Find Your Place: Simple Distributed Algorithms for Community Detection

Given an underlying network, the "averaging dynamics" is the following distributed process: Initially, each node sets its local value to an element of {-1,1}, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its value to the average of its neighbors. We show that when the graph is organized into two equal-sized well-connected "communities" separated by a sparse cut, the temporal behavior of values of nodes in different communities are different and such a difference can be independently checked by each node. Depending on our assumptions on the underlying graph, this can either hold for all or for most nodes. Either way, this behavior allows nodes to locally identify (exactly or approximately) the structure of the two communities. We show that the conditions under which the above happens are satisfied with high probability in graphs sampled according to the stochastic block model "G_{2n,p,q}", provided that "(p-q)n >> \sqrt{(p+q) n \log n}" and "q" is large enough to guarantee that the graph is connected w.h.p. (i.e., "q = \Omega(\log n / n^2)"). In this case the nodes are able to perform approximate reconstruction of the community structure in "O(\log n)" time. With respect to an analog of the stochastic block model for regular graphs, our algorithm based on the averaging dynamics performs exact reconstruction in a logarithmic number of rounds for the full range of parameters for which exact reconstruction is known to be doable in polynomial time.