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.
Highlights
Networks and graphs are increasingly popular and effective tools for visualising and modelling large and complex processes and ‘big’ data sets
We know that many important graphical structures in the world are not tree-like; they tend to have many short loops, and we know that processes on graphs are affected significantly by the presence of such loops
Some methods were extended with loop corrections [7, 8, 9, 10], but all tend to fail for graphs with many short loops
Summary
Networks and graphs are increasingly popular and effective tools for visualising and modelling large and complex processes and ‘big’ data sets. It is problematic that most of our tools and algorithms for analysing (processes on) finitely connected graphs, such as cavity methods [1], belief propagation type algorithms [2, 3, 4], and conventional replica analyses [5, 6], require topologies that are locally tree-like. It is designed to handle analytically ensembles of large and sparse random graphs with prescribed degree sequences and prescribed loop statistics (via their adjacency spectra), and stochastic processes on such graphs. It is based on an alternative flavour of the replica method, with imaginary replica dimensions, and produces explicit closed equations in the infinite size limit, leading to Shannon entropies and expressions for spectra of ensembles of sparse loopy graphs. The familiar equations describing tree-like graphs are recovered as a simple limiting case
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