Constrained Markovian dynamics of random graphs

TL;DR

This paper develops a statistical mechanics framework for constrained graph evolution using Markov processes, providing formulas for mobility and conditions for uniform invariant measures, with applications to biological and synthetic graphs.

Contribution

Introduces a formalism for constrained graph dynamics, deriving mobility formulas and conditions for uniform measures, enabling controlled Markov chain design for graph sampling.

Findings

Derived an exact formula for graph mobility based on adjacency spectrum.

Designed Glauber-type Markov chains with controllable invariant measures.

Validated the theory on synthetic and biological graphs.

Abstract

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.