Finding and Testing Network Communities by Lumped Markov Chains

Carlo Piccardi,Yamir Moreno

doi:10.1371/journal.pone.0027028

Abstract

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition that is based on a quality threshold. By means of a lumped Markov chain model of a random walker, a quality measure called “persistence probability” is associated to a cluster, which is then defined as an “-community” if such a probability is not smaller than . Consistently, a partition composed of -communities is an “-partition.” These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired -level allows one to immediately select the -partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the quality of each single community. Given its ability in individually assessing each single cluster, this approach can also disclose single well-defined communities even in networks that overall do not possess a definite clusterized structure.

Highlights

Complex networks are currently one of the most extensively studied subjects in the field of applied mathematics
Since high modularity values are obtained in presence of groups of nodes with comparatively large intra-community edge density, maximizing modularity should put in evidence the ‘‘best’’ partition
An important relationship between random walks and modularity is put forward by Kim et al [28] who propose their LinkRank modularity Qlr, a variation to the standard modularity aimed at obtaining a better performance on directed graphs

Summary

Introduction

Complex networks are currently one of the most extensively studied subjects in the field of applied mathematics. One of the most promising but challenging tasks in network science is community analysis, which is aimed at revealing possible partitions of a network into subsets of nodes (communities, or clusters) with dense intra- but sparse inter-group connections Finding and analyzing such partitions often provides invaluable help in deeply understanding the structure and function of a network, as widely demonstrated by several case studies in social sciences [5,6], biology [7], ecology [8], economics [9], or information science [10,11], just to name a few. Since high modularity values are obtained in presence of groups of nodes with comparatively large intra-community edge density, maximizing modularity should put in evidence the ‘‘best’’ partition This method has been proven successful in many circumstances but, on the other hand, it has been widely demonstrated that, due to intrinsic limitations, it does not necessarily always yield a significant partition [12,15,19]. Many other methods for community analysis have been put forward in the last few years, trying to simultaneously finding a meaningful network partition and assessing its significance (we recall, e.g., [20,21,22])

Methods

Results

Discussion

Conclusion