Replica methods for loopy sparse random graphs

A. C C Coolen*

*Corresponding author for this work

Research output: Contribution to journalConference paperpeer-review

11 Citations (Scopus)
128 Downloads (Pure)


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.

Original languageEnglish
Article number012022
Number of pages11
JournalJournal of Physics: Conference Series
Issue number1
Publication statusPublished - 6 Apr 2016


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