Complex network description of phase transitions in the classical and quantum disordered Ising Model*

Complex~network analysis is a powerful tool to describe and characterize classical systems such as~the~Ising model in a transverse magnetic field. Measuring spin-spin correlations gives~rise to the adjacency matrix, representing a weighted network. In this study, the spin-spin correlations at different temperatures~are~analytically calculated, yielding~phase-dependent~complex~networks,~from simple networks in the low temperature ferromagnetic limit to random ones at high temperature.~ The network structure varies as the transverse field and temperature~change,~recovering the phase diagram and providing initial insight into correlations in the critical region.~Analyzing the resulting complex network using a variety of network measures such as the degree histogram, average clustering, betweenness centrality and the graph entropy, the complexity is characterized.~ This method is applied for both the disordered classical Ising and quantum Ising lattice, demonstrating the role of finite temperature and disorder in generation of complexity.