Ising phase transitions in networks with tunable degree-degree correlations.

We present a systematic numerical study of the ferromagnetic Ising model on complex networks with tunable degree-degree correlations. The networks are constructed with a truncated power-law degree distribution, ensuring finite moments, and correlations are introduced via the Xulvi-Brunet-Sokolov rewiring algorithm, allowing continuous control from assortative to disassortative mixing. Using finite-size scaling analysis, we find that the system undergoes second-order phase transitions for all correlation levels. Assortative correlations increase the critical temperature, whereas disassortative correlations decrease it. For strong correlations, the network exhibits a segregation into degree-based subgroups, leading to multiple cascading transitions before the system fully loses magnetic order. For the continuous transitions, we compute the critical exponents and find that they remain robust across all correlation regimes, taking values consistent with mean-field predictions for the Ising model. This result establishes that the universality class of the system is preserved despite significant changes in network structure. Our results highlight the distinct roles of degree correlations in shaping nonuniversal features, such as transition temperatures and structural organization, while leaving the underlying universality class unchanged. This work provides insights into the interplay between topology and critical phenomena in heterogeneous networks.