Tolerating the community detection resolution limit with edge weighting

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 Barthelemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than root 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 Barthelemy'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 less than root W is an element of/2 total edge weight, where W is the total edge weight in the network and is an element of is the maximum weight of an intercommunity edge. If is an element of 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 is an element of, we modify the Clauset, Newman, and Moore (CNM) community detection algorithm to maximize weighted modularity, and we 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.