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 Barthélemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than p 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 Barthélemy'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 p W /2 total edge weight, where W is the total edge weight in the network and is the maximum weight of an inter-community edge. If 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 , 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.
In REF the authors pointed out that by weighting a network the resolution limit amounts to W ε/2, being W the sum of the weights in the network and ε the maximum weight of an edge connecting vertices located in two different communities.
Tolerating the Community Detection Resolution Limit with Edge Weighting
Computer Science Mathematics Physics Medicine