Tolerating the Community Detection Resolution Limit with Edge Weighting

TL;DR

This paper addresses the resolution limit problem in community detection by extending modularity to weighted networks, proposing a method to improve detection accuracy of small communities through edge weighting and algorithm modification.

Contribution

It introduces a novel approach to mitigate the resolution limit in community detection by deriving new edge weights and adapting the CNM algorithm for weighted modularity maximization.

Findings

Weighted modularity can resolve smaller communities when inter-community edge weights are minimized.

Modified CNM algorithm achieves accuracy comparable to state-of-the-art methods.

Experimental results show significant improvement over traditional modularity-based detection.

Abstract

Communities of vertices within a giant network such as the World-Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barth\'{e}lemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than $\sqrt{L/2}$ edges, where $L$ is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barth\'{e}lemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with fewer than $\sqrt{W \epsilon/2}$ total edge weight, where $W$ is the total edge weight in the network and $\epsilon$ is the maximum weight of an inter-community edge. If $\epsilon$ is small, then small communities can be resolved. Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low $\epsilon$, we modify the ``CNM'' community detection algorithm to maximize weighted modularity, and show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed.