Two phase transitions in modular multiplex networks

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 Random Regular network or as a two-dimensional square lattice. We show here that these systems exhibit diverse transition behaviors, with some transitions occurring continuously and others abruptly; notably, one realistic model could display two distinct first-order transitions in the same system. For the modular Random Regular multiplex, we further characterize the spatial transition through its scaling behavior, revealing signatures of a mixed-order phase transitions. 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.

💡 Research Summary

The paper investigates how modular multiplex networks—systems composed of densely connected modules (e.g., cities) that are sparsely linked to each other—fail under random node removal (percolation). Two distinct phase transitions are identified. The first “spatial” transition occurs when inter‑module links are sufficiently damaged that the modules become isolated from one another; the second “module‑collapse” transition follows when the internal connectivity of each module (modeled as an Erdős–Rényi (ER) network) disintegrates, causing the whole system to fragment completely.

Two multiplex constructions are studied. Model (a) places modules on a two‑dimensional square lattice; each module is linked only to its four nearest‑neighbor modules. Model (b) arranges modules on a Random Regular (RR) graph, where each module connects to a fixed number D of other modules irrespective of spatial distance. Within every module, intra‑module links follow a Poisson distribution with mean k_intra, and inter‑module links have mean k_inter; the total number of inter‑links per module is Q = k_inter·ζ² (ζ is the linear size of a module). Nodes belong to two layers simultaneously, and a node is functional only if it belongs to the giant component in both layers, introducing inter‑layer dependency.

Simulations of the largest connected component P∞ as a function of the surviving node fraction p reveal three characteristic behaviours. In a single‑layer lattice modular network both transitions are continuous (second‑order), matching classic 2‑D percolation. In the multiplex lattice, the spatial transition remains continuous while the module‑collapse transition becomes abrupt (first‑order) because failures cascade across layers. In the multiplex RR network both transitions are abrupt; moreover, the spatial transition exhibits mixed‑order characteristics: the order parameter vanishes with a critical exponent β ≈ 0.506 and the cascade duration diverges with exponent ζ ≈ 0.506, indicating a hybrid of discontinuity and critical scaling.

The authors develop an analytical theory for the spatial transition threshold p_spc. They compare the probability that two modules lack any inter‑link connecting their local giant components, X =