Phase diagram and metastability of the Ising model on two coupled networks

TL;DR

This paper analyzes the phase transitions and metastability in a coupled Ising model on two networks, deriving analytical phase diagrams and confirming them with simulations, revealing ergodicity breaking at low temperatures.

Contribution

It provides the first analytical characterization of phase diagrams and metastable states in a coupled network Ising model, including stability analysis and transition nature.

Findings

Metastable states appear discontinuously with coexistence regions.

The system's escape time from metastable states grows exponentially with inverse temperature.

Analytical results are validated by Monte-Carlo simulations.

Abstract

We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erd\"os-R\'enyi random graphs. We derive analytical expressions for the free-energy of the system and the magnetization of each graph, from which the phase diagrams, the stability of the different states, and the nature of the transitions among them, are clearly characterized. We show that a metastable state appears discontinuously by varying the model parameters, yielding a region in the phase diagram where two solutions coexist. By performing Monte-Carlo simulations, we confirm the exactness of our main theoretical results and show that the typical time the system needs to escape from a metastable state grows exponentially fast as a function of the temperature, characterizing ergodicity breaking in the thermodynamic limit.