Hierarchical block structures and high-resolution model selection in large networks

Discovering the large-scale topological features in empirical networks is a crucial tool in understanding how complex systems function. However most existing methods used to obtain the modular structure of networks suffer from serious problems, such as the resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs for the very popular approach of modularity optimization, but also for more principled ones based on statistical inference and model selection. Here we construct a nested generative model which, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible by current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable, and can be combined with advanced techniques of community detection to yield an efficient algorithm which scales well for very large networks.