Annealed inhomogeneities in random ferromagnets

TL;DR

This paper investigates the annealed ferromagnetic Ising model on complex networks, revealing how annealing alters critical temperatures and exponents, differing significantly from quenched systems, especially in networks with degree fluctuations.

Contribution

It provides a detailed analysis of the annealed Ising model on complex networks, highlighting differences from quenched models and uncovering new critical behaviors and exponents.

Findings

Three distinct annealed critical temperatures for Poissonian degree networks.

Annealed critical temperatures can be finite even when quenched ones are infinite.

Annealing affects universality classes, especially in power-law degree distributions.

Abstract

We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erd\H{o}s-R\'enyi), or with prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite, since then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment, and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random i.i.d. degrees washes away the universality class with power-law critical exponents.