Performance analysis of spectral community detection in realistic graph models

Abstract

This article proposes a spectral analysis of dense random graphs generated by (a modified version of) the degree-corrected stochastic block model, for a setting where the inter block probabilities differ by O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−</sup> ) with n the number of nodes. We study a normalized version of the graph modularity matrix which is shown to be asymptotically well approximated by an analytically tractable (spiked) random matrix. The analysis of the latter allows for the precise evaluation of (i) the transition phase where clustering becomes asymptotically feasible and (ii) the alignment between the dominant eigenvectors and the block-wise canonical basis, thus enabling the estimation of mis-classification rates (prior to post-processing) in simple scenarios.