Spectral partitioning in equitable graphs

Abstract

Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e. random graphs with a block-regular structure, is studied, for which analytical results can be obtained. In particular, the spectral density of this ensemble is computed exactly for a modular and bipartite structure. Kesten-McKay's law for random regular graphs is found analytically to apply also for modular and bipartite structures when blocks are homogeneous. Exact solution to graph partitioning for two equal-sized communities is proposed and verified numerically, and a conjecture on the absence of an efficient recovery detectability transition in equitable graphs is suggested. Final discussion summarizes results and outlines their relevance for the solution of graph partitioning problems in other graph ensembles, in particular for the study of detectability thresholds and resolution limits in stochastic block models.

Spectral gap in random bipartite biregular graphs and applications

“…Since the rank of X is r, it follows that the nullity of X * is n − r, so there are n − r independent vectors w for which X * w = 0. Now, note that picking v 1 = w, v 2 = 0, and λ = ±i √ d 1 − 1, we satisfy equations (4) and (5). Thus, ±i √ d 1 − 1 are eigenvalues of B, both with multiplicity n − r. The remaining 4r eigenvalues of B determined by A come from nonzero eigenvalues of A.…”

“…Barrett et al [4] studied the effect of these symmetries from a group theory standpoint. The work of [5] is closest to ours: they consider spectral properties of such graphs and their implications for spectral community clustering. In particular, they show that the spectrum of what we call the 'frame' (in their words, the discrete spectrum, which is deterministic) is contained in that of the random graph.…”

Spectral density of equitable core-periphery graphs

“…We derive the cavity equations [16,17] to compute the continuous part of the spectrum of the adjacency matrices in equitable graphs, as shown in [18,13]. Given an ensemble of N × N symmetric matrices the set of eigenvalues of a given adjacency matrix A is denoted by {λ A i } N i=1 .…”

“…Between each pair of blocks the edges are drawn according to a k-regular graph, where the value of k equals the corresponding element of the connectivity matrix, then the total set of edges is given by the union of the sets for each of the m 2 regular and biregular graphs [Algorithm 1]. This ensemble has been successfully studied in the context of community detection [15] and an algorithm was found that is able to identify blocks simply by looking at the list of edges in the graph [13], exploiting the symmetry of the eigenvectors of the adjacency matrix.…”

“…The paper is organized as follows: in Section 1 the equitable graph ensemble is defined and a generative algorithm is introduced. In Section 2 a brief introduction of the cavity approach to the computation of the spectral density of random matrices is provided and the general expression for the cavity variances of equitable graphs is shown, as previously derived in [11,13]. In Section 3 we derive the main result of the paper, that is the analytic expression for the spectral density of a large class of equitable core-periphery graphs.…”