Imaginary replica analysis of loopy regular random graphs

Abstract

We present an analytical approach for describing spectrally constrained maximum entropy ensembles of finitely connected regular loopy graphs, valid in the regime of weak loop-loop interactions. We derive an expression for the leading two orders of the expected eigenvalue spectrum, through the use of infinitely many replica indices taking imaginary values. We apply the method to models in which the spectral constraint reduces to a soft constraint on the number of triangles, which exhibit `shattering' transitions to phases with extensively many disconnected cliques, to models with controlled numbers of triangles and squares, and to models where the spectral constraint reduces to a count of the number of adjacency matrix eigenvalues in a given interval. Our predictions are supported by MCMC simulations based on edge swaps with nontrivial acceptance probabilities.

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Cited by in corpus (7)

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• Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs
• Analytic solution of the two-star model with correlated degrees
• Phase transitions in atypical systems induced by a condensation transition on graphs
• Satisfiability transition in asymmetric neural networks
• Top eigenpair statistics of diluted Wishart matrices
• Free-energy density functional for Strauss's model of transitive networks