A Maximum Entropy approach to Massive Graph Spectra

Machine Learning Research Group and Oxford-Man Institute for Quantitative Finance, Department of Engineering Science, University of Oxford Abstract Graph spectral techniques for measuring graph similarity, or for learning the cluster number, require kernel smoothing. The choice of kernel function and bandwidth are typically chosen in an ad-hoc manner and heavily affect the resulting output. We prove that kernel smoothing biases the moments of the spectral density. We propose an information theoretically optimal approach to learn a smooth graph spectral density, which fully respects the moment information. Our method's computational cost is linear in the number of edges, and hence can be applied to large networks, with millions of nodes. We apply our method to the problems to graph similarity and cluster number learning, where we outperform comparable iterative spectral approaches on synthetic and real graphs. Keywords: Networks, Information Theory, Maximum Entropy, Graph Spectral Theory, Random matrix theory, iterative methods, kernel smoothing 1. Introduction: networks, their graph spectra and importance Many systems of interest can be naturally characterised by complex networks; examples include social networks (Mislove et al., 2007b; Flake et al., 2000; Leskovec et al., 2007), biological networks (Palla et al., 2005) and technological networks.