Replica methods for loopy sparse random graphs

Abstract

I report on the development of a novel statistical mechanical formalism for the analysis of random graphs with many short loops, and processes on such graphs. The graphs are defined via maximum entropy ensembles, in which both the degrees (via hard constraints) and the adjacency matrix spectrum (via a soft constraint) are prescribed. The sum over graphs can be done analytically, using a replica formalism with complex replica dimensions. All known results for tree-like graphs are recovered in a suitable limit. For loopy graphs, the emerging theory has an appealing and intuitive structure, suggests how message passing algorithms should be adapted, and what is the structure of theories describing spin systems on loopy architectures. However, the formalism is still largely untested, and may require further adjustment and refinement.

11 pages, no figures. To be published in Proceedings of The International Meeting on High-Dimensional Data-Driven Science (HD3-2015), Kyoto, Japan, on 14-17 December, 2015

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• Imaginary replica analysis of loopy regular random graphs
• Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles
• Analytic solution of the two-star model with correlated degrees
• Belief propagation on networks with cliques and chordless cycles
• Theory for the conditioned spectral density of non-invariant random matrices
• Uncovering the non-equilibrium stationary properties in sparse Boolean networks