On the Analysis of a Label Propagation Algorithm for Community Detection

This paper initiates formal analysis of a simple, distributed algorithm for community detection on networks. We analyze an algorithm that we call \textscMax-LPA, both in terms of its convergence time and in terms of the "quality" of the communities detected. \textscMax-LPA is an instance of a class of community detection algorithms called label propagation algorithms. As far as we know, most analysis of label propagation algorithms thus far has been empirical in nature and in this paper we seek a theoretical understanding of label propagation algorithms. In our main result, we define a clustered version of \er random graphs with clusters $V_1, V_2,..., V_k$ where the probability $p$, of an edge connecting nodes within a cluster $V_i$ is higher than $p'$, the probability of an edge connecting nodes in distinct clusters. We show that even with fairly general restrictions on $p$ and $p'$ ($p = Ω(\frac1n^1/4-ε)$ for any $ε > 0$, $p' = O(p^2)$, where $n$ is the number of nodes), \textscMax-LPA detects the clusters $V_1, V_2,..., V_n$ in just two rounds. Based on this and on empirical results, we conjecture that \textscMax-LPA can correctly and quickly identify communities on clustered \er graphs even when the clusters are much sparser, i.e., with $p = \fracc\log nn$ for some $c > 1$.