Find Your Place: Simple Distributed Algorithms for Community Detection

Given an underlying network, the \emph{averaging dynamics} is the following distributed process: Initially, each node locally chooses a value in $\{-1,1\}$, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. Under various models of graphs with a sparse balanced cut, that include the stochastic block model, we show that the process resulting from this simple protocol produces in logarithmic time a clustering that exactly or approximately (depending on the model) reflects the underlying cut. We also prove that a natural extension of this algorithm solves the community detection problem on a regular version of the stochastic block model with more than two communities. Rather surprisingly, our results provide a rigorous evidence of the ability of natural dynamics to solve a computational problem that is non-trivial in a centralized setting.