Two phase transitions in modular multiplex networks

TL;DR

The paper investigates how modular structure and inter-module connectivity shape percolation-driven failures in multiplex networks. It introduces two inter-module topologies—lattice-based and Random Regular—and compares them to a single-layer modular baseline, revealing two phase transitions: inter-module disconnection and global collapse within modules. A key contribution is an analytical expression for the module-disconnection threshold and a derived formula for the spatial transition threshold $p_c^{sp}$, which agrees with simulations across models; the RR case exhibits a mixed-order spatial transition with exponents near $1/2$. The results highlight the crucial role of connectivity range and interdependence in determining resilience, with implications for the design and protection of critical infrastructures.

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 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.

Figures (6)

• Figure 1: A schematic representation of our models. The red and blue lines represent the links in the first and second layer in the multiplex. Each circle represents a module and each dot represents a node. (a) and (b) Modular multiplexes, while drawing only inter-nodes and inter-links. In (a) the inter-links connect each module to some of its $4$ closet modules in the lattice of modules, while in (b) the interlinks connect any module to any $D$ modules (Here, $D = 1$) which may be at any distance. (c) Demonstrating a close-up of a module in both models (a) and (b), with interlinks and intra-links which are distributed Poissonian as an $ER$ network.
• Figure 2: Simulation results for the largest cluster $P_{\infty}$ as a function of $p$. We compare three models: (i) a single-layer lattice model, which exhibits two continuous phase transitions; (ii) a multiplex lattice model, displaying one continuous and one abrupt (mixed-order) transition; and (iii) a multiplex random regular ($RR$) network, which exhibits two abrupt transitions. Notably, the lower transition point is identical for both multiplex models, as it corresponds to the fragmentation of an individual multiplex Erdős–Rényi ($ER$) module. Inset: The change of the giant component of the multiplex $RR$ network from its value at criticality as a function of the distance of $p$ from $p^{sp}_c$ in log log scale. From that we obtain $\beta=0.506$. Parameters used are $L=10^4$, $k_{intra} = 4$, $\zeta=100$, $k_{inter}= 10^{-3}$ and $Q=10$.
• Figure 3: Simulation results for the largest cluster $P_{\infty}$ in the two multiplex models as a function of $p$ for different values of $\zeta$ on semi log scale. Two distinct transitions are observed. The first (higher p) transition at $p^{sp}_{c}$ of the lattice. The second (lower p) transition occurs when the small $ER$ modules collapse at, $p^{ER}_{c}$. (a) Multiplex of a modular network where the modules are connected as in 2D lattice, and (b) multiplex of a modular network where the modules are connected as a $RR$ network. Parameters used are $L= 10^4$, $k_{intra} = 4$ and $k_{inter}= 10^{-3}$.
• Figure 4: Multiplex modular lattice (model (a)). (a)$G(p^{sp}_{c})$ and (b)$p^{sp}_{c}$ as a function of $\zeta$ for various values of $k_{inter}$. The circles represent simulation results and the lines are the theory obtained from (a) equation (3) and (b) equation (4). In the limit of $\zeta \rightarrow \infty$ the system approaches a single ER network and $p^{sp}_c$ approaches $p^{ER}_{c}$. Parameters used are $L= 10^4$ and $k_{intra} = 4$.
• Figure 5: Simulations of the largest cluster $P_{\infty}$ as a function of $p$ for different values of $\zeta$ on semi log scale with a fixed module degree, $Q=10=k_{inter}\cdot \zeta^2$. Two distinct transitions are observed. The first (higher) transition at $p^{sp}_{c}$ of the lattice. The second (lower) transition occurs when the small $ER$ modules break apart, $p^{ER}_{c}$. (a) multiplex of a modular network where the models are connected as in 2D lattice, and (b) multiplex of a modular network where the models are connected as in a Random Regular network. Parameters used are $L= 10^4$ and $k_{intra} = 4$.