Replica methods for loopy sparse random graphs*

International Meeting on High-Dimensional Data-Driven Science 2015

Coolen, A.C.C.^1

Department of Mathematics, King's College London, The Strand, London

WC2R 2LS, United Kingdom^1

关键词: Adjacency matrices;    Emerging theories;    Hard constraints;    Message passing algorithm;    Random graphs;    Replica method;    Soft constraint;    Spin systems;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/699/1/012022/pdf
DOI  :  10.1088/1742-6596/699/1/012022

来源: IOP

【 摘 要 】

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.