Find Your Place: Simple Distributed Algorithms for Community Detection

TL;DR

This paper introduces a simple distributed algorithm where nodes iteratively update values based on neighbors to detect communities in graphs, proving it effectively uncovers underlying structures in various models.

Contribution

It demonstrates that a straightforward local averaging process can efficiently detect communities, including in complex stochastic block models, with rigorous theoretical guarantees.

Findings

Detects communities in logarithmic time in various graph models

Works on regularized stochastic block models with multiple communities

Provides theoretical proof of effectiveness for simple local dynamics

Abstract

Given an underlying graph, we consider the following \emph{dynamics}: 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 the values held by 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. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reflects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics to address a computational problem that is non-trivial even in a centralized setting.