the overall network performance can be improved by saving time-frequency resources. Specifically, the required time-frequency resources $L$ decreases to $\Delta_{\rm EC}(G)$ from $\Delta(G)$ (See Theorem \ref{th_vizing}), where $\Delta_{\rm EC}(G)$ can be controlled as an algorithm parameter.


There are two noticeable points in our edge cutting algorithm. The first one is that selecting the edges to cut has a significant effect to the performance of the algorithm. A desirable way is selecting edges inversely proportional to the area of the corresponding 2nd-order Voronoi region. For instance, assume that we have two candidate edges for cutting, $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ and $E\left({\bf{d}}_i, {\bf{d}}_k\right)$. If $\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)) > \lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_k \right))$, we cut $E({\bf{d}}_i, {\bf{d}}_k)$ since its 2nd-order Voronoi area is smaller than the other's area. By doing this, a large cooperative area remains while saving the resources.
The second point is that we can restore edges after finishing the edge cutting. The intuition is that the ideal situation for the proposed method is that every BS has the same degree, as in a symmetric network case Fig.~4 (a). For this reason, we find a vertex whose degree is less than $\Delta_{\rm EC}(G)$ and re-connect edges while satisfying the maximum degree of the graph is $\Delta_{\rm EC}(G)$. The edge restore order is reverse of the edge cut order, so that $E\left({\bf{d}}_i, {\bf{d}}_j\right)$ is restored if $\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)) > \lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_k \right))$.
%There are two noticeable points in the edge cutting algorithm. The first one is that how to select edges to cut has a significant effect to the performance of the algorithm. A desirable way is selecting edges inversely proportional to the area of the corresponding 2nd-order Voronoi region. For instance, assume that we have two candidate edges for cutting, $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ and $E\left({\bf{d}}_i, {\bf{d}}_k\right)$. If $\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)) > \lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_k \right))$, we cut $E({\bf{d}}_i, {\bf{d}}_k)$ since its 2nd-order Voronoi area is smaller than the other's area. By doing this, large cooperative area remains while saving the resources.
%The second point is that we can restore edges after finishing the edge cutting. The intuition for this point is that the ideal situation for the proposed method is that every BS has the same degrees, as in a symmetric network case Fig.~\ref{net_example_sym_asym} (a). For this reason, we find a vertex whose degree is less than $\Delta_{\rm EC}(G)$ and re-connect edges while satisfying the maximum degree of the graph is $\Delta_{\rm EC}(G)$. The edge restore order is reverse of the edge cut order, so that $E\left({\bf{d}}_i, {\bf{d}}_j\right)$ is restored if $\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)) > \lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_k \right))$.
%Specifically, under the assumption that $L > \Delta_{\rm EC}$ and the tagged user is served when the edge-cutting algorithm is used, the ergodic spectral efficiency is 
%\begin{align}
%R^{\ell}_{{\rm{cut}}} \left(N, K, \beta\right) \ge \frac{1}{\Delta_{{\rm EC}}} \log_2 \left(1 + \frac{\left(\beta^2 - 4 \right)}{{8K}} {\exp\left( \psi\left(N-2K+1 \right)\right)} \right).
%\end{align}
%it is removed and we re-draw $G\left(\mathcal{N}\backslash {\bf{d}}_i \right)$. By repeating this, we construct $G\left(\mathcal{N}'\right)$ whose $\Delta \le \Delta_{\max}$. This makes $L = \Delta_{\max}$ if $\Delta_{\max} >6$.
%Since the required time-frequency resources are determined by $\Delta$, 
%and it is possible to make $L=\Delta_{\max}$. 
%If users are selected in the cut 2nd-order Voronoi region, i.e., $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ if $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ is cut, they can be served without using interference management techniques, i.e., single cell operation. With this method, the used time-frequency resources by the overall network are able to be reduced to $\Delta_{\max}$.
%, where $\Delta_{\max}$ time-frequency resources are for the proposed clustering and $1$ time-frequency resource is for serving the cut region.
%To relieve this, it is desirable to cut some edges of the $G\left(\mathcal{N}\right)$ so as to make $\Delta$ a reasonable value. Specifically,
%if the edge between ${\bf{d}}_i$ and ${\bf{d}}_j$ is cut, the BSs pair $\{ {\bf{d}}_i, {\bf{d}}_j \}$ is not coordinated. Then, users in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j\right)$ are not protected from the dominant interference, which leads to performance degradation of those users. Through this modification, however, the overall network performance is able to be enhanced by saving the time-frequency resources use. 
%We call this edge-cutting network modification. 
%To make $\Delta = \Delta_{\rm EC}$, if the number of edges is $e > \Delta_{{\rm EC}}$, we randomly select $e - \Delta_{\rm EC}$ adjacent BSs around the corresponding BS and simply do not form a BS cluster with them. Assuming $E\left({\bf{d}}_i, {\bf{d}}_j \right)$ is cut, the users located in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ are served without interference management technique, i.e., single cell operation. Applying this to the whole network, $\Delta = \Delta_{\rm EC}$ holds. 
%We call this method the edge-cutting. 
%Then, the overall network can be operated with $L = \Delta_{\rm EC}+1$ time-frequency resource, where one additional time-frequency resource is for users in an cut edge.
%are served by without interference management technique.
%By the edge-cutting, the performance of some users is degraded due to the interference, but it saves time-frequency resources. 
%In the later section, we will compare the performance of the proposed method with the edge-cutting algorithm.
% and find that the proposed clustering with appropriate $\Delta_{\rm EC}$ improves the ergodic spectral efficiency.
The pseudo code description of the edge-cutting algorithm is described in the following using a MATLAB style notation.

\begin{algorithm}
\caption{Edge-cutting algorithm}
\begin{algorithmic}
\State {\bf{Input}}: $G\left(\mathcal{N}\right)$, $N_{\rm BS} = \left| \mathcal{N}\right| $, $\Delta_{\rm EC}(G)$.
\State {\bf{Initialize}}: Let $S_{\rm Neighbor}({\bf{d}}_i)$, $i \in \mathcal{N}$ be a set of vertices connected via the edge to ${\bf{d}}_i$ in $G(\mathcal{N})$\\
%${\bf{E}} \in \mathbb{N}^{N_{\rm BS} \times N_{\rm BS}}$, ${\bf{E}}{\left(i,j\right)}= 1$ if and only if ${\bf{d}}_i \in \mathcal{N}$ and ${\bf{d}}_j \in \mathcal{N}$ is connected by an edge in $G{\left(\mathcal{N}\right)}$, otherwise ${\bf{E}}{\left(i,j\right)}= 0$. ${\bf{ E}}_{\rm cut} = {\bf E}$.\\
(Area-based edge cutting)
\FOR {$i =1:N_{\rm BS}$} 
\IF {$\left|S_{\rm Neighbor}({\bf{d}}_i) \right| > \Delta_{\rm EC}(G) $}
\State Find ${\bf{d}}_j \in S_{\rm Neighbor}({\bf{d}}_i)$
% and sort them according to $(\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_{j} \right)))$
\State Cut ${\bf{d}}_j$ in order of $(\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_{j} \right)))$ (ascending order) so that {$\left|S_{\rm Neighbor}({\bf{d}}_i) \right| \le \Delta_{\rm EC}(G) $}
%\State ${\rm Cand\_set} = {\rm find}\left({\bf{E}}_{\rm cut}{\left(:, i \right)} ==1 \right)$
%\FOR  {$i_{\rm cand} = 1:{\rm{length}({\rm Remove\_candidates})}$}
%\State ${\rm{Area\_set}}$ = ${\rm{sort}}(\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_{\rm Cand\_set} \right)))$
%\ENDFOR
%\State ${\bf{E}}_{\rm cut}{\left({\rm Area\_set}\left(1:{\rm sum}\left({\bf{E}}_{\rm cut}{\left(:, i \right)}\right) - \Delta_{\rm EC}(G)  \right), i\right) }= 0$
%\State ${\bf{E}}_{\rm cut}{\left(i, {\rm Area\_set}\left(1:{\rm sum}\left({\bf{E}}_{\rm cut}{\left(:, i \right)}\right) - \Delta_{\rm EC}(G)  \right)\right) }= 0$
\ENDIF
\ENDFOR
\\ (Area-based edge restore)
\FOR {$i =1:N_{\rm BS}$} 
\IF {$\left|S_{\rm Neighbor}({\bf{d}}_i) \right| < \Delta_{\rm EC}(G) $}
\State Find ${\bf{d}}_j$ cut in the area-based edge cutting
% and sort them according to $(\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_{j} \right)))$
\State Restore ${\bf{d}}_j$ in order of $(\lambda(\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_{j} \right)))$ (descending order) satisfying $\max(S_{\rm Neighbor}({\bf{d}}_k)) \le \Delta_{\rm EC}(G),k\in \mathcal{N}$
%while satisfying $\max(S_{\rm Neighbor}({\bf{d}}_k)) \le \Delta_{\rm EC}(G),k\in \mathcal{N}$
%\ENDFOR
%\WHILE  {$\max({\rm sum}({\bf{E}}_{\rm cut}(:,j))) == \min(\Delta_{{\rm EC}}(G), {\rm sum}({\bf{E}}(:,j))),j\in\{i, {\rm Cand\_set}\}$}
%\State $k = k+1$
%\State ${\bf{E}}_{\rm cut}{\left({\rm Area\_set}(k), i\right) }= 1$
%\State ${\bf{E}}_{\rm cut}{\left(i,  {\rm Area\_set}(k) \right) }= 1$
%\ENDWHILE
\ENDIF
\ENDFOR
%\State In $G^{\rm cut}\left(\mathcal{N} \right)$, ${\bf{d}}_i$ and ${\bf{d}}_j$ are connected if and only if ${\bf{E}}_{\rm cut}\left(i,j \right) = 1$.
\State Output: $G^{\rm cut}\left(\mathcal{N} \right)$
\end{algorithmic}
\end{algorithm}
\begin{remark}
Since users in the cut region are not served in the proposed algorithm, there can be a fairness issue. This is able to be resolved by using the proposed method and conventional single cell operation method alternatively. For example, we can use the proposed method during time fraction $\alpha$ and conventional single cell operation during time fraction $(1-\alpha)$. 
Every region is covered during time fraction $(1-\alpha)$. In addition to that, more advanced scheduling algorithm such as proportional fairness can be also incorporated. 
\end{remark}

\begin{remark}
In the proposed edge cutting algorithm, $\Delta_{\rm EC}(G)$ can be determined in various ways. One promising way is to make that all vertices in $G(\mathcal{N})$ have the same degree, so that time-frequency resources are equally allocated to every BS. To do this, $\Delta_{\rm EC}(G)$ should be equal to the minimum degree of $G$, i.e, $\min D({\bf{d}}_i), \; \forall {\bf{d}}_i \in \mathcal{N}$. Other ways to determine $\Delta_{\rm EC}(G)$ can be incorporated depending on BS geometrical features and user density.
For instance, one can formulate an optimization problem as in \cite{6615901}, where $\Delta_{\rm EC}(G)$ is maximized while guaranteeing the difference between $\max D({\bf{d}}_i), \; \forall {\bf{d}}_i \in \mathcal{N}$ and $\min D({\bf{d}}_i), \; \forall {\bf{d}}_i \in \mathcal{N}$ is less than a particular value.
\end{remark}

Finally, we summarize the procedure of the proposed method including the edge cutting algorithm as in the following:
\begin{enumerate}[1.]
\item Tessellate a network plane into 2nd-order Voronoi regions $\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)$ for ${\bf{d}}_i, {\bf{d}}_j \in \mathcal{N}$.
\item Make a BS cluster $\{{\bf{d}}_i, {\bf{d}}_j\}$ according to the corresponding $\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)$ for ${\bf{d}}_i, {\bf{d}}_j \in \mathcal{N}$.
\item Draw a graph $G(\mathcal{N})$ by Delaunay triangulation.
\item Perform the edge cutting algorithm.
\item By using edge-coloring, make BS cluster pattens $\mathcal{P}_{\ell}$ for $\ell \in \{1,..,L\}$ and assign different time-frequency resource to each BS cluster pattern.
\end{enumerate}
Now we investigate computational complexity of each step. Assuming we have $\left|\mathcal{N}\right|$ points, constructing $\left|\mathcal{N}\right|$ points 2nd-order Voronoi region requires $\mathcal{O}(4\left|\mathcal{N}\right| \log \left|\mathcal{N}\right|)$ \cite{DTL_kNVD}, drawing Delaunay triangulation demands $\mathcal{O}(\left|\mathcal{N}\right| \log \left|\mathcal{N}\right|)$ \cite{Berg:2008:CGA:1370949}, and the edge cutting algorithm needs $\mathcal{O}(\left|\mathcal{N}\right| \Delta(G))$. In general, edge-coloring is NP-complete. 
The complexity is not important part in the proposed method since BS clusters and cluster patterns made by the proposed method are preserved until the geometry of the BSs changes, which occurs on the order of months or years conventionally.
% and the BS ge
%Conventionally, the BS geometry would not be changed on the order of months or years.


\subsection{Signal Model}
%For tractability of the analysis, it is assumed that $K$ single antenna users are selected to be associated with a BS in each cooperative area, resulting in that there are total $2K$ users to be served in each 2nd-order Voronoi region irrespective of the size of the region. For instance, $2K$ users are in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$, where $K$ users are associated with the BS located at ${\bf{d}}_i$, and the other $K$ users are associated with the BS located at ${\bf{d}}_j$.
%This underlying assumption is justified by assuming that the number of users on a field is far larger than the number of deployed BSs, so that there are at least $2K$ users in each 2nd-order Voronoi region. More specific user scheduling algorithms are beyond the scope of this paper. 

Now we describe the signal model. We focus on a user associated to the BS located at ${\bf{d}}_0$. We denote this user as the tagged user. 
%Since there are $K$ users for each BS and each BS has $L$ different 2nd-order Voronoi regions, we assume that there are $K/L$ users in each 2nd-order Voronoi region. For simplicity, we assume $K/L \in \mathbb{N}$. The symmetric network assumption can be generalized to the asymmetric network by specifying the possibly different number of users selected in each 2nd-order Voronoi region.
We also assume that the tagged user is located on the origin. This assumption is easily generalized to arbitrary user location by shifting each BS's location to ${\bf{d}}_i - {\bf{u}}$ for $i \in \mathbb{Z}^+$, where ${\bf{u}}$ is a location of the tagged user.
Under the assumption that the tagged user is in $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1\right)$ and $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1\right)$ is not cut in the edge cutting algorithm, a BS pair $\{{\bf{d}}_0, {\bf{d}}_{1} \}$ forms a cluster according to the proposed cluster pattern. 
% In a cluster pattern $P_{\ell}$, it is assumed that a BS pair located at $\{{\bf{d}}_0, {\bf{d}}_{i} \}$ forms a cluster according to the proposed cluster pattern.
Let's also assume $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1\right) \in \mathcal{P}_{\ell}$.
% and $\mathcal{C}_{\ell} = \{{\bf{d}}_0, {\bf{d}}_1 \}$. 
In each BS located at ${\bf{d}}_i$, an information symbol vector ${\bf{s}}^{\ell}_{i} \in \mathbb{C}^{K}$, where $ {\bf{s}}^{\ell}_{i} = \left[s^{\ell}_{i,1}, ..., s^{\ell}_{i,K} \right]^T$ for $i \in \mathbb{Z}^+$ is encoded to be transmitted to $K$ users. 
%Each BS at ${\bf{d}}_j$ for $j\in \{0,i\}$ separately encodes an information symbol $s^{\ell}_{j,k}$ and sends it to the $k$-th user of the BS at ${\bf{d}}_j$.
%An information symbol vector is written as ${\bf{s}}^{\ell}_{j} = \left[s^{\ell}_{j,1}, ..., s^{\ell}_{j,K/L} \right]^T$.
%, where $K'$ is the number of selected users in a cooperative area $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_i \right)$.
The average power of this transmit symbol vector satisfies $\mathbb{E}\left[ \left\| {\bf{s}}^{\ell}_{i} \right\|^2 \right] \le P$.
When transmitting the symbol vector, a linear beamforming matrix ${\bf{V}}_{i}^{\ell} = \left[{\bf{v}}_{i,1}^{\ell}, ..., {\bf{v}}_{i,K}^{\ell} \right]$, whose ${\bf{v}}_{i,k}^{\ell} \in \mathbb{C}^{N}$ and  $\left\| {\bf{v}}_{i,k}^{\ell}\right\|=1$ for $k = 1,...,K$ is used. %for the multi-user communication. 
%Without loss of generality, we describe the signal model received by $u_{\left(i,j\right),k}^{\ell}$ for arbitrary $k = 1,...,K/4$ and arbitrary $i,j\in \mathbb{Z}$.  
%Among $K/L$ users associated with the BS at ${\bf{d}}_0$,  For simplicity of the notations, we assume that the tagged user is located on the origin.
%For simplicity of notation, the subscript notations of the tagged user are omitted and the associated BS of the tagged user is denoted by $B_0$. The BS cluster set including $B_0$ is written as $\mathcal{C}^{\ell}_0$. 
The observation from the tagged user is given by
%\begin{align} \label{2d_sig_model}
%y^{\ell}=& \underbrace{\left\| {\bf{d}}_0 \right\|^{-\beta/2}\left({{\bf{h}}}_0^{\ell}\right)^T {\bf{V}}^{\ell}_0{{\bf{s}}}_0^{\ell}}_{ {\rm desired\;signal \;} {\rm +\;inter\;user\;interference} } +  \underbrace{
%\sum_{{\bf{d}}_j \in \mathcal{C}_{\ell} \backslash {\bf{d}}_0}
% \left\| {\bf{d}}_{1} \right\|^{-\beta/2} \left( {{\bf{h}}}^{\ell}_{1}\right)^T {\bf{V}}^{\ell}_{1} {{\bf{s}}}_{1}^{\ell}}_{\rm intra\;cluster\;interference}   
% +\underbrace{\sum_{{\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1 \}} \left\| {\bf{d}}_j \right\|^{-\beta/2} \left( {{{\bf{h}}}}^{\ell}_{j} \right)^T {\bf{V}}^{\ell}_{j} {{\bf{s}}}^{\ell}_{j}}_{\rm out\;of\;cluster\;interference}   + n^{\ell} ,
%\end{align}
\begin{align} \label{2d_sig_model}
y^{\ell}=& \underbrace{\left\| {\bf{d}}_0 \right\|^{-\beta/2}\left({{\bf{h}}}_0^{\ell}\right)^T {\bf{V}}^{\ell}_0{{\bf{s}}}_0^{\ell}}_{ {\rm desired\;signal \;} {\rm +\;inter\;user\;interference} } +  \underbrace{
%\sum_{{\bf{d}}_j \in \mathcal{C}_{\ell} \backslash {\bf{d}}_0}
 \left\| {\bf{d}}_{1} \right\|^{-\beta/2} \left( {{\bf{h}}}^{\ell}_{1}\right)^T {\bf{V}}^{\ell}_{1} {{\bf{s}}}_{1}^{\ell}}_{\rm intra\;cluster\;interference} \nonumber \\   
& +\underbrace{\sum_{{\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1 \}} \left\| {\bf{d}}_j \right\|^{-\beta/2} \left( {{{\bf{h}}}}^{\ell}_{j} \right)^T {\bf{V}}^{\ell}_{j} {{\bf{s}}}^{\ell}_{j}}_{\rm out\;of\;cluster\;interference}   + n^{\ell} ,
\end{align}
where ${{\bf{h}}_{i}^{\ell}} \in \mathbb{C}^{N}$ is the channel coefficient vector from the BS at ${\bf{d}}_i$ to the tagged user. 
The channel coefficients are assumed to have independent and identically distributed (IID) complex Gaussian entries with zero mean and unit variance, i.e., $\mathcal{CN}\left(0,1\right)$. 
%The channel vector that has no subscript notation, $\left({\bf{h}}^{\ell}\right)^T \in \mathbb{C}^{N}$, denotes that the channel coefficient vector from the $B_0$ to the tagged user. 
%The entries of ${\bf{h}}^{\ell}$ is also assumed to be IID complex Gaussian with zero mean and unit variance.
%$D_{\left(v,w\right)}$ is the distance from $B_{\left(v,w\right)}$ to the tagged user, and $D$ is the distance from $B_0$ to the tagged user. 
The distance dependent pathloss is with a reference at $1$m and has an exponent of $\beta$ and the additive Gaussian noise is $n^{\ell}$ that follows $\mathcal{CN}\left(0, \sigma^2\right)$. $\mathcal{N}_{\ell} \subseteq \mathcal{N}$ is a set of BSs included in the same cluster pattern with the BS at ${\bf{d}}_0$, i.e., $\mathcal{N}_{\ell} = \{ \left. {\bf{d}}_i \right| \mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right) \in \mathcal{P}_{\ell}, \forall j\}$. 
The received signal can be separated into three parts: the desired signal and the inter user interference from the associated BS at ${\bf{d}}_0$, the intra-cluster interference from a BS set ${\bf{d}}_1$, and the unmanageable out-of-cluster interference, respectively.

Before transmission of an information symbol, pilot symbols are exploited to learn the downlink channel coefficient vector ${\bf{h}}^{\ell}_0$ and ${\bf{h}}_{1}^{\ell}$.
% if ${\bf{d}}_j \in \mathcal{C}_\ell$
%, i.e., the BS is included in the corresponding BS cluster. 
Then, the user sends the obtained CSI back to the corresponding BSs
%$\mathcal{C}^\ell_{\left(i,j\right)}$ 
via an error-free feedback link. CSI at the transmitter (CSIT) within the same BS cluster is allowed. 
To mitigate intra-cluster interference, CBF is employed by using obtained CSIT.
%The term ``weighted" is used because we consider a distance term when designing ZF beamformer. 
Assuming that our tagged user's index is $k$, the beamforming matrix ${\bf{V}}^{\ell}_{0}$ and ${\bf{V}}^{\ell}_{1}$ are designed to satisfy the following condition.
\begin{align}
{\rm{maximize}}: &\left| \left({\bf{h}}^{\ell}_0\right)^T {\bf{v}}^{\ell}_{0,k} \right|^2   \\
{\rm{subject\;to}}: & \left| \left( {\bf{h}}^{\ell}_0\right)^T {\bf{v}}^{\ell}_{0, k'} \right|^2 =0\;\;{\rm{for}}\;\; k' \ne k  \\
%& \left| \left( {\bf{h}}^{\ell}_{1}\right)^T {\bf{v}}^{\ell}_{1, k''} \right|^2  =0 \;\;{\rm{for}}\;\;
%{\bf{d}}_1 \in \mathcal{C}_{\ell}\;\;{\rm{and}}\;\; 1\le k'' \le K, 
& \left| \left( {\bf{h}}^{\ell}_{1}\right)^T {\bf{v}}^{\ell}_{1, k''} \right|^2  =0 \;\;{\rm{for}}\;\;
 1\le k'' \le K, 
\end{align}
where the constraints of multi-user CBF are for nullifying the inter-user interference and for removing intra-cluster interference, respectively.
Under the conventional zero forcing (ZF) conditions, 
it is always possible to find such ${\bf{V}}^{\ell}_0$ and ${\bf{V}}_{j}^{\ell}$ if $2K \le N$ with high enough probability if each channel coefficient is IID.

\subsection{Performance Metrics}
If perfect CSIT within the same BS cluster is allowed, then the inter-user interference and intra-cluster interference terms in \eqref{2d_sig_model} are zero. 
Assuming that $\tilde {h}^{\ell}_0$ indicates a modified channel coefficient after multiplying with a beamforming matrix ${\bf{V}}^{\ell}_0$ and $s^{\ell}_{0,0}$ is the information symbol for the tagged user, we have the following modified received signal
%After removing intra-cluster interference by using CBF, the resultant received signal $\tilde y^{\ell}$ for $\ell = 1,2,3,4$ is given by
%\begin{align}
%\tilde y^{\ell} =&  \left\|{\bf{d}}_0 \right\|^{-\beta/2} \tilde {h}^{\ell}_0 {s}^{\ell}_{0,0} +\sum_{{\bf{d}}_j  \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\}} \left\| {\bf{d}}_j \right\|^{-\beta/2} \left( {{\bf{h}}}^{\ell}_{j}\right)^T {\bf{V}}^{\ell}_{j} {{\bf{s}}}^{\ell}_{j}   + n^{\ell}.
%\end{align}
\begin{align}
\tilde y^{\ell} =&  \left\|{\bf{d}}_0 \right\|^{-\beta/2} \tilde {h}^{\ell}_0 {s}^{\ell}_{0,0} +\sum_{{\bf{d}}_j  \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\}} \left\| {\bf{d}}_j \right\|^{-\beta/2} \left( {{\bf{h}}}^{\ell}_{j}\right)^T {\bf{V}}^{\ell}_{j} {{\bf{s}}}^{\ell}_{j}  \nonumber \\
& + n^{\ell}.
\end{align}
The SINR of the information symbol $s^{\ell}_{0,0}$ is
\begin{align} \label{sinr}
{{\rm SINR}}_{|{\ell}} &=  \frac{ \left\| {\bf{d}}_0 \right\|^{-\beta} \left| \tilde h^{\ell}_0 \right|^2 }{{ \sum_{{\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1
\}} \left\| {\bf{d}}_j \right\|^{-\beta}\left|\left( {{\bf{h}}}^{\ell}_{j} \right)^T {\bf{V}}^{\ell}_{j} \right|^2}+\sigma^2/P/K  } \nonumber \\
&= \frac{  \left| \tilde h^{\ell}_0 \right|^2 }{I_{{\ell}}+ \left\|{\bf{d}}_0 \right\|^{\beta}/{\rm{SNR}}/K }, 
\end{align}
where $I_{{\ell}} = { \sum_{ {\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\}} \left( \left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta}\left| \left({{\bf{h}}}^{\ell}_{j}\right)^T {\bf{V}}^{\ell}_{j} \right|^2}$, and $P/\sigma^2 = {\rm{SNR}}$. 
%the CCDF of the corresponding SINR in phase $\ell$ is defined as
%\begin{align} 
%&F_{{{\rm SIR}}_{|\mathcal{P}_{\ell}}}^{c}\left(P, D, \mathcal{D}^{\ell}_{0}, \beta;\gamma\right) = \mathbb{P}\left[ \frac{P \left| \tilde h^{\ell} \right|^2 }{{P \sum_{B_{\left(v,w\right)} \in \mathcal{N} \backslash \mathcal{C}^{\ell}_{0}}\left(d_{\left(v,w\right)} \right)^{-\beta}\left|{{h}}^{\ell}_{\left(v,w\right)} {\bf{v}}^{\ell}_{\left(v,w\right)}{{\bf{s}}}^{\ell}_{\left(v,w\right)} \right|^2} + \left(D\right)^{\beta} } > \gamma \right],  \label{ccdf_sinr} \\  
%&{\rm{where}}\;\mathcal{D}^{\ell}_{0} = \left\{\left. d_{\left(v,w\right)} \right| B_{\left(v,w\right)} \in \mathcal{N} \backslash \mathcal{C}^{\ell}_{0} \right\}. \label{sinr_d}
%\end{align}
%From the defined SINR model, 
Given the system assumptions, the rate coverage probability is given as
\begin{align}
&P^{\ell} \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, \beta, \gamma \right) = \mathbb{P}\left[ \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) > \gamma \right], \label{rate_cov_def} 
%&{\rm{where}}\;\mathcal{D}_{\ell} = \left\{\left.  {\bf{d}}_j  \right| {\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\} \right\}, \nonumber
\end{align}
where $\mathcal{D}_{\ell} = \left\{\left.  {\bf{d}}_j  \right| {\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\} \right\}$ and $\gamma$ is the rate threshold. The ergodic spectral efficiency is then
\begin{align} 
&R^{\ell}  \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, L, \beta\right) = \frac{1}{L}\mathbb{E}\left[ \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right]. \label{rate_def}
%&{\rm{where}}\;\mathcal{D}_{\ell} = \left\{\left.  {\bf{d}}_j \right| {\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\} \right\}. \nonumber
\end{align}
%where $\mathcal{D}_{\ell} = \left\{\left.  {\bf{d}}_j \right| {\bf{d}}_j \in \mathcal{N}_{\ell} \backslash \{{\bf{d}}_0, {\bf{d}}_1\} \right\}$. 
The pre-log term $1/L$ is used because $L$ time-frequency resources are used in a network. $L$ is determined as an algorithm parameter in the edge cutting algorithm.

\section{Rate Coverage Analysis}
In this section, we derive the rate coverage probability in a closed form under the interference limited assumption. To this end, we first introduce the following lemma.
%Lemma \ref{lem_ch}. 
Lemma \ref{lem_ch} provides distributions of the desired channel gain and the interference power.

%as in the following theorem.
%, i.e., SNR is high enough, so that ${\rm{SINR}}_{|\ell} \approx {\rm{SIR}}_{|\ell}$.

\begin{lemma} \label{lem_ch}
The desired channel gain $\left|\tilde h^{\ell}_0 \right|^2$ follows Chi-squared distribution with $2\left(N - 2K +1\right)$ degrees of freedom.  The out-of-cluster interference $I_{{\ell}}$ follows weighted sum of Chi-squared distribution with $2K$ degrees of freedom. 
%The weights are the ratio of the distances from the associated BS to the user and the interfering BS to the user.
\end{lemma}
\begin{IEEEproof}
See \cite{NY:dynamic} and reference therein.
\end{IEEEproof}

By using this Lemma, we derive the rate coverage. The following Theorem is the main result of this section. 

\begin{theorem}
In the interference limited regime, when the BSs form BS clusters by using the proposed method, the rate coverage probability of the tagged user is 
%\begin{align} \label{rate_coverage}
%\mathbb{P}\left[ \log_2\left(1 + {\rm{SIR}}_{|{\ell}}\right)  > \gamma\right] 
%&P^{\ell} \left({\rm SNR} \rightarrow \infty, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, \beta, \gamma \right) =  \nonumber \\
%&\sum_{m=0}^{N-2K}\frac{\left(2^{\gamma}-1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m }{\partial s^m} \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left( \frac{1}{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K}   \right.,
%\end{align}
\begin{align} \label{rate_coverage}
%\mathbb{P}\left[ \log_2\left(1 + {\rm{SIR}}_{|{\ell}}\right)  > \gamma\right] 
&P^{\ell} \left({\rm SNR} \rightarrow \infty, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, \beta, \gamma \right) =  \nonumber \\
&\sum_{m=0}^{N-2K}\frac{\left(2^{\gamma}-1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m }{\partial s^m} \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left( \frac{1}{1+s \left( \frac{\left\|{\bf{d}}_j \right\| }{ \left\| {\bf{d}}_{0} \right\|} \right) ^{-\beta}} \right) ^{K}   \right.,
\end{align}
where $s = 2^{\gamma} - 1$ and $\gamma$ is the rate threshold.
\end{theorem}
\begin{IEEEproof}
As defined in \eqref{rate_cov_def}, the rate coverage probability is rewritten as
%\begin{align}
%\mathbb{P}\left[ \log_2\left(1 + {\rm{SINR}}_{|{\ell}}\right)  > \gamma\right] & \mathop = ^{(a)}\mathbb{P}\left[ \frac{  \left| \tilde h^{\ell}_0 \right|^2 }{I_{{\ell}}}  >\left(  2^{\gamma} - 1 \right) \right] \nonumber \\
%&=\mathbb{P}\left[ {  \left| \tilde h^{\ell}_0 \right|^2 }  > \left(2^{\gamma} -1\right)  {I_{{\ell}} } \right]\nonumber \\
%& \mathop =  ^{(b)}\mathbb{E}_{I_{\ell}}\left[\left. \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!}  \left( I_{\ell}\right) ^{m}  e^{-\left( 2^{\gamma} - 1\right) I_{\ell}}  \right. \right], \label{rate_coverage_pt1}
%\end{align}
\begin{align}
&\mathbb{P}\left[ \log_2\left(1 + {\rm{SINR}}_{|{\ell}}\right)  > \gamma\right] \nonumber \\
& \mathop = ^{(a)}\mathbb{P}\left[ \frac{  \left| \tilde h^{\ell}_0 \right|^2 }{I_{{\ell}}}  >\left(  2^{\gamma} - 1 \right) \right] \nonumber \\
%&=\mathbb{P}\left[ {  \left| \tilde h^{\ell}_0 \right|^2 }  > \left(2^{\gamma} -1\right)  {I_{{\ell}} } \right]\nonumber \\
& \mathop =  ^{(b)}\mathbb{E}_{I_{\ell}}\left[\left. \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!}  \left( I_{\ell}\right) ^{m}  e^{-\left( 2^{\gamma} - 1\right) I_{\ell}}  \right. \right], \label{rate_coverage_pt1}
\end{align}
where (a) comes from that $\left\| {\bf{d}} \right\|^{\beta}/{\rm SNR} \rightarrow 0$ when ${\rm{SNR}} \rightarrow \infty$, and (b) follows Lemma \ref{lem_ch} and the complement cumulative distribution function of a Chi-squared random variable with degrees of freedom $2\left(N-2K + 1\right)$.
Using the derivation of the Laplace transform, i.e., 
$\mathbb{E}\left[ X^m e^{-sX} \right] = \left(-1 \right)^m \frac{d^m \mathcal{L}_X \left( s\right)}{ds^m}$, \eqref{rate_coverage_pt1} is represented as
%\begin{align}
%\mathbb{E}_{I_{\ell}}\left[ \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!}  \left( I_{\ell}\right) ^{m}  e^{-\left(2^{\gamma} - 1 \right)I_{\ell}}  \right] &= \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m \mathcal{L}_{I_{\ell}} \left( s\right)}{\partial s^m}\right|_{s = 2^{\gamma} - 1},
%\end{align}
\begin{align}
&\mathbb{E}_{I_{\ell}}\left[ \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!}  \left( I_{\ell}\right) ^{m}  e^{-\left(2^{\gamma} - 1 \right)I_{\ell}}  \right] \nonumber \\
&= \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma} - 1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m \mathcal{L}_{I_{\ell}} \left( s\right)}{\partial s^m}\right|_{s = 2^{\gamma} - 1},
\end{align}
where $\mathcal{L}_{I_{\ell}}\left(s\right)$ is the Laplace transform of $I_{\ell}$, defined as 
\begin{align}
\mathcal{L}_{I_{\ell}}\left(s\right) = \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left( \frac{1}{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K},
\end{align}
which completes the proof.
\end{IEEEproof}

Computing the rate coverage probability \eqref{rate_coverage} is not easy since computing the derivative of the product of $\left( {1}/{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K}$ results in many terms.
% to compute when $\left|\mathcal{N} \backslash \mathcal{C}^{\ell}_0\right| \gg 1$, 
To provide a simpler form of the rate coverage probability, the following Corollary is used to approximate the rate coverage probability.
%we rather use the approximated form of the rate coverage probability \eqref{rate_coverage_approx}.f

\begin{corollary}
The rate coverage probability \eqref{rate_coverage} can be approximated as in the following form.
%\begin{align} \label{rate_coverage_approx}
%&P^{\ell} \left({\rm SNR} \rightarrow \infty, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, \beta, \gamma \right) \approx \nonumber \\
%& \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma}-1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m }{\partial s^m} \left(\frac{ e^{-\sum_{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}s \left(\frac{\left\|{\bf{d}}_j \right\|}{\left\| {\bf{d}}_0 \right\|} \right)^{-\beta}  }}{1 + s\left( \left\| {\bf{d}}_{\rm min} \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} }\right)^{K}   \right.,
%\end{align}
\begin{align} \label{rate_coverage_approx}
&P^{\ell} \left({\rm SNR} \rightarrow \infty, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, \beta, \gamma \right) \approx \nonumber \\
& \sum_{m=0}^{N-2K}\frac{\left(2^{\gamma}-1\right)^m}{m!} \left(-1 \right)^m  \left.\frac{\partial^m }{\partial s^m} \left(\frac{ e^{-\sum_{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}s \left(\frac{\left\|{\bf{d}}_j \right\|}{\left\| {\bf{d}}_0 \right\|} \right)^{-\beta}  }}{1 + s\left(\frac{ \left\| {\bf{d}}_{\rm min} \right\| }{ \left\| {\bf{d}}_0 \right\| }\right)^{-\beta} }\right)^{K}   \right.,
\end{align}
where $s = 2^{\gamma} - 1$ and ${\bf{d}}_{\min} = \mathop {\arg \min} \limits_{{\bf{d}}_j \in \mathcal{D}_{\ell}} \left\| {\bf{d}}_j\right\|$.
% and $B_{\rm min}$ indicates a BS whose distance to the user is $d_{\rm min}$.
\end{corollary}
\begin{IEEEproof}
Using the Talyor expansion of the exponential function $e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots$, when $x$ is small enough, $e^{x} \approx 1 + x$, therefore $e^{-x} \approx 1/\left(1 + x\right)$. Then we have
%\begin{align}
%&\prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left( \frac{1}{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K}\nonumber \\
%& = \left( \frac{1}{1 + s\left( \left\| {\bf{d}}_{\rm min} \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} } \right) ^{K} \prod_{{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}}\left( \frac{1}{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K} \nonumber \\
%&\approx  \left(\frac{1}{1 + s\left( \left\| {\bf{d}}_{\rm min} \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} }\right)^{K}\left( e^{-\sum_{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}s \left(\frac{\left\|{\bf{d}}_j \right\|}{\left\| {\bf{d}}_0 \right\|} \right)^{-\beta}  } \right) ^{K},
%\end{align}
\begin{align}
&\prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left( \frac{1}{1+s \left( \frac{\left\|{\bf{d}}_j \right\| }{\left\| {\bf{d}}_{0} \right\|} \right) ^{-\beta}} \right) ^{K}\nonumber \\
& = \left( \frac{1}{1 + s\left(\frac{ \left\| {\bf{d}}_{\rm min} \right\| }{ \left\| {\bf{d}}_0 \right\| }\right)^{-\beta} } \right) ^{K} \prod_{{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}}\left( \frac{1}{1+s \left( \frac{\left\|{\bf{d}}_j \right\| }{ \left\| {\bf{d}}_{0} \right\|} \right) ^{-\beta}} \right) ^{K} \nonumber \\
&\approx  \left(\frac{1}{1 + s\left(\frac{ \left\| {\bf{d}}_{\rm min} \right\| }{ \left\| {\bf{d}}_0 \right\| }\right)^{-\beta} }\right)^{K}\left( e^{-\sum_{{\bf{d}}_j \in \mathcal{D}_{\ell} \backslash {\bf{d}}_{\rm min}}s \left(\frac{\left\|{\bf{d}}_j \right\|}{\left\| {\bf{d}}_0 \right\|} \right)^{-\beta}  } \right) ^{K},
\end{align}
which completes the proof.
\end{IEEEproof}

%Unlike the exact form of the rate coverage probability obtained in \eqref{rate_coverage}, t
Using the Taylor expansion, we approximate a product of $\left( {1}/{1+s \left( \left\|{\bf{d}}_j \right\| / \left\| {\bf{d}}_{0} \right\| \right) ^{-\beta}} \right) ^{K}$ in an exponential form, so that we are able to calculate the derivative of \eqref{rate_coverage_approx} more simply than that of \eqref{rate_coverage}. 

Now we verify the approximated form of the rate coverage probability \eqref{rate_coverage_approx} by comparing to the Monte-Carlo simulation results.
For the verification, we use a network model described in Fig.~\ref{network_snapshot} (b) in Section VI.
%The approximated rate coverage probability has a form of 
%n arbitrary network model whose BS locations are irregular similar to Fig.~\ref{2d_net_model}. In this network model, each BS has a protection range 1km, i.e., no BS is placed within a radius of 1km. 
%In this network model, $L = 5$, i.e., six different cluster patterns are exploited alternatively with different time-frequency resources.
%different cluster pattern $\mathcal{P}_{\ell}$ for $\ell \in \{1,2,...,5\}$. 
%Since only $1/5$ time resource is used for serving the tagged user, the rate threshold is tuned by $\gamma' = L \gamma$. 
As illustrated in Fig.~\ref{rate_coverage_verification}, the analytical approximation tightly matches the simulated rate coverage probability over entire rate threshold region and for different numbers of BS antennas and users settings. 
%Comparing to the performance of the case where $N=3$ and $K = 1$, the rate coverage performance is improved when $N=4$ and $K = 1$. The reason is that when the number of the BS antennas is increased, the desired channel gain obtained by the user is increased due to CBF, which leads to improve the rate coverage performance. In contrast, when $N=4$ and $K = 2$, the rate coverage performance is degraded since the the desired channel gain is divided by the number of users.
%Since the significant performance degradation occurs when the power gain is 
%We are able to see that the power gain is more significant factor than the desired channel gain for the rate coverage performance from this observation.
%One should note that the case $N=6$ and $K/L = 2$ is not appropriate assumption since $N < K$, though we allow the assumption only for this simulation to verify the rate coverage probability analysis.

\begin{figure}[t]
\centerline{\resizebox{0.7\columnwidth}{!}  {\includegraphics{rate_coverage_verification_eps.eps}}}  
\caption{The comparison of the rate coverage probability obtained by Monte-Carlo simulations and the analytical approximation \eqref{rate_coverage_approx}. 
It is assumed that $\beta = 4$, $N \in \{3, 4, 10\}$, and $K \in \{1,2, 5\}$.
% The analytical approximation tightly matches to the simulation results over the entire rate threshold region, for all the cases of the numbers of BS antennas and users.
}    
   \label{rate_coverage_verification}
\end{figure} 

One noticeable point in the rate coverage analysis is that since we do not assume any particular random network, the rate coverage is a function of distances between BSs and the tagged user. Prior work \cite{6967804, 5953530, 5951699} also characterized performance in a fixed network model, where performance was derived as a function of the distances between BSs and a user. 
% In this work, they also presented the performance as a function of distances between BSs and a user.

\section{Ergodic Spectral Efficiency Analysis}
In this section, we derive an exact expression of the ergodic spectral efficiency in an integral form and provide a lower bound on the ergodic spectral efficiency in a closed form. As in the rate coverage analysis, the ergodic spectral efficiency is characterized as a function of distances between BSs and the tagged user.

\subsection{Exact Characterization}
Our exact expression is derived by using 
Lemma \ref{lem_useful}, which yields a general expression of the ergodic spectral efficiency in terms of moment generating functions.
%introduced for deriving the ergodic spectral efficiency of the CM.
%to characterize the  rate for the CM, we use the following lemma.

%\begin{lemma}[\cite{useful}, Lemma 1] \label{lem_useful}
\begin{lemma} \label{lem_useful}
Let $x_1,...,x_N,y_1,...,y_M$ be arbitrary non-negative random variables. Then
%\begin{align}
%\mathbb{E}\left[\ln \left( 1+ \frac{\sum_{n=1}^{N}x_n  }{\sum_{m=1}^{M} y_m + 1} \right) \right]  = \int_{0}^{\infty} \frac{\mathcal{M}_y\left(z\right) - \mathcal{M}_{x,y}\left(z\right)}{z}{\rm{exp}}\left(-z\right) {\rm d}z,
%\end{align}
\begin{align}
&\mathbb{E}\left[\ln \left( 1+ \frac{\sum_{n=1}^{N}x_n  }{\sum_{m=1}^{M} y_m + 1} \right) \right]  \nonumber \\
&= \int_{0}^{\infty} \frac{\mathcal{M}_y\left(z\right) - \mathcal{M}_{x,y}\left(z\right)}{z}{\rm{exp}}\left(-z\right) {\rm d}z,
\end{align}
where $\mathcal{M}_y\left(z\right) = \mathbb{E}\left[e^{-z\sum_{m=1}^{M}y_m} \right]$ and $\mathcal{M}_{x,y}\left(z\right) = \mathbb{E} \left[ e^{-z\left( \sum_{n=1}^{N} x_n + \sum_{m=1}^{M} y_m \right)} \right]$.
\end{lemma}
\begin{IEEEproof}  
See Lemma 1 in reference \cite{useful}.
\end{IEEEproof}

Leveraging Lemma \ref{lem_useful}, an exact expression of the ergodic spectral efficiency is obtained as in the following Theorem.
%Now we derive the  spectral efficiency of the CM. 
%To obtain it in a closed form, the approximated ergodic rate is characterized.
%The exactness of this approximation will be verified in the later section.
\begin{theorem}
%Recalling the SINR model in phase $\ell$ for $\ell \in \{1,2,3,4\}$ \eqref{sinr}, we denote the desired signal $P\left| \tilde h^{\ell} \right|^2$ by $S$ and the interference $P{ \sum_{B_{\left(v,w\right)} \in \mathcal{N} \backslash \mathcal{C}^{\ell}_{0}}\left(d_{\left(v,w\right)} \right)^{-\beta}\left|{{h}}^{\ell}_{\left(v,w\right)} {\bf{v}}^{\ell}_{\left(v,w\right)}{{\bf{s}}}^{\ell}_{\left(v,w\right)} \right|^2} $ by $I_{\mathcal{P}_{\ell}}$. 
When the BSs form BS clusters by using the proposed method, 
the ergodic spectral efficiency of the tagged user is
%\begin{align} \label{ergodic_rate_th}
%&R^{\ell}  \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, L, \beta\right) \nonumber \\
%&=  \frac{\log_2\left(e\right)}{L} \int_{0}^{\infty} \frac{{\rm{exp}}\left(-z \frac{\left\| {\bf{d}}_0 \right\|^{\beta}}{{\rm{SNR}}/K} \right)}{z}  \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}} \left( \frac{1}{1+ z \left(\left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta}}\right)^{K}  \left( 1 -  \left( \frac{1}{ 1 + z} \right)^{N-2K+1} \right) {\rm d}z.
%\end{align}
\begin{align} \label{ergodic_rate_th}
&R^{\ell}  \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, L, \beta\right) \nonumber \\
&=  \frac{\log_2\left(e\right)}{L} \int_{0}^{\infty} \frac{{\rm{exp}}\left(-z \frac{\left\| {\bf{d}}_0 \right\|^{\beta}}{{\rm{SNR}}/K} \right)}{z} \cdot  \nonumber \\
&  \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}} \left( \frac{1}{1+ z \left(\frac{\left\| {\bf{d}}_j \right\| }{ \left\| {\bf{d}}_0 \right\|} \right)^{-\beta}}\right)^{K}  \left( 1 -  \left( \frac{1}{ 1 + z} \right)^{N-2K+1} \right) {\rm d}z.
\end{align}
%where $\mathcal{M}_S\left(z\right)$ and $\mathcal{M}_{I_{\mathcal{P}_{\ell}}}\left(z\right)$ indicate the moment generating function of the desired signal and the interference, respectively. $\mathcal{M}_S\left(z\right)$ is
\end{theorem}
\begin{IEEEproof}
We start by characterizing the moment generating function of the desired channel gain. Since the desired channel gain is a Chi-squared random variable with $2\left(N-2K+1 \right)$ degrees of freedom from Lemma \ref{lem_ch},
%i.e., $\left|\tilde h ^{\ell} \right|^2 \sim \chi^2\left(2\left(N-K/2+1 \right) \right)$, 
its moment generating function is given by
\begin{align}
\mathcal{M}_S\left(z\right) =& \mathbb{E}\left[e^{-z \left|\tilde h ^{\ell}_0 \right|^2} \right] \nonumber \\
=& \left( \frac{1}{ 1 + z} \right)^{N-2K+1}.
\end{align}
Similarly, the moment generating function of the interference is
\begin{align}
\mathcal{M}_{I_{\ell}}\left(z\right) = \prod_{{\bf{d}}_j \in \mathcal{D}_{\ell}} \left( \frac{1}{1+ z \left(\left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta}}\right)^{K}.
\end{align}
By applying Lemma \ref{lem_useful}, the Theorem is completed.
\end{IEEEproof}
Given distances from the tagged user to each BS, the ergodic spectral efficiency of the tagged user is obtained by numerically calculating \eqref{ergodic_rate_th}.
%With the derived expression, the averaged  rate during four phases is obtained by \eqref{averge_rate}.
%Although the ergodic spectral efficiency is represented simply in an integral form, it still 

\subsection{A Lower Bound Characterization}
We also derive a lower bound on the ergodic spectral efficiency with a closed form which is useful in obtaining insight on how system parameters change the ergodic spectral efficiency. To derive a lower bound, the following Lemma is claimed.
\begin{lemma} \label{lem_lower}
For any non-negative independent random variables $S$ and $I$,
\begin{align}
\mathbb{E}\left[\log_2\left(1 + \frac{S}{I+1} \right) \right] \ge \log_2\left( 1 + \frac{e^{\mathbb{E}[\ln S]}}{\mathbb{E}\left[I+1\right]}\right).
\end{align}
\end{lemma}
\begin{IEEEproof}
%We are able to complete the proof by using Jensen's inequality two times.
\begin{align}
\mathbb{E} \left[\log_2 \left( 1+ \frac{S}{I+1} \right) \right] &  {\ge}  \mathbb{E}_{S} \left[ \mathbb{E} \left[ \left. \log_2\left( 1 + \frac{S}{I+1}\right) \right| S \right] \right]   \nonumber \\
& \mathop {\ge} \limits^{(a)} \mathbb{E}\left[   \log_2 \left( 1 + \frac{e^{\ln S}}{\mathbb{E}\left[I+1\right]}\right) \right]  \nonumber \\
& \mathop {\ge} \limits^{(b)} \log_2\left( 1 + \frac{e^{\mathbb{E}[\ln S]}}{\mathbb{E}\left[I+1\right]}\right)
\end{align}
where (a) follows Jensen's inequality and $\log_2 \left( 1 + \frac{1}{x}\right)$ is a convex function of non-negative variable $x$, and (b) also follows Jensen's inequality and $\log_2 \left( 1 + e^{y}\right)$ is a convex function of non-negative variable $y$.
\end{IEEEproof}
Applying Lemma \ref{lem_lower}, a lower bound of the ergodic spectral efficiency is obtained in the following Theorem.
\begin{theorem} \label{th_low}
When the BSs form a BS coordination set according to the proposed cluster pattern, 
the ergodic spectral efficiency of the tagged user is lower bounded as
%\begin{align}
%R^{\ell}  \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, L, \beta\right)  \ge \frac{1}{L}\log_2\left(1 + \frac{{\rm{exp}}\left( \psi\left(N-2K+1 \right)  \right)  }{ K \sum_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left(\left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} + \left\| {\bf{d}}_0 \right\|^{\beta} / {\rm{SNR}}/K}  \right), \label{lower_bound_rate}
%\end{align}
\begin{align}
&R^{\ell}  \left({\rm{SNR}}, \left\| {\bf{d}}_0 \right\|, \mathcal{D}_{\ell},N, K, L, \beta\right)  \nonumber \\
&\ge \frac{1}{L}\log_2\left(1 + \frac{{\rm{exp}}\left( \psi\left(N-2K+1 \right)  \right)  }{ K \sum_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left(\frac{\left\| {\bf{d}}_j \right\| }{ \left\| {\bf{d}}_0 \right\|} \right)^{-\beta} + \left\| {\bf{d}}_0 \right\|^{\beta} / {\rm{SNR}}/K}  \right), \label{lower_bound_rate}
\end{align}
where $\psi\left(z\right)$ is the digamma function defined as $\Gamma'\left(z\right) / \Gamma\left(z\right)$, where $\Gamma\left(z\right)$ is the gamma function defined as
\begin{align}
\Gamma\left(z\right) = \int_{0}^{\infty} x^{z-1}e^{-z} {\rm d} z.
\end{align}
\end{theorem}
\begin{IEEEproof}
First, we characterize $\mathbb{E}\left[ \ln \left|\tilde h ^{\ell}_0 \right|^2\right]$.
Since $\left|\tilde h ^{\ell}_0 \right|^2 \sim \chi^2\left(2\left(N-2K+1\right) \right)$, 
\begin{align}
\mathbb{E}\left[ \ln \left|\tilde h ^{\ell}_0 \right|^2\right] =  \psi\left(N-2K+1 \right),
\end{align}
where $\psi\left(z\right)$ is the digamma function defined as $\Gamma'\left(z\right) / \Gamma\left(z\right)$. Next, we compute $\mathbb{E}\left[I_{{\ell}} \right]$. Since the interference is a sum of independent Chi-squared random variables with different weights corresponding to the ratio of the distances, we have
\begin{align}
\mathbb{E}\left[I_{{\ell}} \right] = K \sum_{{\bf{d}}_j \in \mathcal{D}_{\ell} }\left(\frac{\left\| {\bf{d}}_j \right\|}{ \left\| {\bf{d}}_0 \right\|} \right)^{-\beta}.
\end{align}
By applying Lemma \eqref{lem_lower}, we complete the proof. 
%Verification of the obtained lower bound is provided in Section VI.
\end{IEEEproof}

\begin{figure}[t]
\centerline{\resizebox{0.7\columnwidth}{!}  {\includegraphics{ergodic_verification_eps.eps}}}  
\caption{The comparison of the ergodic spectral efficiency obtained by Monte-Carlo simulations and the analytical lower bound \eqref{lower_bound_rate}. 
It is assumed that $\beta = 4$, $N \in \{3, 4, 10\}$, and $K \in \{1,2, 5\}$.
% The analytical approximation tightly matches to the simulation results over the entire rate threshold region, for all the cases of the numbers of BS antennas and users.
}    
   \label{ergodic_verification}
\end{figure} 

Now we verify the derived lower bound by comparing with the ergodic spectral efficiency obtained by using Monte-Carlo simulations. We use a network model described in Fig.~\ref{network_snapshot} (b) in Section VI for the verification. It is assumed that $L = \Delta_{\rm EC}(G) = 4$. The comparison result is in Fig.~\ref{ergodic_verification}. In Fig.~\ref{ergodic_verification}, we observe two separate regime depending on SNR, which are referred as the DoF regime and the saturation regime in \cite{lozano:2013:limitcoop}. In both regimes, the lower bound is reasonably tight for different numbers of BS antennas and users settings.
%Theorem \ref{th_low} gives intuition about how the proposed method 
%affects the ergodic spectral efficiency. For example, from the approximation 
%\begin{align}
%\log_2\left(1+x \right) \approx x \log_2 e
%\end{align}
%when $x \ll 1$,
%the following approximation is possible in a low SINR regime.
%\begin{align} \label{ergodic_lower_approx}
%\frac{1}{L}\log_2\left(1 + \frac{{\rm{exp}}\left( \psi\left(N-2K+1 \right)  \right)  }{ K \sum_{{\bf{d}}_j \in \mathcal{D}_{\ell}}\left(\left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} + \left\| {\bf{d}}_0 \right\|^{\beta} / {\rm{SNR}}/K}  \right) \nonumber \\
%\approx \frac{\log_2 e}{L}\cdot \frac{{\rm{exp}}\left( \psi\left(N-2K+1 \right)  \right)  }{ K \sum_{{\bf{d}}_{j} \in \mathcal{D}_{\ell}}\left(\left\| {\bf{d}}_j \right\| / \left\| {\bf{d}}_0 \right\| \right)^{-\beta} + \left\| {\bf{d}} \right\|^{\beta} / {\rm{SNR}}/K}.
%\end{align}
%By using the proposed method, one dominant denominator of \eqref{ergodic_lower_approx} is eliminated, which leads significant increase of the ergodic spectral efficiency.


\section{Ergodic Spectral Efficiency in Random Networks}

%\begin{figure}[t]
%\centerline{\resizebox{0.7\columnwidth}{!}{\includegraphics{voronoi_random.pdf}}}  
%\caption{An illustration of the proposed cluster pattern description in a random network. 
%Once the 2nd-order Voronoi regions are defined on the plane, the same coordination strategy is applied with the case of a fixed network model.
%Since the cluster pattern is designed by using the 2nd-order Voronoi region, a user scheduling time and the corresponding BS coordination set is determined by the 2nd-order Voronoi region.
%}    
%\label{voronoi_random_fig}   
%\end{figure}  

So far, we characterize the performance when the BSs are coordinated with the proposed cluster patterns in a fixed network model. In this section, we analyze the ergodic spectral efficiency when applying the proposed method in a random network.
We skip characterizing the rate coverage probability of the proposed method in a random network since it is equivalent to a special case of (26) and (27) in \cite{NY:dynamic}. 
%modeled by using a homogeneous PPP, and analyze the performance as in the square grid network model. For a random network 
We consider a downlink cellular network where BSs equipped with $N$ antennas are distributed according to a homogeneous PPP, $\Phi = \left\{ \left. {\bf{d}}_{i} \right| i \in \mathbb{N} \right\}$ with density $\lambda$ on the plane $\mathbb{R}^2$. 
For easy of notation, we start the BS location index with $1$ so that the $k$-th closest BS from the origin is denoted as ${\bf{d}}_{k}$.
Single antenna users are also distributed as a homogeneous PPP, $\Phi_{\rm{U}} = \left\{ \left. {\bf{u}}_{i} \right| i \in \mathbb{N} \right\}$ with density $\lambda_{{\rm{U}}}$, which is independent with $\Phi$. We assume that $\lambda_{\rm U} \gg \lambda$ so that there exist at least $K$ users in each region with high enough probability. 
Under this assumption, $K$ users are selected in each region.
In this network model, the proposed method is applied in the same way with the case of a fixed network model, i.e., make BS clusters by using 2nd-order Voronoi region, draw a graph $G\left(\Phi \right)$ by Delaunay triangulation, and design BS cluster patterns by using the corresponding edge-coloring.
%Similarly to the fixed network model case, the 
%In this network model, we are able to find the 2nd-order Voronoi region according to the definition \eqref{voronoi_defn} with probability 1. Then, as in the case of a fixed network model, the BS coordination pattern is designed by using the notion of the 2nd-order Voronoi region. 
%We illustrate a simple example of the cluster pattern in Fig.~\ref{voronoi_random} focusing on ${\bf{d}}_0$. As described in Fig.~\ref{voronoi_random_fig}, we are able to find 5 different Voronoi regions including ${\bf{d}}_0$, denoted by $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1 \right)$, $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_2 \right)$, $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_8 \right)$, $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_{9} \right)$, and $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_{10} \right)$, respectively. By using the proposed cluster pattern, we form five BS coordination set $\{{\bf{d}}_0, {\bf{d}}_1 \}$, $\{{\bf{d}}_0, {\bf{d}}_2 \}$, $\{{\bf{d}}_0, {\bf{d}}_8 \}$, $\{{\bf{d}}_0, {\bf{d}}_{9} \}$, and $\{{\bf{d}}_0, {\bf{d}}_{10} \}$ and alternatively exploit each cluster pattern in different time duration $T_{\ell}$ for $\ell \in \{1,2,...,5\}$ as illustrated in Fig.~\ref{voronoi_random_fig}. For instance, assume that user 1 is in $\mathcal{V}_2\left({\bf{d}}_0, {\bf{d}}_1 \right)$. Then it is served in the time duration $T_1$ by a cooperative BS set $\left\{{\bf{d}}_0,{\bf{d}}_1\right\}$, which leads to protection from the interference of the BS located in ${\bf{d}}_1$, which causes the most dominant interference to user 1.
%Thanks to this, user 1 get the performance gain equivalent to that of the pair-wise dynamic clustering method \cite{NY:dynamic} in the time duration $T_1$. Unlike the dynamic clustering method, however, the considered clustering method is operated by a network-centric way, which causes lower overheads since the cluster pattern is formed regardless of users' conditions. 
%We denote that the cluster patterns $\mathcal{P}_{\ell}$ for $\ell \in \{1,...,L\}$.
Under the assumption that the proposed method is used,
we derive a lower bound on the ergodic spectral efficiency in a closed form.
%The rate coverage distribution is omitted since the SIR distribution when the dynamic clustering method was applied was characterized in \cite{NY:dynamic}. 
One should note that the randomness of the network model in this section is not a core part of the proposed method. Rather, it is a tool for analyzing the ergodic spectral efficiency performance.


Without loss of generality, we focus on the typical user located at the origin, and assume that it is associated with the BS located at ${\bf{d}}_1$. Under the premise that the typical user is in $\mathcal{V}_2\left({\bf{d}}_1, {\bf{d}}_2\right) \in \mathcal{P}_{\ell}$, the typical user is served by a BS cluster $\{ {{\bf{{d}}}_1},{{ \bf{{d}}}_2} \}$.
%in time duration $T_{\ell}$ for $\ell \in \left\{1,2,...,L\right\}$, which means there are $L$ %cluster patterns including the BS at ${\bf{d}}_0$. 
%Similarly to the fixed network case, we assume that $K$ users in $\mathcal{V}_2\left({\bf{d}}_1, {\bf{d}}_2\right)$ are selected to be associated with the BS located at ${\bf{d}}_1$, resulting in that there are $2K$ users in each 2nd-order Voronoi region.
%, and $N \ge LK'$.
Similarly to the fixed network case, the typical user communicates with a BS pair $\{ {{\bf{{d}}}_1},{{ \bf{{d}}}_2} \}$ by applying CBF, so that the intra-cluster interference from the BS at ${\bf{d}}_2$ to the typical user is removed.
%After forming the cluster set, the typical user is protected from interference by applying CBF as in the fixed network model case.
Then, equivalently with \eqref{sinr}, the SINR of the typical user is given by
\begin{align} \label{sir_random}
{\rm SINR}_{\left| \ell \right.} = \frac{\left\| {\bf{d}}_1 \right \|^{-\beta} \left| \tilde h^{\ell}_1 \right|^2}{\sum_{{\bf{d}}_j \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_j \right\|^{-\beta}\left| \left({{\bf{h}}}^{\ell}_{j}\right)^T {\bf{V}}^{\ell}_{j} \right|^2} + 1/ {\rm{SNR}}/K },
\end{align}
where $\tilde h^{\ell}_1$ is the modified channel coefficient from the BS at ${\bf{d}}_1$ to the typical user after applying CBF, ${\bf{h}}_j^{\ell}$ is the channel coefficient vector from the BS at ${\bf{d}}_j$ to the typical user, and ${\bf{V}}^{\ell}_{j}$ is a beamforming matrix of the BS at ${\bf{d}}_j$, respectively. 
%For analytical tractability, we assume that there are $K'$ users in each region, therefore $LK' = K$. 
The entries of the channel coefficient vector follows $\mathcal{CN}\left(0,1\right)$. From \eqref{sir_random}, the ergodic spectral efficiency is defined as
\begin{align} \label{rate_random}
&R^{\ell} \left({\rm{SNR}},N, K, \beta\right) = \frac{1}{L}\mathbb{E}\left[\log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right],
\end{align}
where the pre-log term $1/L$ is used because $1/L$ time-frequency resources are exploited to serve the typical user. 
%$L$ is determined as an algorithm parameter in the edge cutting algorithm.

\subsection{Lower Bound on the Ergodic Spectral Efficiency}
Now we derive a lower bound on the ergodic spectral efficiency. 
%We first obtain a lower bound for unmodified asymmetric network and then obtain a modified asymmetric network where some edges are cut for reducing the time-frequency resources use. 
%For analytical tractability, we assume that the edge cutting is randomly performed. 
To do this, we need the following Lemma. 
%Lemma \ref{Lem_Ber} is for obtaining $\mathbb{E}\left[L \right]$, and 
Lemma \ref{lem_joint_pdf} provides probability density function (PDF) of distances of the closest BS and $k$-th closest BS to the origin.

%first obtain $\mathbb{E}\left[\Delta \right]$
%, which is the average of the maximum degree in $G\left( \Phi \right)$ to 
%resources. 
%Since $L = \Delta$ if $\Delta > 6$, 
%Recalling from Section II, $L$ is given by
%\begin{align}
%L = \left\{\begin{array}{cc} \Delta+1, & {\rm if}\; \Delta \le 6, \\  \Delta,  & {\rm if}\; \Delta > 6 \end{array} \right.
%\end{align}
%Calculating $\mathbb{E}\left[\Delta\right]$ of a random graph is a hard problem and only scaling laws are available in general \cite{BerEppYao91}.
%where $\Delta$ is the maximum degree of a vertex in $G\left(\Phi\right)$, 
%we obtain the the average of the maximum degree of a vertex in $G\left(\Phi\right)$ in the following lemma.
%the condition of existence of the 2nd-order Voronoi region for two arbitrary points.
%
%\begin{lemma}[The expected maximum degree of a vertex, \cite{BerEppYao91}] \label{Lem_Ber}
%Let $S$ be the cube $\left[0, n^{1/2} \right] \times \left[0, n^{1/2} \right]$, and let $G\left(\Phi \right) \cap S$ denote the embedded graph consisting of vertices and edges of the Delaunay triangulation that lie entirely in $S$. Then, the expected maximum degree $\Delta$ in $G\left(\Phi\right) \cap S$ is $\Theta\left(\log n/\log\log n \right)$, with a constant depending only on dimension on which $\Phi$ is defined.
%Assuming arbitrary two distinct points ${\bf{d}}_i$ and ${\bf{d}}_j$ for $i \ne j$, the 2nd-order Voronoi region of the considered points $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ exists if $\mathcal{V}_1\left({\bf{d}}_i \right)$ and $\mathcal{V}_1 \left({\bf{d}}_j \right)$ has a shared edge.
%Given a set of points and a subset $H = \left\{i_1, i_2, ..., i_k \right\}$ of $I$, $k>1$, if $H_s = H - \{j_s\}$, where $j_s \in H$, for $s = 1,2,...,m$, $m \le k$, are the only subsets of $H$ such that $\mathcal{V}_{k-1}\left(H_s, I\right)$ exists
%\end{lemma}
%\begin{proof}
%See the reference \cite{BerEppYao91}.
%If $\mathcal{V}_1\left({\bf{d}}_i \right)$ and $\mathcal{V}_1 \left({\bf{d}}_j \right)$ has a shared edge, any point on that edge is included in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$ by the definition of the 2nd-order Voronoi region. 
%Now, consider a point ${\bf{d}}$ in $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$. Then, ${\bf{d}}$ should be included in either $\mathcal{V}_1\left({\bf{d}}_i \right)$ or $\mathcal{V}_1\left({\bf{d}}_j \right)$. Without loss of generality, assume that ${\bf{d}} \in \mathcal{V}_1\left({\bf{d}}_i \right)$. Then, 
%\end{proof}
%Although obtaining the exact $\mathbb{E}\left[ \Delta \right]$ is challenging, 
%Since the scaling law of $\Delta$ is independent of $\lambda$ from Lemma \ref{Lem_Ber}, we conjecture that $\Delta$ is also independent of $\lambda$. 
%Based on this conjecture, we numerically obtain $\mathbb{E}\left[L\right]$ instead of obtaining it analytically. By using Monte-Carlo simulations, we find $\mathbb{E}\left[L\right] \approx 11.1$. Next, Lemma \ref{lem_joint_pdf} gives the joint PDF of distances of two different BSs to the origin.

\begin{lemma} \label{lem_joint_pdf}
The joint probability density function (PDF) of $\left\| {\bf{d}}_1 \right\|$ and $\left\|{\bf{d}}_k \right\|$ is 
%\begin{align}
%f_{\left\| {\bf{d}}_1 \right\|, \left\| {\bf{d}}_k \right\|}\left(r_1, r_k \right) = \left\{ \begin{array}{cc} \frac{4\left(\lambda \pi \right)^k}{\left(k-2 \right)!} r_1 r_k \left(r_k^2 - r_1^2 \right)^{k-2} e^{-\lambda \pi r_k^2} & {\rm if}\; r_1 \le r_k \\ 0&{\rm otherwise.}  \end{array} \right.
%\end{align}
\begin{align}
&f_{\left\| {\bf{d}}_1 \right\|, \left\| {\bf{d}}_k \right\|}\left(r_1, r_k \right) \nonumber \\
&= \left\{ \begin{array}{cc} \frac{4\left(\lambda \pi \right)^k}{\left(k-2 \right)!} r_1 r_k \left(r_k^2 - r_1^2 \right)^{k-2} e^{-\lambda \pi r_k^2} & {\rm if}\; r_1 \le r_k \\ 0&{\rm otherwise.}  \end{array} \right.
\end{align}
\end{lemma}
\begin{proof}
See Appendix C in \cite{NY:dynamic}.
\end{proof}

Leveraging Lemma \ref{lem_joint_pdf}, Theorem \ref{theo_ppp_ergodic} is claimed. 
For analytical tractability, we assume the interference limited regime.

\begin{theorem} \label{theo_ppp_ergodic}
When BSs' locations are distributed by a homogeneous PPP and the proposed method is applied,
the ergodic spectral efficiency of the typical user is lower bounded as
%The ergodic spectral efficiency when the CM is applied with the proposed method \eqref{rate_random} is lower bounded by
%\begin{align} \label{theo_low_random}
%R^{\ell} \left( N, K, \beta\right) \ge \frac{1}{L} \log_2 \left(1 + \frac{\left(\beta^2 - 4 \right)}{{8K}} {\exp\left( \psi\left(N-2K+1 \right)\right)} \right),
%\end{align}
\begin{align} \label{theo_low_random}
&R^{\ell} \left( N, K, \beta\right) \nonumber \\
&\ge \frac{1}{L} \log_2 \left(1 + \frac{\left(\beta^2 - 4 \right)}{{8K}} {\exp\left( \psi\left(N-2K+1 \right)\right)} \right),
\end{align}
where $\psi \left( z \right)$ is the digamma function, defined as $\Gamma'\left(z\right) / \Gamma\left(z\right)$, where $\Gamma\left(z\right)$ is the gamma function defined as
\begin{align}
\Gamma\left(z\right) = \int_{0}^{\infty} x^{z-1}e^{-z} {\rm d} z.
\end{align}
\end{theorem}
\begin{proof}
%From our numerical finding that $\mathbb{E}\left[L \right] = 11.1$, 
The ergodic spectral efficiency \eqref{rate_random} is written by
%\begin{align} \label{theorem4_frame}
%&\mathbb{E}\left[ \frac{1}{L} \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right] 
%= \frac{1}{L} \mathbb{E} \left[ \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right] \nonumber \\
%\mathop {\ge} \limits^{(a)} & \frac{1}{\mathbb{E}\left[ L \right] } \mathbb{E} \left[ \log_2\left(1+  {{\rm SIR}}_{|{\ell}} \right) \right] \nonumber \\
%\mathop {\ge} \limits^{(a)} &\frac{1}{L} \log_2\left(1 + \frac{e^{\mathbb{E} \left[ \ln\left( \left| \tilde h^{\ell}_1 \right|^2 \right)  \right]}}{\mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}\left| \left({{\bf{h}}}^{\ell}_{i}\right)^T {\bf{V}}^{\ell}_{i} \right|^2}\right]} \right),
%\end{align}
\begin{align} \label{theorem4_frame}
&\mathbb{E}\left[ \frac{1}{L} \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right] \nonumber \\ 
&= \frac{1}{L} \mathbb{E} \left[ \log_2\left(1+  {{\rm SINR}}_{|{\ell}} \right) \right] \nonumber \\
%\mathop {\ge} \limits^{(a)} & \frac{1}{\mathbb{E}\left[ L \right] } \mathbb{E} \left[ \log_2\left(1+  {{\rm SIR}}_{|{\ell}} \right) \right] \nonumber \\
&\mathop {\ge} \limits^{(a)} \frac{1}{L} \log_2\left(1 + \frac{e^{\mathbb{E} \left[ \ln\left( \left| \tilde h^{\ell}_1 \right|^2 \right)  \right]}}{A} \right),
\end{align}
%where (a) follows Lemma \ref{lem_lower} and $L = \Delta_{\rm EC}(G)$. Note that $\Delta_{\rm EC}(G)$ is determined as an algorithm parameter of the edge-cutting algorithm.
%From Lemma \ref{Lem_Ber}, $\rho = \Theta \left(\log n/\log\log n \right)$. 
%Since the expected number of the edge of Voronoi region is $6$, $\rho = 6$. 
%For sufficiently large $n$, $\rho \approx $
where $A = \mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}\left| \left({{\bf{h}}}^{\ell}_{i}\right)^T {\bf{V}}^{\ell}_{i} \right|^2}\right]$, (a) follows Lemma \ref{lem_lower}, and $L = \Delta_{\rm EC}(G)$. Note that $\Delta_{\rm EC}(G)$ is determined as an algorithm parameter of the edge-cutting algorithm.
Now, we obtain the expectation in the nominator and denominator of \eqref{theorem4_frame}.
Since the interference is mitigated by CBF, as in the case of the deterministic square grid network model, $\left|\tilde h ^{\ell}_1 \right|^2$ follows $\chi^2\left(2\left(N-2K+1\right) \right)$, which leads to 
\begin{align} \label{theorem4_X}
\mathbb{E} \left[ \ln\left( \left| \tilde h^{\ell}_1 \right|^2 \right)  \right] = \psi\left(N-2K+1 \right),
\end{align}
where $\psi\left(\cdot\right)$ is the digamma function. 
Next, we calculate the expectation of the interference part. We write the expectation of the denominator of the SIR as
%\begin{align} 
%&{\mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}\left|\left({{\bf{h}}}^{\ell}_{i}\right)^T {\bf{V}}^{\ell}_{i} \right|^2}\right]}
%\mathop = \limits^{(a)}  K {\mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}\right]} \nonumber \\
%\mathop = \limits^{} & K \mathbb{E}_{r_1, r_2} \left[ {\mathbb{E}_{\Phi \backslash \mathcal{B}\left(0, r_2 \right)} \left[\left. r_1^{\beta} \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, r_2 \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}
%2 \pi \lambda \int_{r_2}^{\infty} r^{1-\beta}  {\rm d} r 
%\right| \left\| {\bf{d}}_1 \right\| = r_1,  \left\| {\bf{d}}_2 \right\| = r_2\right]} 
%\right] \nonumber \\
%\mathop = \limits^{} & K \mathbb{E}_{r_1, r_2} \left[ {\mathbb{E}_{\Phi \backslash \mathcal{B}\left(0, r_2 \right)} \left[\left. r_1^{\beta} \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, r_2 \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}
%\right| \left\| {\bf{d}}_1 \right\| = r_1,  \left\| {\bf{d}}_2 \right\| = r_2\right]} 
%\right] \nonumber \\
%\mathop = \limits^{(b)} & K \mathbb{E}_{r_1, r_2} \left[ r_1^{\beta} 2 \pi \lambda \int_{r_2}^{\infty} r^{1-\beta}  {\rm d} r \right] \nonumber \\
%\mathop = \limits^{} &  \frac{ 2 K\pi \lambda}{\beta-2} \mathbb{E}_{r_1, r_2} \left[  r_1^{\beta} r_2^{2-\beta}  \right] \nonumber \\
%\mathop = \limits^{(c)} & \frac{ 2 K\pi \lambda}{\beta-2} \int_{r_2 = 0}^{\infty} \int_{r_1 = 0} ^{r_2}  4\left(\lambda \pi \right)^2   e^{-\lambda \pi r^2_2} r _1^{\beta+1} r_2^{3-\beta}{\rm d} r_1 {\rm d} r_2 \nonumber \\
%=& \frac{8K}{\beta^2 - 4} \label{theorem4_Y}
%\end{align}
\begin{align} 
&{\mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}\left|\left({{\bf{h}}}^{\ell}_{i}\right)^T {\bf{V}}^{\ell}_{i} \right|^2}\right]} \nonumber \\
&\mathop = \limits^{(a)}  K {\mathbb{E} \left[\left\| {\bf{d}}_1 \right \|^{\beta}   \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, \left\| {\bf{d}}_2 \right\| \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}\right]} \nonumber \\
%\mathop = \limits^{} & K \mathbb{E}_{r_1, r_2} \left[ {\mathbb{E}_{\Phi \backslash \mathcal{B}\left(0, r_2 \right)} \left[\left. r_1^{\beta} \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, r_2 \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}
%2 \pi \lambda \int_{r_2}^{\infty} r^{1-\beta}  {\rm d} r 
%\right| \left\| {\bf{d}}_1 \right\| = r_1,  \left\| {\bf{d}}_2 \right\| = r_2\right]} 
%\right] \nonumber \\
\mathop = \limits^{} & K \mathbb{E}_{r_1, r_2} \left[ {\mathbb{E}_{\Phi \backslash \mathcal{B}\left(0, r_2 \right)} \left[\left. r_1^{\beta} \sum_{{\bf{d}}_i \in \Phi \backslash \mathcal B\left(0, r_2 \right)}{ \left\| {\bf{d}}_i \right\|^{-\beta}}
\right| \begin{array}{c}{\left\| {\bf{d}}_1 \right\| = r_1}, \\{ \left\| {\bf{d}}_2 \right\| = r_2}\end{array}\right]} 
\right] \nonumber \\
\mathop = \limits^{(b)} & K \mathbb{E}_{r_1, r_2} \left[ r_1^{\beta} 2 \pi \lambda \int_{r_2}^{\infty} r^{1-\beta}  {\rm d} r \right] \nonumber \\
%\mathop = \limits^{} &  \frac{ 2 K\pi \lambda}{\beta-2} \mathbb{E}_{r_1, r_2} \left[  r_1^{\beta} r_2^{2-\beta}  \right] \nonumber \\
\mathop = \limits^{(c)} & \frac{ 2 K\pi \lambda}{\beta-2} \int_{r_2 = 0}^{\infty} \int_{r_1 = 0} ^{r_2}  4\left(\lambda \pi \right)^2   e^{-\lambda \pi r^2_2} r _1^{\beta+1} r_2^{3-\beta}{\rm d} r_1 {\rm d} r_2 \nonumber \\
=& \frac{8K}{\beta^2 - 4} \label{theorem4_Y}
\end{align}
where (a) follows $\left|\left({{\bf{h}}}^{\ell}_{i}\right)^T {\bf{V}}^{\ell}_{i} \right|^2 \sim \chi^2\left(2K \right)$ and it is independent to $\Phi$, (b) follows the Campbell's theorem, and (c) follows that the joint PDF in Lemma \ref{lem_joint_pdf}. 
%The joint PDF of $\left\| {\bf{d}}_1 \right\|$ and $\left\| {\bf{d}}_2 \right\|$ can be directly obtained from Lemma \ref{lem_joint_pdf}.
%The proof of the above PDF is provided in \cite{NY:dynamic}. 
Plugging \eqref{theorem4_X} and \eqref{theorem4_Y} into \eqref{theorem4_frame} the proof is completed.
\end{proof}

%Since it is challenging to find $\mathbb{E}\left[1/L\right]$ analytically in a homogeneous PPP, it is possible to use numerically found $\mathbb{E}\left[1/L\right]$. For instance, we find that $\mathbb{E}\left[1/L\right] \approx 0.09$ by Monte-Carlo simulations when $\lambda = 3\times 10^{-7}$. Further, 
One should note that Theorem \ref{theo_ppp_ergodic} considers the worst case for the out-of-cluster interference. This is because, in an irregular and asymmetric network model, e.g., a homogeneous PPP model, some BSs do not use the specific time-frequency resource exploited to serve the typical user. Then, the interference from those BSs should not be counted in the out-of-cluster interference term. Analyzing this, however, is highly difficult because it depends on the correlation of the distributed points. It is not tractable in the known PPP framework. For this reason, we rather consider the worst case scenario, which assumes that the out of cluster interference comes from all the BSs in a network.



%In an asymmetric network, there can be too many
%is that asymmetric network geometry can cause too much 
%time-frequency resources use and this can degrade the performance.
%the total time-frequency resource used by each BS can be different. 
%Focusing on the BS at ${\bf{d}}_0$, the cluster patterns $\mathcal{P}_6$, $\mathcal{P}_7$, $\mathcal{P}_8$, and $\mathcal{P}_9$ are not exploited, which means the BS at ${\bf{d}}_0$ does not use the time-frequency resources assigned to $\mathcal{P}_6$, $\mathcal{P}_7$, $\mathcal{P}_8$, and $\mathcal{P}_9$. This unbalanced resource assignment can cause inefficiency compared to the symmetric network case.
\section{Performance Comparisons}
%In this section, the performance comparisons by using Monte-Carlo simulations are described. We first introduce baseline methods and explain simulation results.
\subsection{Simulation Setup}
We provide performance comparison results to demonstrate the superiority of the proposed method via system level simulations. As a simulation setup, we consider a $1400{\rm m} \times 1400{\rm m}$ square network plane, that consists of $49$ ($= 7 \times 7$) square grids whose a side length is $200 {\rm m}$. 
%For the wrap-around, we only consider users inside $1000 {\rm m} \times 1000 {\rm m}$. 
%tessellated by square grids whose side length is $200 \rm m$. The whole network plane consists of $49$ ($= 7 \times 7$) squares, so that the size of the whole network plane is. 
Within each square grid, one BS is randomly located. A range of a BS location is restricted by the perturbation square whose a side length is $p$, $0 \le p \le 200$, and a center is same with the square grid. Given $p$, a BS can randomly located inside $p {\rm m}\times p{\rm m}$ perturbation square. For example, if the perturbation square has $p = 0 $, 
each BS is regularly located on a center of each square grid. This is equivalent to a regular network. In other extreme case, if the perturbation parameter has $p = 200 $, each BS can be located in any place in each square grid, which causes maximal randomness in a network. Fig.~\ref{perturb_square} describes the perturbation square and the corresponding BS location. This setting mimics an actual BS deployment scenario, where a BS is built to satisfy a certain guard region to other BSs, preventing that any two BSs are closely located each other.

%We have total $49$ BSs in a network. 
For the simulation, we consider two cases, $p = 100$ and $p = 200$. 
A case of $p=200$ represents more irregularity of BSs locations than a case of $p=100$. In these networks, $K_{\rm perBS} \times 49$ users are uniformly dropped. 
%We repeat the user dropping $1000$ times and average it. 
Network snapshots for these two cases are illustrated in Fig.~\ref{network_snapshot}.
For these snapshots, $K_{\rm perBS}=10$. 
Clearly, we observe that BSs are more irregularly located when $p=200$.
In the performance comparison, we only consider users inside $1000 {\rm m} \times 1000 {\rm m}$ for wrap-around. 
The number of BS antennas and selected users are assumed to be $N = 3$ and $K = 1$. The path-loss exponent is $4$.

\begin{figure}[t]
\centerline{\resizebox{0.5\columnwidth}{!}{\includegraphics{perturb_square_eps.eps}}}    
\caption{A description of the perturbation square whose a side length is $p$. In the described case, a BS is randomly located inisde a region marked by grey shade. }  
   \label{perturb_square} 
\end{figure}

Applying the proposed method in a given network, it is set that $L = \Delta_{\rm EC}(G) = 4$ in both cases of $p=100$ and $p=200$ for the edge cutting algorithm. Recalling the edge cutting algorithm, we cut edges inversely proportional to the area of the corresponding 2nd-order Voronoi regions. However, measuring the area of 2nd-order Voronoi region is a challenging task. For this reason, instead of directly measuring each area, we uniformly drop dummy users and count the number of users associated with the certain two closest BSs pair. Denoting that the number of users whose the two closest BSs are $\{{\bf{d}}_i, {\bf{d}}_j\}$ as $N({\bf{d}}_i, {\bf{d}}_j )$, we approximate $\lambda(\mathcal{V}_2({\bf{d}}_i, {\bf{d}}_j)) \approx N({\bf{d}}_i, {\bf{d}}_j )$. In the simulation, we drop $5000$ dummy users for the approximation.


\begin{figure}[t] 
\centering 
$\begin{array}{cc}  
{\resizebox{0.4\columnwidth}{!}
{\includegraphics{net_snapshot_100_eps.eps}}}  &  
{\resizebox{0.4\columnwidth}{!}
{\includegraphics{net_snapshot_200_eps.eps}}}  \\  
\mbox{(a)} &
\mbox{(b)}
\end{array}$
\caption{Network snapshots for $p=100$ in (a) and $p=200$ in (b). It is assumed that $K_{\rm perBS} = 10$. Each network plane is tessellated by 1st order Voronoi region of each BS. }
\label{network_snapshot}  
\end{figure} 

For the performance comparison, we consider two conventional coordination methods: dynamic clustering and static clustering. We explain as follows:
\subsubsection{Dynamic Clustering} 
In this method, selected users communicate with their two closest BSs.
%Since there is no known user scheduling 
For comparison, we propose a simple user scheduling method by modifying FlashLinQ \cite{flashlinq} for dynamic clustering.
Assuming a global scheduler, it assigns random orders to dropped users and the orders can be changed over time for user fairness. Then, according to the assigned order, a user finds the two closest BSs and create a BS cluster. If two users select the same BS, i.e., the BS selection conflict problem occurs, a user whose an order is late gives up communication in this particular time slot and wait for the next time slot. Since two users are able to communicate with one BS cluster ($K=1$ for one BS and a BS cluster has two BSs), once a BS cluster is formed by a user, a global scheduler immediately finds the other user to communicate with the corresponding BS cluster. This process continues until the number of made BS clusters is maximal.
%This is similar to maximal packing in a network since BS clusters are maximally packing to a network. 
Once all the BS clusters are made, selected users communicate with their BS clusters by using CBF, same with the proposed method.
If there are remaining BSs that could not make a BS cluster, a global scheduler finds a user to communicate with them. No BS coordination is used for these users. 
 

\subsubsection{Static Clustering}
In this method, BS clusters are predetermined irrespective of users' conditions. A BS randomly selects one BS among its neighbor BSs, and makes a BS cluster with the selected one. Then, a user is independently selected by each BS and communicates with the predetermined BS cluster. Similarly to the proposed method and dynamic clustering, the intra-cluster interference is mitigated by using CBF. In the static clustering, there is non-zero probability that a user cannot communicate with its two closest BSs since BS clusters are made independently to users' locations.

%Fig.~\ref{baseline_explain} illustrates the baseline methods applied in a square grid network where each BS is located on a square grid. In single cell operation described in Fig.~\ref{baseline_explain} (a), the user associated with BS 1 receives interference from all the BSs since there is no interference management technique. In fractional frequency reuse described in Fig.~\ref{baseline_explain} (b), the interference only comes from BS 4, since BS 2 and BS 3 (marked by darker shade) are allocated by different sub-band from the sub-band of BS 1. In random clustering describe in Fig.~\ref{baseline_explain} (c), BS 1 forms a BS cluster with BS 2 (marked by green dotted line), so that the interference from BS 2 is mitigated.
%In particular, in random clustering, there is a non-zero possibility that the performance of the user is degraded by the dominant interference. This is mainly because a BS cluster is formed by a random manner. For instance, in Fig.~\ref{baseline_explain} (c), the dominant interference comes from BS 3 since BS 3 is the closest interfering BS for the user, but BS 1 forms a BS cluster with BS 2, causing the dominant interference still degrades the user performance. 
%The random clustering is considered as a baseline method since it is a conventional way to form a BS cluster in irregular networks without exploiting the proposed clustering strategy.
%there is no way to guarantee that every user is free from the dominant interference without using the proposed clustering strategy at the best of authors' knowledge.





\subsection{Simulation Results}
First, we compare the edge users sum throughput depending on $K_{\rm perBS}$. 
An edge user is defined as a user whose a distance ratio between the closest BS and the second closest BS is larger than $2/3$, i.e., $\left\| {\bf{d}}_0\right\| / \left\|{\bf{d}}_1 \right\| > 2/3$. 
%The rationale behind this setting is on the conventional edge user definition in a square grid BS topology. In a square grid BS topology, an edge user is defined as a user a user located in the outside of the square whose a side length is $2/3R$, where $R$ is a inter-site distance. In irregular BS topologies, however, this definition cannot be applied since a shape of each cell is not 
For obtaining the edge users sum throughput, we only pick the active edge users and sum their throughput. 
In the simulation, SNR is set to be $100{\rm dB}$. Fig.~\ref{sum_throughput_compare} illustrates the edge users sum throughput. As observed in the figure, when the dropped users are dense enough, the performance of the proposed method achieves the performance of the dynamic clustering. Specifically, the proposed method have the same performance as the dynamic clustering when $K_{\rm perBS}>20$ in the $p=100$ case, and when $K_{\rm perBS}>40$ in the $p=200$ case. Comparing to the performance of the static clustering in the dense user environment, the edge users throughput is improved by $28\%$ in the $p=100$ case and $24\%$ in the $p=200$ case.
The rationale behind this improvement obviously comes from that users communicate their two closest BSs in the proposed method, where the strongest interference is mitigated. In the static clustering, the strongest interference still can degrade the performance. 
In contrary to that, when the dropped users are not dense enough, the performance of the proposed method is even lower than that of the static clustering. This is because the proposed method uses predefined cooperative regions (2nd-order Voronoi regions) and time-frequency allocation. For intuition, let's consider an example cooperative region $\mathcal{V}_2\left({\bf{d}}_i, {\bf{d}}_j \right)$. For serving users in this region, a BS cluster $\{ {\bf{d}}_i, {\bf{d}}_j\}$ is made and a specific time-frequency resource is allocated to this cluster. If there is no user in this region, however, the allocated time-frequency resource is wasted, and this degrades the performance. In the dynamic clustering, this resource waste does not occur since a BS cluster is flexibly made by a global scheduler, therefore there is always a user to communicate with the BS cluster. 
When the users are dense enough, all the predefined region has at least $K$ users therefore no resource is wasted. Then, the proposed method provides the same performance as the dynamic clustering. This intuition also justifies the difference between the results of the $p=100$ and $p=200$ cases. Since the BSs locations are more regular when $p=100$, each 2nd-order Voronoi region has similar area. When $p=200$, however, the BSs locations are highly irregular, therefore some 2nd-order Voronoi regions can be very small. Then, the probability that each region has at least $K$ users is reduced. This is the reason why in the $p=200$ case, more users are required to achieve the same performance as the dynamic clustering compared to the $p=100$ case. 

\begin{figure}[!t] 
\centering
$\begin{array}{cc}  
{\resizebox{0.465\columnwidth}{!}
{\includegraphics{sum_throughput_100_eps.eps}}}  &
{\resizebox{0.47\columnwidth}{!} 
{\includegraphics{sum_throughput_200_eps.eps}}}  \\   
\mbox{(a)} &
\mbox{(b)}
\end{array}$
\caption{The edge users sum throughput depending on $K_{\rm perBS}$ in the $p=100$ case (a) and in the $p=200$ case (b). It is assumed that $N=3$, $K=1$, and $\beta = 4$. The edge cutting is performed with $\Delta_{\rm EC}(G)=4$.}
%Four users are located in the cell-edge region as described in Fig.~\ref{cell_edge_region}}    
\label{sum_throughput_compare}  
\end{figure} 

From this observation, we make some intuitive explanations about the effect of $\Delta_{\rm EC}(G)$ on the network performance. Considering $\Delta_{\rm EC}(G) = 1$ as an extreme case, no time-frequency resource are wasted. 
A large user density, however, would be required to achieve the same level of the coordination benefit with that of dynamic clustering. This is mainly because many 2nd-order Voronoi regions are cut, therefore users exist in the remaining 2nd-order Voronoi region with only small probability. In another extreme case, if the edge-cutting algorithm is not applied, i.e., $\Delta_{\rm EC}(G) = \Delta(G)$, the performance of the proposed method is not affected by the user density since all regions are covered by the proposed method, while the network performance degradation is unavoidable due to the resource waste. In summary, less user density is required as $\Delta_{\rm EC}(G)$ increases, while more resource waste is expected. 



We can incorporate the channel estimation overhead into the sum throughput performance. From \cite{NY:dynamic}, the pilot overhead is 
\begin{align}
\alpha = \frac{L_p(2, N, {\rm SINR})}{L_b},
\end{align}
where $L_p(2, N, {\rm SINR})$ is the number of symbols for pilots and $L_b$ is a given fading coherence time in symbols. The number of symbols for pilots is $L_p(2, N, {\rm SINR}) = 2\eta N$, and $\eta$ is 
\begin{align}
\eta = \max\left\{1, \left\lfloor \frac{1}{\rm SINR} \left(\frac{1}{\rm MMSE } -1\right)\right\rfloor \right\},
\end{align}
where ${\rm MMSE}$ is the minimum mean square error of the pilot signals. Incorporating the pilot overhead into the throughput, the performance is degraded by $(1-\alpha)$. 
Since all the considered coordination methods are pair-wise BS coordination where a BS cluster consists of two BSs, the amount of channel estimation overhead is same. 


%\begin{remark}
%From the simulation result, 
%\end{remark}



\begin{figure}[!t] 
\centering
$\begin{array}{cc}  
{\resizebox{0.47\columnwidth}{!}
{\includegraphics{coverage_100_eps.eps}}} &
{\resizebox{0.455\columnwidth}{!}
{\includegraphics{coverage_200_eps.eps}}}  \\  
\mbox{(a)} &
\mbox{(b)}
\end{array}$
\caption{The rate coverage probability in the $p=100$ case (a) and the $p=200$ case (b). It is assumed that $N=3$, $K=1$, and $\beta = 4$. The edge cutting is performed with $\Delta_{\rm EC}(G)=4$.}
\label{rate_compare_fig}  
\end{figure}


Now we compare the rate coverage probability. For obtaining the rate coverage probability, we calculate the probability that the selected user's throughput is larger than rate threshold $\gamma$. It is assumed that $\rm {SNR}=100 {\rm dB}$ and $K_{\rm perBS} = 40$.
Fig.~\ref{rate_compare_fig} illustrates the rate coverage probability. As observed in this figure, the proposed method provides the same or similar coverage probability with that of the dynamic clustering at low rate threshold, while the coverage probability degrades to the similar level of the static clustering at high rate threshold. Focusing on $\gamma = 1 {\rm bit}$, the rate outage probability is reduced by $50\%$ comparing to the static clustering in the $p=100$ case, and it is reduced by $15\%$ in the $p=200$ case. Since the rate coverage performance at low rate threshold is determined by edge users, the simulation results show that the proposed method is able to provide the same or similar level of coordination benefit with the dynamic clustering to edge users.
At high rate threshold, however, the rate coverage performance is degraded. The reason of this observation is on the edge cutting algorithm. As a result of the edge cutting algorithm, we cut some regions and do not serve users in the cut regions even if users' locations in the cut region are very close to their associated BS. Then, we miss a chance to have a high rate, which degrades the rate coverage at high rate threshold.



\section{Conclusion}
In this paper, we proposed a method for designing BS clusters and BS cluster patterns in 
irregular BS topologies. The core idea of the proposed method is to form BS clusters according to the 2nd-order Voronoi regions, and use edge-coloring for the graph induced by Delaunay triangulation to assign time-frequency resources (colors) to each BS cluster.
%assign made BS clusters into a cluster patterns by using edge-coloring in graph theory. 
By using the proposed method, users in the uncut regions communicate with their two closest BS while avoiding the BS selection conflict problem.
% To avoid a BS selection conflict problem, we use the notion of edge-coloring in graph theory. We assign different time-frequency resources (colors) to each edge of the graph induced by the Delaunay triangulation. Using the BS cluster patterns by the proposed method, every selected user in the plane is guaranteed to communicate with the two closest BSs. 
%This coordination benefit is achieved without a BS conflict problem.
%a user is guaranteed to be in a cluster-center region. 
%By alternatively exploiting the proposed cluster patterns, every user in the plane have a chance to communicate in a cluster-center region.
% method exploits multiple different clustering patterns alternatively so that every user is near the cluster-center. 
In a fixed network with irregular BSs' locations, we provided analytical expressions of the 
rate coverage probability and the ergodic spectral efficiency in terms of the number of antennas per BS, the number of users, distance, SNR, and the pathloss exponent. 
In a random network modeled by a homogeneous PPP,
we also derived a lower bound of the ergodic spectral efficiency as a function of the pathloss exponent and the number of associated users.
Through system level simulations, we showed that the proposed method achieves the same edge users sum throughput performance with dynamic clustering when the users are dense enough. In the rate coverage probability performance, the proposed method provides the same or similar performance with that of dynamic clustering at low rate threshold.
%Through Monte-Carlo simulations, we showed that the proposed clustering outperforms other conventional strategies.
%with less coordination making coordination feasible even with a large number of BSs. 
Future work should consider the application of more advanced coordination strategies going beyond coordinated beamforming.







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