Ferromagnetic ordering in graphs with arbitrary degree distribution

TL;DR

This paper analyzes the phase transition of the Ising model on random graphs with arbitrary degree distributions, deriving critical temperatures and exponents using the replica method, especially for power-law graphs.

Contribution

It provides an exact computation of critical temperatures and exponents for the Ising model on arbitrary degree distribution graphs, including non-trivial scaling for heavy-tailed distributions.

Findings

Ferromagnetic transition occurs when the average degree is finite.

Critical exponents depend on the moments of the degree distribution.

Power-law graphs exhibit non-trivial scaling exponents.

Abstract

We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided <k> <= <k^2> < \infty. However, if the fourth moment of the degree distribution is not finite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs.