  From data centers to IoT devices to Internet-based applications, overlay
networks have become an important part of modern computing. Many of these
overlay networks operate in fragile environments where processes are
susceptible to faults which may perturb a node's state and the network
topology. Self-stabilizing overlay networks have been proposed as one way to
manage these faults, promising to build or restore a particular topology from
any initial configuration or after the occurrence of any transient fault. To
date there have been several self-stabilizing protocols designed for overlay
networks. These protocols, however, are either focused on a single specific
topology, or provide very inefficient solutions for a general set of overlay
networks.
  In this paper, we analyze an existing algorithm and show it can be used as a
general framework for building many other self-stabilizing overlay networks.
Our analysis for time and space complexity depends upon several properties of
the target topology itself, providing insight into how topology selection
impacts the complexity of convergence. We then demonstrate the application of
this framework by analyzing the complexity for several existing topologies.
Next, using insights gained from our analysis, we present a new topology
designed to provide efficient performance during convergence with the general
framework. Our process demonstrates how the implications of our analysis help
isolate the factors of interest to allow a network designer to select an
appropriate topology for the problem requirements.
