The statistical mechanics of networks

TL;DR

This paper explores a class of network models based on maximum entropy principles that match real-world network measurements, providing exact solutions and methods for more complex cases.

Contribution

It introduces a comprehensive framework for modeling networks using statistical mechanics, including exact solutions for models with arbitrary degree distributions and independent edges.

Findings

Exact solutions for models with arbitrary degree distributions.

Methods for solving models with correlated edges.

Application of statistical mechanics techniques to network modeling.

Abstract

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of the ensemble. Models of this type play the same role in the study of networks as is played by the Boltzmann distribution in classical statistical mechanics; they offer the best prediction of network properties subject to the constraints imposed by a given set of observations. We give exact solutions of models within this class that incorporate arbitrary degree distributions and arbitrary but independent edge probabilities. We also discuss some more complex examples with correlated edges that can be solved approximately or exactly by adapting various familiar methods, including mean-field theory, perturbation theory, and saddle-point expansions.