Double phase transition of the Ising model in core-periphery networks

TL;DR

This paper investigates the Ising model on core-periphery networks, revealing a double phase transition with an intermediate inhomogeneous phase and analyzing how connection scaling affects the nature of these transitions.

Contribution

The study uncovers a double phase transition in the Ising model on core-periphery networks and develops a mean-field theory to explain the phenomena.

Findings

Inhomogeneous intermediate phase with core nodes more ordered

Double peaks in susceptibility at two distinct temperatures

Scaling of core-periphery connections influences divergence of susceptibility peaks

Abstract

We study the phase transition of the Ising model in networks with core-periphery structures. By Monte Carlo simulations, we show that prior to the order-disorder phase transition the system organizes into an inhomogeneous intermediate phase in which core nodes are much more ordered than peripheral nodes. Interestingly, the susceptibility shows double peaks at two distinct temperatures. We find that, if the connections between core and periphery increase linearly with network size, the first peak does not exhibit any size-dependent effect, and the second one diverges in the limit of infinite network size. Otherwise, if the connections between core and periphery scale sub-linearly with the network size, both peaks of the susceptibility diverge as power laws in the thermodynamic limit. This suggests the appearance of a double transition phenomenon in the Ising model for the latter case. Moreover, we develop a mean-field theory that agrees well with the simulations.