Atomic subgraphs and the statistical mechanics of networks

TL;DR

This paper introduces a flexible framework for generating complex networks using atomic subgraphs, enabling models with realistic motifs and broad applicability across various network types, all derived from a unified entropy expression.

Contribution

It develops a general maximum entropy model for networks based on atomic subgraphs, unifying many existing models and allowing for customizable network motifs and structures.

Findings

Models can generate networks with realistic motifs like triangles.

Unified entropy expression for diverse network models.

Framework encompasses many existing network models.

Abstract

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.