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.

广义模块化的最大化在网络上的所有社区的成员在统计上彼此之间是无法区分的。但是，在更现实的网络中，没有理论限制最大化性能，在这些网络中，边缘在社区内和社区之间异构分布。利用随机图的性质，我们在分辨率参数上建立了渐进的理论界线，广义模块化最大化对这些参数表现良好。从这个关于随机图模型的新观点，我们发现可以以令人惊讶的简单直接的方式来解释模块化最大化的分辨率极限。给定由随机块模型生成的网络，其分辨率参数大于其密度的社区可能会散布在多个群集中，而分辨率参数小于其背景的社区间边缘密度的社区将合并为一个大组件。因此，当社区内边缘没有合适的分辨率参数退出密度在一个子图是比社区间边缘较低的密度在一些其他子图。对于这样的网络，我们提出了一种渐进的聚集启发式算法，以在多个尺度上检测实际重要的社区。