p-star models, mean field random networks and the heat hierarchy

TL;DR

This paper explores the mean field approximation of p-star models for homogeneous random networks, revealing connections to the heat hierarchy and phase transition phenomena through PDE analysis and simulations.

Contribution

It introduces the mean field analog of p-star models, linking their partition functions to the heat hierarchy and analyzing phase transitions via PDE methods.

Findings

Mean field models satisfy the heat hierarchy of PDEs.

Phase transitions correspond to singularities and shocks in solutions.

Monte Carlo simulations confirm macroscopic agreement, but local discrepancies exist.

Abstract

We consider the mean field analog of the p-star model for homogeneous random networks, and compare its behaviour with that of the p-star model and its classical mean field approximation in the thermodynamic regime. We show that the partition function of the mean field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading order solution develops singularities in the space of parameters that evolve as classical shocks regularised by a viscous term. Shocks are associated with phase transitions and stable states are automatically selected consistently with the Maxwell construction. The case p = 3 is studied in detail. Monte Carlo simulations show an excellent agreement between the p-star model and its mean field analog at the macroscopic level, although significant discrepancies arise when local features are compared.