Learning multifractal structure in large networks

Generating random graphs to model networks has a rich history. In this paper, we analyze and improve upon the multifractal network generator (MFNG) introduced by Palla et al. We provide a new result on the probability of subgraphs existing in graphs generated with MFNG. From this result it follows that we can quickly compute moments of an important set of graph properties, such as the expected number of edges, stars, and cliques. 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 a new method of moments algorithm for fitting large networks to MFNG. 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 multi- fractal 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

Network Density of States

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.Comment: 10 pages, 7 figure