On the Analysis of a Label Propagation Algorithm for Community Detection

Abstract

This paper initiates formal analysis of a simple, distributed algorithm for community detection on networks. We analyze an algorithm that we call Max-LPA, both in terms of its convergence time and in terms of the “quality” of the communities detected. Max-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 Erdös-Rényi 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′ ($p = \Omega\left(\frac{1}{n^{1/4-\epsilon}}\right)$ for any ε > 0, p′ = O(p ²), where n is the number of nodes), Max-LPA detects the clusters V ₁, V ₂, …, V _n in just two rounds. Based on this and on empirical results, we conjecture that Max-LPA can correctly and quickly identify communities on clustered Erdös-Rényi graphs even when the clusters are much sparser, i.e., with $p = \frac{c\log n}{n}$ for some c > 1.

This work was done when the first author (KK) was visiting The University of Iowa on an Indo-US Science and Technology Forum Fellowship. The work of the second author (SP) was partially supported by National Science Foundation grant CCF 0915543.