Phase Transitions in a Network with Assortative Mixing

TL;DR

This paper investigates how degree correlations in a network influence phase transitions in an Ising model with a power-law degree distribution, revealing the impact of assortativity on critical behavior and phase transition points.

Contribution

It introduces a study of phase transitions in a network with assortative mixing using the Ising model, focusing on how degree correlations affect critical phenomena.

Findings

Calculated phase transition points for varying assortativity.

Determined critical exponents for magnetization, susceptibility, and correlation length.

Showed that degree correlations influence the critical behavior of the system.

Abstract

In this work, we employed the Ising model to identify phase transitions in a magnetic system where the degree distribution of the network follows a power-law and the connections are assortatively mixed. In the Ising model, the spins assume only two values, $\sigma = \pm 1$, and interact through ferromagnetic coupling $J$. The network is characterized by four variable parameters: $\alpha$ denotes the degree distribution exponent, the minimum degree $k_0$, the maximum degree $k_m$, and the $p_r$ represents the assortativity or disassortativity of the network. To investigate the effect of degree correlations on the critical behavior of the system, we fix $k_0=4$, $k_m=10$, and $\alpha=1$, and vary $p_r$ to obtain an assortative mixing of edges. As result, we have calculated the phase transition points of the system, and the critical exponents related to magnetization $\beta$, magnetic susceptibility $\gamma$, and the correlation length $\nu$.