Learning multifractal structure in large networks

Using random graphs to model networks has a rich history. In this paper, we analyze and improve the multifractal network generators (MFNG) introduced by Palla et al. We provide a new result on the probability of subgraphs existing in graphs generated with MFNG. This allows us to quickly compute moments of an important set of graph properties, such as the expected number of edges, stars, and cliques for graphs generated using MFNG. Specifically, we show how to compute these moments in time complexity independent of the size of the graph and the number of recursive levels in the generative model. We leverage this theory to propose a new method of moments algorithm for fitting MFNG to large networks. Empirically, this new approach effectively simulates properties of several social and information networks. In terms of matching subgraph counts, our method outperforms similar algorithms used with the Stochastic Kronecker Graph model. Furthermore, we present a fast approximation algorithm to generate graph instances following the multifractal structure. The approximation scheme is an improvement over previous methods, which ran in time complexity quadratic in the number of vertices. Combined, our method of moments and fast sampling scheme provide the first scalable framework for effectively modeling large networks with MFNG.