Theoretical analysis of local search strategies to optimize network communication subject to preserving the total number of links

A variety of phenomena such as world wide web, social or business networks, interactions are modelled by various kinds of networks (such as the scale free or preferential attachment networks). However, due to the model‐specific requirements one may want to rewire the network to optimize the communication among the various nodes while not overloading the number of channels (i.e. preserving the number of edges). The purpose of this paper is to present a formal framework for this problem and to examine a family of local search strategies to cope with it.

This is mostly theoretical work. The authors use rigorous mathematical framework to set‐up the model and then we prove some interesting theorems about it which pertain to various local search algorithms that work by rerouting the network.

This paper proves that in cases when every pair of nodes is sampled with non‐zero probability then the algorithm is ergodic in the sense that it samples every possible network on the specified set of nodes and having a specified number of edges with nonzero probability. Incidentally, the ergodicity result led to the construction of a class of algorithms for sampling graphs with a specified number of edges over a specified set of nodes uniformly at random and opened some other challenging and important questions for future considerations.

The measure‐theoretic framework presented in the current paper is original and rather general. It allows one to obtain new points of view on the problem.