Asymptotic resolution bounds of generalized modularity and multi-scale community detection

The maximization of generalized modularity performs well on networks in which the members of all communities are statistically indistinguishable from each other. However, there is no theory bounding the maximization performance in more realistic networks where edges are heterogeneously distributed within and between communities. Using the random graph properties, we establish asymptotic theoretical bounds on the resolution parameter for which the generalized modularity maximization performs well. From this new perspective on random graph model, we find the resolution limit of modularity maximization can be explained in a surprisingly simple and straightforward way. Given a network produced by the stochastic block models, the communities for which the resolution parameter is larger than their densities are likely to be spread among multiple clusters, while communities for which the resolution parameter is smaller than their background inter-community edge density will be merged into one large component. Therefore, no suitable resolution parameter exits when the intra-community edge density in a subgraph is lower than the inter-community edge density in some other subgraph. For such networks, we propose a progressive agglomerative heuristic algorithm to detect practically significant communities at multiple scales.