Distributed community detection in dynamic graphs

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied Planted Bisection Model  dyn-G(n,p,q)dyn-G(n,p,q) where the node set [n][n] of the network is partitioned into two unknown communities and, at every time step, each possible edge (u,v)(u,v) is active with probability p if both nodes belong to the same community, while it is active with probability q   (with q≪pq≪p) otherwise. We also consider a time-Markovian generalization of this model.We propose a distributed protocol based on the popular Label-Propagation   approach and prove that, when the ratio p/qp/q is larger than nbnb (for an arbitrarily small constant b>0b>0), the protocol finds the right “planted” partition in O(log⁡n)O(log⁡n) time even when the snapshots of the dynamic graph are sparse and disconnected (i.e., when p=Θ(1/n)p=Θ(1/n)).