Evaluating Overlapping Communities with the Conductance of their Boundary Nodes

TL;DR

This paper introduces a novel conductance measure for boundary nodes in overlapping communities, derived from graph conductance, and demonstrates its effectiveness using a local greedy algorithm on a well-known social network.

Contribution

It defines a new conductance metric for boundary nodes in overlapping communities and links it to existing graph conductance measures, enabling better community evaluation.

Findings

The boundary node conductance correlates well with community quality.

The method successfully identifies meaningful overlapping communities in Zachary's karate club.

Encouraging initial results suggest potential for broader application.

Abstract

Usually the boundary of a community in a network is drawn between nodes and thus crosses its outgoing links. If we construct overlapping communities by applying the link-clustering approach nodes and links interchange their roles. Therefore, boundaries must drawn through the nodes shared by two or more communities. For the purpose of community evaluation we define a conductance of boundary nodes of overlapping communities analogously to the graph conductance of boundary-crossing links used to partition a graph into disjoint communities. We show that conductance of boundary nodes (or normalised node cut) can be deduced from ordinary graph conductance of disjoint clusters in the network's weighted line graph introduced by Evans and Lambiotte (2009) to get overlapping communities of nodes in the original network. We test whether our definition can be used to construct meaningful overlapping communities with a local greedy algorithm of link clustering. In this note we present encouraging results we obtained for Zachary's karate-club network.