Linear and Optimization Hamiltonians in Clustered Exponential Random Graph Modeling

TL;DR

This paper introduces a non-linear Hamiltonian model for clustered exponential random graphs that avoids unrealistic phase transitions, enabling the generation of networks with controlled intermediate clustering levels.

Contribution

It proposes a novel non-linear Hamiltonian approach that prevents triangle condensation, improving the modeling of clustered networks in exponential random graph theory.

Findings

The non-linear Hamiltonian avoids phase transition into extreme clustering.

The model generates networks with specified intermediate clustering.

Numerical simulations confirm the effectiveness of the approach.

Abstract

Exponential random graph theory is the complex network analog of the canonical ensemble theory from statistical physics. While it has been particularly successful in modeling networks with specified degree distributions, a naive model of a clustered network using a graph Hamiltonian linear in the number of triangles has been shown to undergo an abrupt transition into an unrealistic phase of extreme clustering via triangle condensation. Here we study a non-linear graph Hamiltonian that explicitly forbids such a condensation and show numerically that it generates an equilibrium phase with specified intermediate clustering.