Phase Transitions in a Network with Assortative Mixing

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 $\nu$.