In this paper we study the behavior of maximum out/in-degree of binomial/Poisson random scaled sector graphs in the presence of random vertex and edge faults. We prove the probability distribution of maximum degrees for random faulty scaled sector graphs with $n$ vertices, where each vertex spans a sector of $\alpha$ radians, with radius $r_n\ll \sqrt{\ln n/n}$, become concentrated on two consecutive integers, under some natural assumptions of faulty probabilities. \bigskip skip Keywords: Random scaled sector graphs; Random geometric graphs; Stein-Chen; Extremes
0909.2933.tex
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\begin{document}
\begin{center}
{\LARGE\textbf{Focusing of Maximum Vertex Degrees\\
in Random Faulty Scaled Sector Graphs}}\\
\bigskip
\bigskip
Yilun Shang\footnote{Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, CHINA. email: \texttt{shyl@sjtu.edu.cn}}\\
\end{center}

\begin{abstract}
In this paper we study the behavior of maximum out/in-degree of
binomial/Poisson random scaled sector graphs in the presence of
random vertex and edge faults. We prove the probability distribution
of maximum degrees for random faulty scaled sector graphs with $n$
vertices, where each vertex spans a sector of $\alpha$ radians, with
radius $r_n\ll \sqrt{\ln n/n}$, become concentrated on two
consecutive integers, under some natural assumptions of faulty
probabilities.
\bigskip

\smallskip
\textbf{Keywords:} Random scaled sector graphs; Random geometric
graphs; Stein-Chen; Extremes


\end{abstract}

\bigskip
\normalsize

\bigskip
\normalsize
\noindent{\Large\textbf{1. Introduction}}

\smallskip
Wireless ad-hoc communication network of sensors is gaining
increasing importance in telecommunication society \cite{11,20}. The
random scaled sector graph model proposed in \cite{1} is aiming to
provide a tool for the analysis of routing and transmission of
information in sensor networks communicating through optical devices
or directional antennae. In this setting, a randomly scattered large
amount of transmitters are located in some geographical area. A
transmitter can orient its laser beam in any position of a
prescribed scanning area of $\alpha$ radians, however, the
transmitter can receive light from any point, within distance $r$,
that is ``looking'' to it. Moreover, some hostile unexpected
environments should be taken into account. Every transmitter
(vertex) may have a fail probability understood as the sensor
becomes inoperative resulting from mechanical damages or power
drain. Every connection (out-edge) may also have a fail probability
understood as the failure of communication because of bad weather or
terrain obstacles. Whence a random faulty scaled sector graph (see
Section 2) is dealt with in this paper, which incorporating these
features and serves our object well.

In this paper we investigate extreme degrees of the above mentioned
random digraph model. A focusing result (Theorem 1) turns out that
the maximum out-/in-degrees of random faulty scaled sector graph are
almost determined under some natural assumptions of fault
probabilities (see below for details). A similar focusing phenomenon
has been discovered in classical Erd\"os-R\'enyi random graph
theory, see e.g.\cite{12}(chap.3). Theorem 1 adds to the asymptotic
bound of maximum degree of \cite{1} in the thermodynamic and
sub-connective regimes and by including Poisson point process. It
extends the maximum degree focusing result in \cite{6}(chap.6) by
considering digraphs and including faulty probabilities. It also in
passing gives an answer to the open problems suggested in \cite{2}.

Poisson approximation by Stein-Chen method is used here as in
\cite{7}, where Penrose incorporated the geometric clique number
with scan statistic via a ``clustering rule'' giving a concentration
result. The idea behind can be traced back to \cite{9}, for
instance. We mention that if the clustering rule $h$ is properly
chosen, the geometric maximum degree is also contained in that
framework. In fact, let $h(\mathcal{X})=0$ if $\mathcal{X}$ is not
contained in some ball $B(x,r)$ and otherwise let $h(\mathcal{X})$
be equal to the maximum degree$+1$ of the geometric subgraph induced
by $\mathcal{X}$. This $h$ can be seen suffice the requirements in
\cite{7}.

Compared with the more recent random scaled sector graphs, random
geometric graph models are widely-studied (see \cite{6} and
references therein), in which connections are isotropic and thus
undirected. some properties such as connectivity and layout problems
of geometric graphs including faulty probability have been
addressed, see e.g.\cite{2,16,3,8}. Algorithmic aspects are also
dealt with in various contexts, see e.g.\cite{4,13}. From a
percolation analysis point of view, some concerned results coherent
with ours can be found in \cite{5}.

\bigskip
\noindent{\Large\textbf{2. Statement of main results}}

\smallskip
Given a sequence $\mathcal{X}_n=\{X_1,X_2,\cdots,X_n\}$ of
independently and uniformly distributed $(i.u.d.)$ random points in
$[0,1]^2$ with common density function $f=1_{[0,1]^2}$. We equip
$\mathbb{R}^2$ with Euclidean norm and let $\theta$ be the area of
unit disk. Notice that $\theta=\piup$ in this context, and we may
sometimes retain the sign $\theta$ for readability. Let
$\mathcal{Y}_n=\{Y_1,Y_2,\cdots,Y_n\}$ be $i.u.d.$ random variables
taking values in $[0,2\piup)$ and $\alpha\in(0,2\piup]$ be fixed.
Associate every point $X_i\in \mathcal{X}_n$ a sector, which is
centered at $X_i$, with radius $r_n$, central angle $\alpha$ and
elevation $Y_i$ with respect to the horizontal direction
anticlockwise. This sector is denoted as $S(X_i,Y_i,r_n)$. We denote
by $G_\alpha(\mathcal{X}_n,\mathcal{Y}_n,r_n)$ the digraph with
vertex set $\mathcal{X}_n$, and with arc $(X_i,X_j)$, $i\not=j$,
presents if and only if $X_j\in S(X_i,Y_i,r_n)$. The usual Poisson
version $G(\mathcal{P}_n,\mathcal{Y}_{N_n},r_n)$ is defined
similarly, where $\mathcal{P}_n=\{X_1,X_2,\cdots,X_{N_n}\}$,
$N_n\sim Poi(n)$. Now for every $X_i$, we assign a failure
probability $v_n$, independent of other node and the point process.
Given the presence of $X_i$, we associate every out-edge with a
failure probability $q_n$ independently. This random faulty scaled
sector graph is thus denoted by
$G_{\alpha}(\mathcal{X}_n,\mathcal{Y}_n,v_n,q_n,r_n)$. Likewise, we
can define the Poisson version
$G_{\alpha}(\mathcal{P}_n,\mathcal{Y}_{N_n},v_n,q_n,r_n)$. Set
$\Delta_n^{out}$, $\Delta_n^{in}$ be the maximum out-/in-degree of
$G_{\alpha}(\mathcal{X}_n,\mathcal{Y}_n,v_n,q_n,r_n)$ respectively,
and $\Delta_n^{'out}$, $\Delta_n^{'in}$ be the maximum
out-/in-degree of
$G_{\alpha}(\mathcal{P}_n,\mathcal{Y}_{N_n},v_n,q_n,r_n)$
respectively. Here comes the main result:

\medskip
\noindent\textbf{Theorem 1.}\itshape \quad Suppose $v_n\rightarrow
v\in[0,1)$ and $q_n\rightarrow q\in[0,1)$, as $n\rightarrow\infty$.
Suppose $\mu_n:=\frac{\alpha}{2}nr_n^2(1-v_n)(1-q_n)$, and that
$\inf_{n>0}\mu_n>0$, and that $\mu_n^{1+\varepsilon}=o(\ln n)$ for
some $\varepsilon>0$. Then there exists a sequence
$\{k_n\}_{n\ge1}$, set $\xi_n=P(Poi(\mu_n)\ge k_n)$, such that we
have\\
$$P(\Delta_n^{'out}=k_n-1)-e^{-n(1-v_n)\xi_n}\rightarrow0,$$
$$P(\Delta_n^{'out}=k_n)-e^{-n(1-v_n)\xi_n}\rightarrow1,$$ as
$n\rightarrow\infty$. The same thing holds for $\Delta_n^{'in}$,
$\Delta_n^{out}$ and $\Delta_n^{in}$.\normalfont

\bigskip
\noindent{\Large\textbf{3. Proofs}}

\smallskip
To prove the asymptotic focusing phenomenon, we first give a general
non-asymptotic Poisson approximation lemma, which may be useful in
some other cases. Let $W_{j,n}^{out}(r)$, $W_{j,n}^{in}(r)$ be the
number of vertices of out-/in-degree $j$ in
$G_{\alpha}(\mathcal{X}_n,\mathcal{Y}_n,v,q,r)$ respectively. Let
$W_{j,\lambda}^{'out}(r)$, $W_{j,\lambda}^{'in}(r)$ be the number of
vertices of out-/in-degree $j$ in
$G_{\alpha}(\mathcal{P}_{\lambda},\mathcal{Y}_{N_{\lambda}},v,q,r)$
respectively. For $A\subseteq\mathbb{N}\cup\{0\}$, set
$W_{A,\lambda}^{'out}(r):=\sum_{j\in A}W_{j,\lambda}^{'out}(r)$ and
$W_{A,\lambda}^{'in}(r):=\sum_{j\in A}W_{j,\lambda}^{'in}(r)$. The
total variation distance between the laws of integer valued random
variables $X$, $Y$ is defined by
$$d_{TV}(X,Y)=\sup_{A\subseteq\mathbb{Z}}\{|P(X\in A)-P(Y\in A)|\}.$$
Let $c$, $c'$ be various positive constants, the values may change
from line to line.

\medskip
\noindent\textbf{Lemma 1.}\itshape \quad Suppose a density function
$g$ (not necessarily uniform) is continuous a.e.. Let $0\le v, q<1$,
$r, \lambda>0$ and $A\subseteq\mathbb{N}\cup\{0\}$. Then,
$$d_{TV}(W_{A,\lambda}^{'out}, Poi(EW_{A,\lambda}^{'out}))\le \min\Big(1,\frac{1}{EW_{A,\lambda}^{'out}}\Big)(I_1^{out}+I_2^{out})$$
where,
\begin{multline}
I_1^{out}:=\frac{(1-v)^2\lambda^2}{4\piup^2}\int_{\mathbb{R}^2}\int_0^{2\piup}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_1,y_1,r)\in
A)]\mathrm{d}y_1\\
\int_{B(x_1,3r)}\int_0^{2\piup}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_2,y_2,r)\in
A)]\mathrm{d}y_2g(x_2)\mathrm{d}x_2g(x_1)\mathrm{d}x_1,\nonumber
\end{multline}
\begin{multline}
I_2^{out}:=\frac{(1-v)^2\lambda^2}{4\piup^2}\\
\cdot\int_{\mathbb{R}^2}\int_{B(x_1,3r)}\int_0^{2\piup}\int_0^{2\piup}P\big[\{\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_1,y_1,r))
+1_{[x_2\in S(x_1,y_1,r)]\cap[(x_1,x_2)\ not\ fails]}\in A\}\\
\cap
\{\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_2,y_2,r))+1_{[x_1\in
S(x_2,y_2,r)]\cap[(x_2,x_1)\ not\ fails]}\in
A\}\big]\\
\mathrm{d}y_1\mathrm{d}y_2g(x_2)\mathrm{d}x_2g(x_1)\mathrm{d}x_1\nonumber
\end{multline}
Likewise,
$$d_{TV}(W_{A,\lambda}^{'in}, Poi(EW_{A,\lambda}^{'in}))\le \min\Big(1,\frac{1}{EW_{A,\lambda}^{'in}}\Big)(I_1^{in}+I_2^{in})$$
where,
\begin{multline}
I_1^{in}:=(1-v)^2\lambda^2\int_{\mathbb{R}^2}P[\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_1,r))\in A]\\
\int_{B(x_1,3r)}P[\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_2,r))\in
A]g(x_2)\mathrm{d}x_2g(x_1)\mathrm{d}x_1,\nonumber
\end{multline}
\begin{multline}
I_2^{in}:=(1-v)^2\lambda^2\\
\cdot\int_{\mathbb{R}^2}\int_{B(x_1,3r)}P\big[\{\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_1,r))+1_{[x_1\in
S(x_2,y_2,r)]\cap[(x_2,x_1)\ not\ fails]}\in
A\}\\
\cap\{\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_2,r))+1_{[x_2\in
S(x_1,y_1,r)]\cap[(x_1,x_2)\ not\ fails]}\in
A\}\big]g(x_2)\mathrm{d}x_2g(x_1)\mathrm{d}x_1 \nonumber
\end{multline}
and denote by $\mathcal{\tilde{P}}_{\lambda}$ with intensity
$(\lambda\alpha/2\piup)g$ the thinning of $\mathcal{P}_{\lambda}$,
whose intensity is $\lambda g$. \normalfont

\medskip
\noindent\textbf{Proof}. Given $m\in\mathbb{N}$, partition
$\mathbb{R}^2$ into squares of side $2^{-m}$, with the origin lies
at a square corner. Label these squares as $D_{m,1},D_{m,2},\cdots$,
and denote the center of $D_{m,i}$ as $a_{m,i}$. For each
$x\in\mathbb{R}^2$ and for each $m$, $i$, define $Y_x$, $Y_{m,i}$ as
independent copies of $Y_1$.

For out-degree, set
$$\xi_{m,i}:=1_{[\mathcal{P}_{\lambda(1-v)}(D_{m,i})=1]\cap[\mathcal{P}_{\lambda(1-q)(1-v)}(S(a_{m,i},Y_{m,i},r)\backslash D_{m,i})\in A]}$$
Set $p_{m,i}:=E\xi_{m,i}$, $p_{m,i,j}:=E[\xi_{m,i}\xi_{m,j}]$.
Define an adjacency relation $\sim_m$ on $\mathbb{N}$ by putting
$i\sim_m j$ if and only if $0<||a_{m,i}-a_{m,j}||\le 3r$, and define
the corresponding adjacency neighborhood
$\mathcal{N}_{m,i}:=\{j\in\mathbb{N}|\ ||a_{m,i}-a_{m,j}||\le 3r\}$.
Let $Q_n:=[-n,n]^2$ and
$\mathcal{I}_{m,n}:=\{i\in\mathbb{N}|D_{m,i}\subseteq Q_n\}$. Set
$\mathcal{N}_{m,n,i}:=\mathcal{N}_{m,i}\cap\mathcal{I}_{m,n}$. Thus
$(\mathcal{I}_{m,n},\sim_m)$ is a dependency graph for random
variables $\xi_{m,i}$, $i\in\mathcal{I}_{m,n}$.

Define
$\tilde{W}_{m,n}^{out}:=\sum_{i\in\mathcal{I}_{m,n}}\xi_{m,i}$, then
we observe that
$W_{A,\lambda}^{'out}=\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}\tilde{W}_{m,n}^{out}$.
By theorem 1 of \cite{14},
\begin{equation}
d_{TV}(\tilde{W}_{m,n}^{out},Poi(E\tilde{W}_{m,n}^{out}))\le
\min\Big(1,\frac{1}{E\tilde{W}_{m,n}^{out}}\Big)(a_1(m,n)+a_2(m,n))\label{1}
\end{equation}
where
$$a_1(m,n):=\sum_{i\in\mathcal{I}_{m,n}}\sum_{j\in\mathcal{N}_{m,n,i}}p_{m,i}p_{m,j},\qquad a_2(m,n):=\sum_{i\in\mathcal{I}_{m,n}}\sum_{j\in\mathcal{N}_{m,n,i}\backslash\{i\}}p_{m,i,j}.$$
Define $w_m(x):=2^{2m}p_{m,i}1_{[x\in D_{m,i}]}$, wherefore
$\int_{Q_n}w_m(x)\mathrm{d}x=\sum_{i\in\mathcal{I}_{m,n}}p_{m,i}$.
If $f$ is continuous at $x$, we have
$\lim_{m\rightarrow\infty}w_m(x)=(1-v)\lambda
g(x)P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x,Y_x,r))\in A]$, by mean
value theorem. Observe that $w_m(x)\le
2^{2m}E\mathcal{P}_{\lambda(1-v)}(D_{m,i})\le \lambda g_{\max}$, so
by the dominated convergence theorem, we obtain
$$\lim_{m\rightarrow\infty}E\tilde{W}_{m,n}^{out}=(1-v)\lambda\int_{Q_n}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x,Y_x,r))\in A]g(x)\mathrm{d}x$$
and by Fubini theorem and Palm theory for (marked) Poisson point
process \cite{6}(Sect. 1.7), we have
\begin{eqnarray*}
\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}E\tilde{W}_{m,n}^{out}&\hspace{-7pt}=&\hspace{-7pt}(1-v)\lambda\int_{\mathbb{R}^2}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x,Y_x,r))\in
A]g(x)\mathrm{d}x\\
&\hspace{-7pt}=&\hspace{-7pt}\frac{(1-v)\lambda}{2\piup}\int_{\mathbb{R}^2}\int_0^{2\piup}P[Poi\big(\int_{S(x,y,r)}\lambda(1-v)(1-q)g(z)\mathrm{d}z\big)\in
A]g(x)\mathrm{d}y\mathrm{d}x\\
&\hspace{-7pt}=&\hspace{-7pt}EW_{A,\lambda}^{'out}
\end{eqnarray*}
For $x_1\in D_{m,i}$, $x_2\in D_{m,j}$, define
$u_m(x_1,x_2):=2^{4m}p_{m,i}p_{m,j}1_{[j\in \mathcal{N}_{m,i}]}$ and
$v_m(x_1,x_2):=2^{4m}p_{m,i,j}1_{[j\in
\mathcal{N}_{m,i}\backslash\{i\}]}$. Therefore, we have
$a_1(m,n)=\int_{Q_n}\int_{Q_n}u_m(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2$
and
$a_2(m,n)\\=\int_{Q_n}\int_{Q_n}v_m(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2$.
For different continuous points $x_1$, $x_2$ of $g$, if also
$||x_1-x_2||\not=r$ and $||x_1-x_2||\not=3r$, then
\begin{multline}
\lim_{m\rightarrow\infty}u_m(x_1,x_2)=\frac{(1-v)^2\lambda^2}{4\piup^2}g(x_1)g(x_2)\int_0^{2\piup}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_1,y_1,r)\in
A)]\mathrm{d}y_1\\
\cdot\int_0^{2\piup}P[\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_2,y_2,r)\in
A)]\mathrm{d}y_2\cdot1_{[B(x_1,3r)]}(x_2)\nonumber
\end{multline}
Similarly,
\begin{multline}
\lim_{m\rightarrow\infty}v_m(x_1,x_2)=\frac{(1-v)^2\lambda^2}{4\piup^2}g(x_1)g(x_2)\\
\cdot\int_0^{2\piup}\int_0^{2\piup}P\big[\{\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_1,y_1,r))
+1_{[x_2\in S(x_1,y_1,r)]\cap[(x_1,x_2)\ not\ fails]}\in A\}\\\cap
\{\mathcal{P}_{\lambda(1-q)(1-v)}(S(x_2,y_2,r))+1_{[x_1\in
S(x_2,y_2,r)]\cap[(x_2,x_1)\ not\ fails]}\in
A\}\big]\mathrm{d}y_1\mathrm{d}y_2\cdot1_{[B(x_1,3r)]}(x_2)\nonumber
\end{multline}
For $x_1\in D_{m,i}$, $x_2\in D_{m,j}$, we have
$$u_m(x_1,x_2)\le
2^{4m}E\mathcal{P}_{\lambda(1-v)}(D_{m,i})E\mathcal{P}_{\lambda(1-v)}(D_{m,j})\le
g_{\max}^2$$ and $$v_m(x_1,x_2)\le
2^{4m}E\mathcal{P}_{\lambda(1-v)}(D_{m,i})E\mathcal{P}_{\lambda(1-v)}(D_{m,j})1_{[i\not=j]}\le
g_{\max}^2.$$ Whence by the dominated convergence theorem we have
$\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}a_1(m,n)=I_1^{out}$
and
$\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}a_2(m,n)=I_2^{out}$.
Hence, the out-degree case is proved by taking limit in both sides
of (\ref{1}).

For in-degree, set
$$\eta_{m,i}:=1_{[\mathcal{P}_{\lambda(1-v)}(D_{m,i})=1]\cap[\#\{x'\in\mathcal{P}_{\lambda(1-q)(1-v)}|D_{m,i}\subseteq S(x',Y_{x'},r)\}\in A]}$$
Set $q_{m,i}:=E\eta_{m,i}$, $q_{m,i,j}:=E[\eta_{m,i}\eta_{m,j}]$.
Define the dependency graph for random variables $\eta_{m,i}$,
$i\in\mathcal{I}_{m,n}$ just as above.

Define
$\tilde{W}_{m,n}^{in}:=\sum_{i\in\mathcal{I}_{m,n}}\eta_{m,i}$, then
we observe that
$W_{A,\lambda}^{'in}=\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}\tilde{W}_{m,n}^{in}$.
By theorem 1 of \cite{14},
\begin{equation}
d_{TV}(\tilde{W}_{m,n}^{in},Poi(E\tilde{W}_{m,n}^{in}))\le
\min\Big(1,\frac{1}{E\tilde{W}_{m,n}^{in}}\Big)(b_1(m,n)+b_2(m,n))\label{2}
\end{equation}
where
$$b_1(m,n):=\sum_{i\in\mathcal{I}_{m,n}}\sum_{j\in\mathcal{N}_{m,n,i}}q_{m,i}q_{m,j},\qquad b_2(m,n):=\sum_{i\in\mathcal{I}_{m,n}}\sum_{j\in\mathcal{N}_{m,n,i}\backslash\{i\}}q_{m,i,j}.$$
Reset $w_m(x):=2^{2m}q_{m,i}1_{[x\in D_{m,i}]}$, then
$\int_{Q_n}w_m(x)\mathrm{d}x=\sum_{i\in\mathcal{I}_{m,n}}q_{m,i}$.
If $f$ is continuous at $x$, we have
$\lim_{m\rightarrow\infty}w_m(x)=(1-v)\lambda
g(x)P[\#\{x'\in\mathcal{P}_{\lambda(1-q)(1-v)}|x\in
S(x',Y_{x'},r)\}\in A]$, by mean value theorem for integrals.
Observe that $w_m(x)\le
2^{2m}E\mathcal{P}_{\lambda(1-v)}(D_{m,i})\le \lambda g_{\max}$, so
by the dominated convergence theorem, we obtain
$$\lim_{m\rightarrow\infty}E\tilde{W}_{m,n}^{in}=(1-v)\lambda\int_{Q_n}P[\#\{x'\in\mathcal{P}_{\lambda(1-q)(1-v)}|x\in
S(x',Y_{x'},r)\}\in A]g(x)\mathrm{d}x$$ and
\begin{eqnarray*}
\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}E\tilde{W}_{m,n}^{in}&\hspace{-7pt}=&\hspace{-7pt}(1-v)\lambda\int_{\mathbb{R}^2}P[Poi\big(\frac{\lambda\alpha(1-q)(1-v)}{2\piup}\int_{B(x,r)}g(z)\mathrm{d}z\big)\in A]g(x)\mathrm{d}x\\
&\hspace{-7pt}=&\hspace{-7pt}EW_{A,\lambda}^{'in}
\end{eqnarray*}
For $x_1\in D_{m,i}$, $x_2\in D_{m,j}$, reset
$u_m(x_1,x_2):=2^{4m}q_{m,i}q_{m,j}1_{[j\in \mathcal{N}_{m,i}]}$ and
$v_m(x_1,x_2):=2^{4m}q_{m,i,j}1_{[j\in
\mathcal{N}_{m,i}\backslash\{i\}]}$. Therefore, we have
$b_1(m,n)=\int_{Q_n}\int_{Q_n}u_m(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2$
and
$b_2(m,n)\\=\int_{Q_n}\int_{Q_n}v_m(x_1,x_2)\mathrm{d}x_1\mathrm{d}x_2$.
For different continuous points $x_1$, $x_2$ of $g$, if also
$||x_1-x_2||\not=r$ and $||x_1-x_2||\not=3r$, then
\begin{multline}
\lim_{m\rightarrow\infty}u_m(x_1,x_2)=(1-v)^2\lambda^2
g(x_1)g(x_2)P[\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_1,r)\in
A)]\\
\cdot P[\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_2,r)\in
A)]1_{[B(x_1,3r)]}(x_2)\nonumber
\end{multline}
Similarly,
\begin{multline}
\lim_{m\rightarrow\infty}v_m(x_1,x_2)=(1-v)^2\lambda^2g(x_1)g(x_2)\\
\cdot P[\{\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_1,r))
+1_{[x_1\in S(x_2,Y_2,r)]\cap[(x_2,x_1)\ not\ fails]}\in A\}\\\cap
\{\mathcal{\tilde{P}}_{\lambda(1-q)(1-v)}(B(x_2,r))+1_{[x_2\in
S(x_1,Y_1,r)]\cap[(x_1,x_2)\ not\ fails]}\in A\}]\cdot
1_{[B(x_1,3r)]}(x_2)\nonumber
\end{multline}
By similar arguments in out-degree case, we have
$\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}b_1(m,n)=I_1^{in}$
and
$\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}b_2(m,n)=I_2^{in}$.
Hence, we conclude the proof by taking limit in both sides of
(\ref{2}). $\Box$
\medskip

For cleanness of the expressions, we will shift our battlefield from
$[0,1]^2$ to $[-1/2,1/2]^2$. From now on, we take $f=1_Q$,
$Q:=[-1/2,1/2]^2$ through this paper. Let $\mathcal{H}_{\lambda}$ be
the homogeneous Poisson point process with intensity $\lambda$ on
$\mathbb{R}^2$ and $|\cdot|$ be Lebesgue measure.

\medskip
\noindent\textbf{Proposition 1.}\itshape \quad Let
$\mu_n:=\frac{\alpha}{2}nr_n^2(1-v_n)(1-q_n)$, and suppose
$\lim_{n\rightarrow\infty}r_n=0$ and $\inf_{n>0}\mu_n>0$. Suppose
$\{j_n\}_{n\ge 1}$ is an $\mathbb{N}-$valued sequence such that for
some $\varepsilon>0$,
\begin{equation}
\lim_{n\rightarrow\infty}j_n/\mu_n^{1+\varepsilon}=\infty \label{3}
\end{equation}
Set $\xi_n:=P(Poi(\mu_n)\ge j_n)$. Then
$$\lim_{n\rightarrow\infty}[P(\Delta_n^{'out}<j_n)-e^{-n(1-v_n)\xi_n}]=0$$
and
$$\lim_{n\rightarrow\infty}[P(\Delta_n^{'in}<j_n)-e^{-n(1-v_n)\xi_n}]=0.$$\normalfont

\medskip
\noindent\textbf{Proof}. For out-degree, set
$$W_n^{'out}:=\sum_{i=1}^{N_{n(1-v_n)}}1_{[\mathcal{P}_{n(1-v_n)(1-q_n)}(S(X_i,Y_i,r_n))\ge j_n+1]\cap[X_i\in
Q]}$$ Then by Palm theory for (marked) Poisson process,
$$EW_n^{'out}\sim (1-v_n)n\int_QP[Poi\big(\frac{\theta\alpha}{2\piup}nr_n^2(1-v_n)(1-q_n)\big)\ge j_n]\mathrm{d}x=(1-v_n)n\xi_n,$$
as $n\rightarrow\infty$. Now take $A=\mathbb{Z}\cap[j_n,\infty)$,
$\lambda=n$, $r=r_n$, $v=v_n$, $q=q_n$, $g=f$ in Lemma 1, we then
obtain
\begin{equation}
\big|P(W_n^{'out}=0)-e^{-EW_n^{'out}}\big|\le
\min\Big(1,\frac{2}{(1-v_n)n\xi_n}\Big)(I_{1,n}^{out}+I_{2,n}^{out}),\label{4}
\end{equation}
for large enough $n$. We have
\begin{eqnarray*}
I_{1,n}^{out}&=&\frac{(1-v_n)^2n^2}{4\piup^2}\int_{\mathbb{R}^2}\int_0^{2\piup}P[\mathcal{P}_{n(1-q_n)(1-v_n)}(S(x_1,y_1,r_n))\ge
j_n]\mathrm{d}y_1\\
&
&\int_{B(x_1,3r_n)}\int_0^{2\piup}P[\mathcal{P}_{n(1-q_n)(1-v_n)}(S(x_2,y_2,r_n))\ge
j_n]\mathrm{d}y_2f(x_2)\mathrm{d}x_2f(x_1)\mathrm{d}x_1\\
&\le&n^2\xi_n^2\theta(3r_n)^2
\end{eqnarray*}
Therefore, by (\ref{3}), $((1-v_n)n\xi_n)^{-1}I_{1,n}^{out}\le
c\mu_n\xi_n\rightarrow0.$ On the other hand, we have,
\begin{eqnarray*}
I_{2,n}^{out}&=&\frac{(1-v_n)^2n^2}{4\piup^2}\int_Q\int_{Q\cap
B(x_1,3r_n)}\int_0^{2\piup}\int_0^{2\piup}P\big[\{\mathcal{P}_{n(1-q_n)(1-v_n)}(S(x_1,y_1,r_n))\\
& &+1_{[x_2\in S(x_1,y_1,r_n)]\cap[(x_1,x_2)\ not\ fails]}\ge
j_n\}\cap\{\mathcal{P}_{n(1-q_n)(1-v_n)}(S(x_2,y_2,r_n))\\
& &+1_{[x_1\in S(x_2,y_2,r_n)]\cap[(x_2,x_1)\ not\ fails]}\ge
j_n\}\big]\mathrm{d}y_1\mathrm{d}y_2\mathrm{d}x_2\mathrm{d}x_1\\
&\le&\frac{(1-v_n)^2n^2}{4\piup^2}\int_Q\int_{Q\cap
B(0,3r_n)}\int_0^{2\piup}\int_0^{2\piup}P\big[\{\mathcal{P}_{n(1-q_n)(1-v_n)}(S(0,y_1,r_n))\ge
j_n-1\}\\
& &\cap\{\mathcal{P}_{n(1-q_n)(1-v_n)}(S(x_2-x_1,y_2,r_n))\ge
j_n-1\}\big]\mathrm{d}y_1\mathrm{d}y_2\mathrm{d}x_2\mathrm{d}x_1\\
&\le&\frac{(1-v_n)^2n^2}{4\piup^2}\int_{B(0,3)}\int_0^{2\piup}\int_0^{2\piup}h_n(z,y_1,y_2)\mathrm{d}y_1\mathrm{d}y_2\mathrm{d}z
\end{eqnarray*}
where,
\begin{multline}
h_n(z,y_1,y_2):=P\big[\{\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))\ge
j_n-1\}\\
\cap\{\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(z,y_2,1))\ge
j_n-1\}\big].\nonumber
\end{multline}
By (\ref{3}), we choose $M\in\mathbb{N}$, such that
$j_n^M/\mu_n^{M+1}\rightarrow\infty$, as $n\rightarrow\infty$. Then
we have
$$P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))\ge j_n+M]\le \xi_n\Big(\frac{\mu_n}{j_n}\Big)^M$$
and
$$P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))\in\{j_n-1,j_n,\cdots,j_n+M-1\}]\le2\xi_n\frac{j_n}{\mu_n}$$
since
$P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))=j_n-1]\le\xi_n(j_n/\mu_n)$
and $P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1\\,1))\ge
j_n]\le\xi_n(j_n/\mu_n)$ when $n$ is large enough.

Let $\eta_z:=2\piup|S(0,y_1,1)\backslash S(z,y_2,1)|/\theta\alpha$,
then we see that the conditional distribution of
$\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(z,y_2,1))$, given that
$\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))=j_n+M$ is the sum of
two independent random variables $j_n+M-U$ and $V$, where $U\sim
Bin(j_n+M,\eta_z)$ and $V\sim Poi(\theta\alpha
nr_n^2\eta_z(1-q_n)(1-v_n)/2\piup)$. Provided $n$ is large enough so
that $M+1<j_n\eta_z/5$, if $U>3j_n\eta_z/5$ and $V<j_n\eta_z/5$ then
$j_n+M-U+V<j_n-1$. Now, note that
$P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(z,y_2,1))\ge
j_n-1|\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))=k]$ is an
increasing function of $k$, and by Chernoff bounds, there exists a
constant $\beta>0$, for any $z\in B(0,3)$ and $n$ large enough, if
$\eta_z>5(M+1)/j_n$ then
\begin{eqnarray*}
\lefteqn{\max_{j_n-1\le k\le
j_n+M-1}P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(z,y_2,1))\ge
j_n-1|\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))=k]}\\
& &\le P[Bin(j_n+M,\eta_z)\le 3j_n\eta_z/5]+P[Poi(\theta\alpha
nr_n^2\eta_z(1-q_n)(1-v_n)/2\piup)\ge j_n\eta_z/5]\\
& &\le2e^{-\beta j_n\eta_z},
\end{eqnarray*}
whereas if $\eta_z\le 5(M+1)/j_n$, then $e^{5\beta(M+1)}e^{-\beta
j_n\eta_z}\ge 1$. Take $c_1=2\vee e^{5\beta(M+1)}$, for any $z\in
B(0,3)$, we have
$$
\max_{j_n-1\le k\le
j_n+M-1}P[\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(z,y_2,1))\ge
j_n-1|\mathcal{H}_{nr_n^2(1-q_n)(1-v_n)}(S(0,y_1,1))=k]$$
$$\le c_1e^{-\beta j_n\eta_z}.$$
Therefore the above discussion gives
$$h_n(z,y_1,y_2)\le \xi_n\Big(\frac{\mu_n}{j_n}\Big)^M+2c_1\xi_n\frac{j_n}{\mu_n}e^{-\beta j_n\eta_z}$$
for $n$ large enough. Now since $\inf_{z\in
B(0,3)}\{\eta_z/||z||\}>0$, there exists a constant $\gamma>0$ such
that
\begin{eqnarray*}
\int_0^{2\piup}\int_0^{2\piup}\int_{B(0,3)}h_n(z,y_1,y_2)-
\xi_n\Big(\frac{\mu_n}{j_n}\Big)^M\mathrm{d}z\mathrm{d}y_1\mathrm{d}y_2
&\le&8\piup^2c_1\xi_n\big(\frac{j_n}{\mu_n}\big)\int_{B(0,3)}e^{-\gamma j_n||z||}\mathrm{d}z\\
&\le&c'\xi_n\big(\frac{j_n}{\mu_n}\big)\Gamma(2)/(\gamma j_n)^2.
\end{eqnarray*}
Accordingly, by the choice of $M$,
$$((1-v_n)n\xi_n)^{-1}I_{2,n}^{out}\le c((1-v_n)n\xi_n)^{-1}\cdot\mu_nn\xi_n\Big[\big(\frac{\mu_n}{j_n}\big)^M+\frac1{\mu_nj_n}\Big]\rightarrow0,$$
as $n\rightarrow\infty$. The out-degree case hereby follows from
(\ref{4}).

For in-degree, set
$$W_n^{'in}:=\sum_{i=1}^{N_{n(1-v_n)}}1_{[\#\{X_j\in\mathcal{P}_{n(1-v_n)(1-q_n)}|X_i\in S(X_j,Y_j,r_n)\}\ge j_n+1]\cap[X_i\in
Q]}$$ Then by Palm theory for Poisson process,
$$EW_n^{'in}\sim (1-v_n)n\int_QP[Poi\big(\frac{\theta\alpha}{2\piup}nr_n^2(1-v_n)(1-q_n)\big)\ge j_n]\mathrm{d}x=(1-v_n)n\xi_n,$$
as $n\rightarrow\infty$. By Lemma 1, we thereby obtain
\begin{equation}
\big|P(W_n^{'in}=0)-e^{-EW_n^{'in}}\big|\le
\min\Big(1,\frac{2}{(1-v_n)n\xi_n}\Big)(I_{1,n}^{in}+I_{2,n}^{in}),\label{5}
\end{equation}
for large enough $n$. We have
\begin{eqnarray*}
I_{1,n}^{in}&=&(1-v_n)^2n^2\int_{\mathbb{R}^2}P[\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(x_1,r_n))\ge
j_n]\\
&
&\int_{B(x_1,3r_n)}P[\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(x_2,r_n))\ge
j_n]f(x_2)\mathrm{d}x_2f(x_1)\mathrm{d}x_1\\
&\le&n^2\xi_n^2\theta(3r_n)^2
\end{eqnarray*}
Thus, by (\ref{3}), $((1-v_n)n\xi_n)^{-1}I_{1,n}^{in}\le
c\mu_n\xi_n\rightarrow0.$ On the other hand, we have,
\begin{eqnarray*}
I_{2,n}^{in}&=&(1-v_n)^2n^2\int_Q\int_{Q\cap
B(x_1,3r_n)}P\big[\{\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(x_1,r_n))\\
& &+1_{[x_1\in S(x_2,Y_2,r_n)]\cap[(x_2,x_1)\ not\ fails]}\ge
j_n\}\cap\{\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(x_2,r_n))\\
& &+1_{[x_2\in S(x_1,Y_1,r_n)]\cap[(x_1,x_2)\ not\ fails]}\ge
j_n\}\big]\mathrm{d}x_2\mathrm{d}x_1\\
&\le&(1-v_n)^2n^2\int_Q\int_{Q\cap
B(0,3r_n)}P\big[\{\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(0,r_n))\ge
j_n-1\}\\
& &\cap\{\mathcal{\tilde{P}}_{n(1-q_n)(1-v_n)}(B(x_2-x_1,r_n))\ge
j_n-1\}\big]\mathrm{d}x_2\mathrm{d}x_1\\
&\le&(1-v_n)^2n^2\int_{B(0,3)}g_n(z)\mathrm{d}z
\end{eqnarray*}
where,
\begin{multline}
g_n(z):=P\big[\{\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))\ge
j_n-1\}\\
\cap\{\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(z,1))\ge
j_n-1\}\big].\nonumber
\end{multline}
Also by (\ref{3}), we choose $M$ as above. Then we have
$$P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))\ge j_n+M]\le \xi_n\Big(\frac{\mu_n}{j_n}\Big)^M$$
and
$$P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))\in\{j_n-1,j_n,\cdots,j_n+M-1\}]\le2\xi_n\frac{j_n}{\mu_n}$$
since
$P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))=j_n-1]\le\xi_n(j_n/\mu_n)$
and
$P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))\\\ge
j_n]\le\xi_n(j_n/\mu_n)$ when $n$ is large enough. Let
$\delta_z:=|B(0,1)\backslash B(z,1)|/\theta$, hereby the conditional
distribution of
$\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(z,1))$,
given that
$\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))=j_n+M$
is the sum of two independent random variables $j_n+M-U$ and $V$,
where $U\sim Bin(j_n+M,\delta_z)$ and $V\sim Poi(\theta\alpha
nr_n^2\delta_z(1-q_n)(1-v_n)/2\piup)$. Provided $n$ is large enough
so that $M+1<j_n\delta_z/5$, if $U>3j_n\delta_z/5$ and
$V<j_n\delta_z/5$ then $j_n+M-U+V<j_n-1$. Now, note that
$P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(z,1))\ge
j_n-1|\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))=k]$
is an increasing function of $k$, and by Chernoff bounds, there
exists a constant $\beta>0$, for any $z\in B(0,3)$ and $n$ large
enough, if $\delta_z>5(M+1)/j_n$ then
\begin{eqnarray*}
\lefteqn{\max_{j_n-1\le k\le
j_n+M-1}P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(z,1))\ge
j_n-1|\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))=k]}\\
& &\le P[Bin(j_n+M,\delta_z)\le 3j_n\delta_z/5]+P[Poi(\theta\alpha
nr_n^2\delta_z(1-q_n)(1-v_n)/2\piup)\ge j_n\delta_z/5]\\
& &\le2e^{-\beta j_n\delta_z},
\end{eqnarray*}
whereas if $\delta_z\le 5(M+1)/j_n$, then $e^{5\beta(M+1)}e^{-\beta
j_n\delta_z}\ge 1$. Take $c_2=2\vee e^{5\beta(M+1)}$, for any $z\in
B(0,3)$, we have
$$
\max_{j_n-1\le k\le
j_n+M-1}P[\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(z,1))\ge
j_n-1|\mathcal{H}_{\frac{\alpha}{2\piup}nr_n^2(1-q_n)(1-v_n)}(B(0,1))=k]$$
$$\le c_2e^{-\beta j_n\delta_z}.$$
Consequently the above discussion gives
$$g_n(z)\le \xi_n\Big(\frac{\mu_n}{j_n}\Big)^M+2c_2\xi_n\frac{j_n}{\mu_n}e^{-\beta j_n\delta_z}$$
for $n$ large enough. Now since $\inf_{z\in
B(0,3)}\{\delta_z/||z||\}>0$, there exists a constant $\gamma>0$
such that
\begin{eqnarray*}
\int_{B(0,3)}g_n(z)- \xi_n\Big(\frac{\mu_n}{j_n}\Big)^M\mathrm{d}z
&\le&2c_2\xi_n\big(\frac{j_n}{\mu_n}\big)\int_{B(0,3)}e^{-\gamma j_n||z||}\mathrm{d}z\\
&\le&c'\xi_n\big(\frac{j_n}{\mu_n}\big)\Gamma(2)/(\gamma j_n)^2.
\end{eqnarray*}
Thus argue as the out-degree case,
$((1-v_n)n\xi_n)^{-1}I_{2,n}^{in}$ tends to $0$, as
$n\rightarrow\infty$. We hereby complete the proof by using
(\ref{5}). $\Box$
\medskip

Now we extend Proposition 1 from $\mathcal{P}_n$ to $\mathcal{X}_n$.

\medskip
\noindent\textbf{Proposition 2.}\itshape \quad Let
$\mu_n:=\frac{\alpha}{2}nr_n^2(1-v_n)(1-q_n)$. Suppose
$\inf_{n>0}\mu_n>0$ and
$\lim_{n\rightarrow\infty}\frac{\mu_n}{n^{1/6}}\\=0$. Suppose
$\{j_n\}_{n\ge 1}$ is an $\mathbb{N}-$valued sequence such that for
some $\varepsilon>0$, (\ref{3}) holds. Set $\xi_n:=P(Poi(\mu_n)\ge
j_n)$. Then
$$\lim_{n\rightarrow\infty}[P(\Delta_n^{out}<j_n)-e^{-n(1-v_n)\xi_n}]=0$$
and
$$\lim_{n\rightarrow\infty}[P(\Delta_n^{in}<j_n)-e^{-n(1-v_n)\xi_n}]=0.$$\normalfont

\medskip
\noindent\textbf{Proof}. For out-degree, we first assume that
$j_n\ge n^{1/5}$. We have
$$P(\Delta_n^{out}\ge n^{1/5})\le nP(Bin(n-1,\frac{\mu_n}{n})\ge n^{1/5})\rightarrow0,$$
as $n\rightarrow\infty$, by Chernoff bounds. Accordingly,
$P(\Delta_n^{out}\ge j_n)$ and $-n(1-v_n)\xi_n$ tend to zero. The
result then follows.

From now on, we thereby assume $j_n<n^{1/5}$ for all $n$, without
loss of generality. Set $\lambda_n:=n+n^{3/4}$, and let
$\mathcal{P}_{\lambda_n}$ be the Poisson point process coupled to
$\mathcal{X}_n$ with intensity $\lambda_nf$. Denote by
$\Delta_n^{+,out}$ the maximum out-degree in
$G_{\alpha}(\mathcal{P}_{\lambda_n},\mathcal{Y}_{N_{\lambda_n}},v_n,q_n,r_n)$.
Set
$\mu_n^+:=\frac{\theta\alpha}{2\piup}\lambda_nr_n^2(1-v_n)(1-q_n)$
and $\xi_n^+:=P(Poi(\mu_n^+)\ge j_n)$. Using Proposition 1 we have
\begin{equation}
\lim_{n\rightarrow\infty}[P(\Delta_n^{+,out}<j_n)-e^{-\lambda_n(1-v_n)\xi_n^+}]=0.\label{6}
\end{equation}
Since $1\le (\mu_n^+/\mu_n)^{j_n}=(1+n^{-1/4})^{j_n}\rightarrow1$
and $0\le\mu_n^+-\mu_n=n^{-1/4}\mu_n\rightarrow0$, as
$n\rightarrow\infty$, we get
$$1\le\frac{\xi_n^+}{\xi_n}\le\frac{e^{-\mu_n^+}(\mu_n^+)^{j_n}[1+\frac{\mu_n^+}{j_n}+(\frac{\mu_n^+}{j_n})^2+\cdots]}{e^{-\mu_n}\mu_n^{j_n}}\rightarrow1.$$
Then by setting $a_n=n(1-v_n)\xi_n$ and
$b_n=\frac{\lambda_n\xi_n^+}{n\xi_n}-1$, we have $b_n>0$ and
$b_n\rightarrow 0$. Observe that
$1-e^{-a_nb_n}\le1-e^{-\sqrt{b_n}}\rightarrow0$, if
$a_n\le1/\sqrt{b_n}$, while $e^{-a_n}<e^{-1/\sqrt{b_n}}\rightarrow0$
if $a_n>1/\sqrt{b_n}$. Consequently,
$$e^{-n(1-v_n)\xi_n}-e^{-\lambda_n(1-v_n)\xi_n^+}=e^{-a_n}(1-e^{-a_nb_n})\rightarrow 0,$$
as $n\rightarrow\infty$. Combing this with (\ref{6}), we have
$$\lim_{n\rightarrow\infty}[P(\Delta_n^{+,out}<j_n)-e^{-n(1-v_n)\xi_n}]=0.$$
Since
$P(\Delta_n^{+,out}<j_n)-P(\Delta_n^{out}<j_n)=P(\Delta_n^{+,out}<j_n\le\Delta_n^{out})-P(\Delta_n^{out}<j_n\le\Delta_n^{+,out}|n\le
N_{\lambda_n}\le n+2n^{3/4})\cdot P(n\le N_{\lambda_n}\le
n+2n^{3/4})$, and $P(\Delta_n^{+,out}<j_n\le\Delta_n^{out})$ tends
to 0, $P(n\le N_{\lambda_n}\le n+2n^{3/4})$ tends to 1 by Chebyshev
inequality, as $n\rightarrow\infty$, to prove the result it suffices
to prove that
$$\lim_{n\rightarrow\infty}P(\Delta_n^{out}<j_n\le\Delta_n^{+,out}|n\le
N_{\lambda_n}\le n+2n^{3/4})=0.$$ Now suppose $\Delta_n^{+,out}\ge
j_n$ and $n\le N_{\lambda_n}\le n+2n^{3/4}$, then there exists a
point in $\mathcal{P}_{\lambda_n}$ of out-degree at least $j_n$ in
$G_{\alpha}(\mathcal{P}_{\lambda_n},\mathcal{Y}_{N_{\lambda_n}},v_n,q_n,r_n)$.
Therefore
$$P[j_n>\Delta_n^{out}|\Delta_n^{+,out}\ge j_n,\ n\le N_{\lambda_n}\le n+2n^{3/4}]\le(j_n+1)\frac{2n^{3/4}}{n}\rightarrow0.$$
The out-degree case thereby follows by multiplication formula of
probability.

For in-degree, the same argument may be applied. $\Box$

\medskip
\noindent\textbf{Proof of Theorem 1}. Let $\xi_n(j):=P(Poi(\mu_n)\ge
j)$, then for $n\in\mathbb{N}$, take $j_n$ satisfying
$n\xi_n(j_n-1)>(1-v_n)^{-1}\ge n\xi_n(j_n)$. Set
$$
k_n:=\left\{\begin{array}{ll} j_n-1&,i\!f\
(1-v_n)n\xi_n(j_n)\le\sqrt{\frac{\xi_n(j_n)}{\xi_n(j_n-1)}}\\
j_n&,otherwise
\end{array}
\right.
$$
Take $\eta>0$ satisfying $(1+\varepsilon)^{-1}=1-2\eta$. Let
$i_n:=\lfloor\mu_n(\ln n)^{\eta}\rfloor$, then $i_n/(\ln
n)^{1-\eta}\rightarrow0$, as $n\rightarrow\infty$. Hence, by
Stirling formula,
$$
(1-v_n)n\xi_n(i_n)\ge(1-v_n)ne^{-1/12i_n}\frac{1}{\sqrt{2\piup
i_n}}e^{-i_n\ln(i_n/\mu_n)}\ge
cni_n^{-1/2}e^{-i_n\ln(i_n/\mu_n)}\rightarrow\infty.
$$
Thereby, $j_n\ge i_n$ and $j_n/\mu_n\rightarrow\infty$, as
$n\rightarrow\infty$. Hence, $\xi_n(j_n)/\xi_n(j_n-1)$,
$\xi_n(j_n+1)/\xi_n(j_n)$ and $\xi_n(j_n-1)/\xi_n(j_n-2)$ all tend
to zero. By the definition of $k_n$,
$(1-v_n)n\xi_n(k_n+1)\rightarrow0$ and
$(1-v_n)n\xi_n(k_n-1)\rightarrow\infty$, as $n\rightarrow\infty$.
Consequently, by Proposition 1, we have
$$P(\Delta_n^{'out}<k_n+1)\rightarrow1,\quad P(\Delta_n^{'out}<k_n-1)\rightarrow0,\quad P(\Delta_n^{'out}<k_n)-e^{-(1-v_n)n\xi_n(k_n)}\rightarrow0,$$
and
$$P(\Delta_n^{'in}<k_n+1)\rightarrow1,\quad P(\Delta_n^{'in}<k_n-1)\rightarrow0,\quad P(\Delta_n^{'in}<k_n)-e^{-(1-v_n)n\xi_n(k_n)}\rightarrow0.$$
Also by Proposition 2, we have
$$P(\Delta_n^{out}<k_n+1)\rightarrow1,\quad P(\Delta_n^{out}<k_n-1)\rightarrow0,\quad P(\Delta_n^{out}<k_n)-e^{-(1-v_n)n\xi_n(k_n)}\rightarrow0,$$
and
$$P(\Delta_n^{in}<k_n+1)\rightarrow1,\quad P(\Delta_n^{in}<k_n-1)\rightarrow0,\quad P(\Delta_n^{in}<k_n)-e^{-(1-v_n)n\xi_n(k_n)}\rightarrow0.$$
The proof thus complete. $\Box$

\bigskip
\noindent{\Large\textbf{4. Open problems}}

\smallskip
A natural question would be to ask what happens for other limiting
regime of $\mu_n$. We believe if $\ln n\ll \mu_n\ll (\ln n)^2$ and
some regular conditions hold for faulty probabilities, then there
exist sequences $i_n$, $j_n$ such that for all $x$:
$$
P\Big(\frac{\Delta_n^{out/in}-i_n}{j_n}<x\Big)\rightarrow
e^{-e^{-x}}.
$$
Therefore the focusing results will hold no longer. Results from
extreme value theory suggest that might be the case. In
Erd\"os-R\'enyi random graph, a similar result holds \cite{15}.

Of course it would be of interest to consider the density function
other than the uniform one.

Note that our random faulty scaled sector graphs are still static
models, so what can be said about the behavior of maximum degrees of
a dynamic model? A direct and meaningful way to get a dynamic faulty
scaled sector graph is to give every point $X_i$ a random lifetime
$T_i$. Suppose that these lifetimes are independent random variables
with common distribution $F(t)=P(T_i\le t)$.

\bigskip

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Computing} 2(2003) pp. 143--154
\bibitem{2}J.\ D\'{\i}az, J.\ Petit, M.\ Serna, Faulty random
geometric networks. \textit{Parallel Processing Letters} 10(2001)
pp. 343--357
\bibitem{5}M.\ Franceschetti, L.\ Booth, M.\ Cook,
R.\ Meester, J.\ Bruck, Continuum percolation with unreliable and
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\bibitem{16}Y.\ Guo, J.\ McNair, Fault tolerant three dimensional environment monitoring using wireless sensor
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\bibitem{10}J.\ F.\ C.\ Kingman, \textit{Poisson Processes}. Oxford
University Press, Oxford, 1993
\bibitem{20}A.\ Kumar, D.\ Manjunath, A tutorial survey of topics in
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pp.645--681
\bibitem{4}E.\ Levy, G.\ Louchard, J.\ Petit, Distributed algorithms
to find Hamiltonian cycles in $G(n,p)$ random graphs.
\textit{Lecture Notes in Computer Science} 3405(2005) pp. 63--74
\bibitem{9}M.\ M{\aa}nsson, Poisson approximation in connection with
clustering of random points. \textit{Ann.\ Appl.\ Prob.} 9(1999) pp.
465--492
\bibitem{6}M.\ D.\ Penrose, \textit{Random Geometric Graphs}. Oxford
University Press, Oxford, 2003
\bibitem{7}M.\ D.\ Penrose, Focusing of the scan statistic and
geometric clique number. \textit{Advances in Applied Probability}
34(2002) pp. 739--753
\bibitem{3}E.\ Peserico, L.\ Rudolph, Robust network connectivity: When it's the big picture that
matters. \textit{Performance Evaluation Review} 34(2006) pp.
299--310
\bibitem{13}J.\ Petit, Hamiltonian cycles in faulty random geometric
networks. \textit{Proceedings in Informatics}, Canada, 12(2002) pp.
97--110
\bibitem{8}C.\ W.\ Yi, P.\ J.\ Wan, X.\ Y.\ Li, O.\ Frieder, Fault
tolerant sensor networks with Bernoulli nodes. \textit{IEEE Wireless
Communication and Networking Conference} New Orleans, Louisiana,
2003


\end{thebibliography}
\end{document}

0909.2933.extracted_bib
\begin{thebibliography}{10}
\bibitem{11}I.\ F.\ Akyildiz, W.\ Su, Y.\ Sankarasubramanian, E.\
Cayirci, Wireless sensor networks: a survey. \textit{Computer
Networks} 38(2002) pp. 393-422
\bibitem{14}R.\ Arratia, L.\ Goldstein, L.\ Gordon, Two moments
suffice for Poisson approximations: the Chen-stein method.
\textit{Ann.\ Prob.} 17(1989) pp. 9--25
\bibitem{12}B.\ Bollobas, \textit{Random Graphs}. Cambridge University
Press, 2001
\bibitem{15}B.\ Bollobas, The distribution of the maximum degree of
a random graph. \textit{Discrete Math.} 32(1980) pp. 201--203
\bibitem{1}J.\ D\'{\i}az, J.\ Petit, M.\ Serna, A random graph model
for optical networks of sensors. \textit{IEEE Transactions on Mobile
Computing} 2(2003) pp. 143--154
\bibitem{2}J.\ D\'{\i}az, J.\ Petit, M.\ Serna, Faulty random
geometric networks. \textit{Parallel Processing Letters} 10(2001)
pp. 343--357
\bibitem{5}M.\ Franceschetti, L.\ Booth, M.\ Cook,
R.\ Meester, J.\ Bruck, Continuum percolation with unreliable and
spread out connections. \textit{Journal of Statistical Physics}
118(2005) pp. 721--734
\bibitem{16}Y.\ Guo, J.\ McNair, Fault tolerant three dimensional environment monitoring using wireless sensor
networks. \textit{IEEE Military Communications Conference} 2007 art.
no. 4086363
\bibitem{10}J.\ F.\ C.\ Kingman, \textit{Poisson Processes}. Oxford
University Press, Oxford, 1993
\bibitem{20}A.\ Kumar, D.\ Manjunath, A tutorial survey of topics in
wireless networking:Part II. \textit{S\=adhan\=a}, vol.32 2007,
pp.645--681
\bibitem{4}E.\ Levy, G.\ Louchard, J.\ Petit, Distributed algorithms
to find Hamiltonian cycles in $G(n,p)$ random graphs.
\textit{Lecture Notes in Computer Science} 3405(2005) pp. 63--74
\bibitem{9}M.\ M{\aa}nsson, Poisson approximation in connection with
clustering of random points. \textit{Ann.\ Appl.\ Prob.} 9(1999) pp.
465--492
\bibitem{6}M.\ D.\ Penrose, \textit{Random Geometric Graphs}. Oxford
University Press, Oxford, 2003
\bibitem{7}M.\ D.\ Penrose, Focusing of the scan statistic and
geometric clique number. \textit{Advances in Applied Probability}
34(2002) pp. 739--753
\bibitem{3}E.\ Peserico, L.\ Rudolph, Robust network connectivity: When it's the big picture that
matters. \textit{Performance Evaluation Review} 34(2006) pp.
299--310
\bibitem{13}J.\ Petit, Hamiltonian cycles in faulty random geometric
networks. \textit{Proceedings in Informatics}, Canada, 12(2002) pp.
97--110
\bibitem{8}C.\ W.\ Yi, P.\ J.\ Wan, X.\ Y.\ Li, O.\ Frieder, Fault
tolerant sensor networks with Bernoulli nodes. \textit{IEEE Wireless
Communication and Networking Conference} New Orleans, Louisiana,
2003


\end{thebibliography}