Comparison of Ising magnet on directed versus undirected Erdos-Renyi and scale-free network

TL;DR

This paper compares Ising models on directed and undirected Erdos-Renyi and scale-free networks, revealing how directionality affects phase transition behaviors like spontaneous magnetization and Curie temperature.

Contribution

It provides a comparative analysis of magnetic properties on directed versus undirected networks across different network topologies, highlighting the impact of directionality.

Findings

Spontaneous magnetization vanishes in directed scale-free networks after equilibration.

Directed ER networks maintain a positive Curie temperature.

Magnetization decay follows an Arrhenius law in certain cases.

Abstract

Scale-free networks are a recently developed approach to model the interactions found in complex natural and man-made systems. Such networks exhibit a power-law distribution of node link (degree) frequencies n(k) in which a small number of highly connected nodes predominate over a much greater number of sparsely connected ones. In contrast, in an Erdos-Renyi network each of N sites is connected to every site with a low probability p (of the orde r of 1/N). Then the number k of neighbors will fluctuate according to a Poisson distribution. One can instead assume that each site selects exactly k neighbors among the other sites. Here we compare in both cases the usual network with the directed network, when site A selects site B as a neighbor, and then B influences A but A does not influence B. As we change from undirected to directed scale-free networks, the spontaneous magnetization vanishes after an equilibration time following an Arrhenius law, while the directed ER networks have a positive Curie temperature.