Two phase transitions in modular multiplex networks

Abstract Modular networks, such as critical infrastructures, are often built from distinct, densely connected modules (e.g., cities) that are sparsely interconnected. When such networks are gradually and randomly disrupted under a percolation process, they undergo two critical phase transitions. The first transition occurs when modules become isolated from one another, while the second corresponds to the collapse of the entire network, including the internal connectivity of the modules. Here, we study these phase transitions in modular multiplex networks and compare them with those observed in single-layer modular networks. We focus on models in which the modules are arranged and connected either as a short-range two-dimensional square lattice or as a long-range Random Regular network. We show here that these systems exhibit diverse transition behaviors, with some transitions occurring continuously and others abruptly; notably, one realistic model display two distinct abrupt transitions in the same system. By tuning the range of the inter-module connectivity between these two limits, we further show that the order of the spatial transition changes gradually from continuous to abrupt. For the modular Random Regular multiplex, we further characterize the spatial transition through its scaling behavior, revealing signatures of a mixed-order phase transition. In addition, we analytically determine the critical threshold at which modules become disconnected. Our results highlight the crucial role of modular organization and the critical role of interdependence in shaping network vulnerabilities under failures.