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.

8 pages, 8 figures

References in corpus (11)

• Fast unfolding of communities in large networks
• Resolution limit in community detection
• Stochastic blockmodels and community structure in networks
• Phase transition in the detection of modules in sparse networks
• Community detection in networks: Modularity optimization and maximum likelihood are equivalent
• Graph spectra and the detectability of community structure in networks
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• Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors
• Equitable random graphs

Cited by in corpus (5)

• ${\ell }_p$-Spread and Restricted Isometry Properties of Sparse Random Matrices
• Spectral gap in random bipartite biregular graphs and applications
• Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
• Spectral density of equitable core-periphery graphs
• Uniqueness of communities in regular stochastic block models