Replica methods for loopy sparse random graphs

TL;DR

This paper introduces a new statistical mechanical formalism for analyzing random graphs with many short loops, extending existing theories for tree-like graphs and suggesting adaptations for message passing algorithms.

Contribution

The paper develops a novel replica-based formalism for loopy sparse random graphs, incorporating degree and spectrum constraints, and explores implications for message passing algorithms.

Findings

Formalism recovers known results for tree-like graphs

Suggests how to adapt message passing algorithms for loopy graphs

Provides insights into spin systems on loopy architectures

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.