Distributed Community Detection via Metastability of the 2-Choices Dynamics

TL;DR

This paper analyzes how simple majority dynamics on networks with community structures lead to stable internal consensus within communities, providing the first analytical insights into non-consensus behaviors on non-complete graphs.

Contribution

It offers the first symmetry-breaking analysis of dynamics on non-complete topologies, revealing community-level consensus and implications for community detection and biological modeling.

Findings

Communities reach internal consensus rapidly and stably.

First theoretical analysis of distributed label propagation algorithms.

Demonstrates potential for species differentiation in evolutionary models.

Abstract

We investigate the behavior of a simple majority dynamics on networks of agents whose interaction topology exhibits a community structure. By leveraging recent advancements in the analysis of dynamics, we prove that, when the states of the nodes are randomly initialized, the system rapidly and stably converges to a configuration in which the communities maintain internal consensus on different states. This is the first analytical result on the behavior of dynamics for non-consensus problems on non-complete topologies, based on the first symmetry-breaking analysis in such setting. Our result has several implications in different contexts in which dynamics are adopted for computational and biological modeling purposes. In the context of Label Propagation Algorithms, a class of widely used heuristics for community detection, it represents the first theoretical result on the behavior of a distributed label propagation algorithm with quasi-linear message complexity. In the context of evolutionary biology, dynamics such as the Moran process have been used to model the spread of mutations in genetic populations [Lieberman, Hauert, and Nowak 2005]; our result shows that, when the probability of adoption of a given mutation by a node of the evolutionary graph depends super-linearly on the frequency of the mutation in the neighborhood of the node and the underlying evolutionary graph exhibits a community structure, there is a non-negligible probability for species differentiation to occur.