Two phase transitions in modular multiplex networks
4 affiliations
- 1Bar-Ilan University
- 2Universitat Bar-Ilan
- 3Northeastern University - Boston Campus
- 4Yeshiva University
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.
Related
- Modular Landau Analysis of Coupled Phase Transitions: Application to Ferroelectric Liquid Crystals
- Dynamics-structure interchangeability in binary opinion models on multiplex networks
- Multiscale Reconstruction of Multiplex Networks with Higher-Order Interactions
- A theory for detecting the Turing instability on static and temporal multiplex networks
- Interaction order controls stochastic-resonance enhancement and redistribution in delayed multiplex neural networks
BibTeX
@article{kfircohen2026phase,
title = {{Two phase transitions in modular multiplex networks}},
author = {Yael Kfir-Cohen and Dana Vaknin Ben Porath and Bnaya Gross and Sergey V Buldyrev and Shlomo Havlin},
journal = {New Journal of Physics},
year = {2026},
month = oct,
doi = {10.1088/1367-2630/aeb4e4},
url = {https://iopscience.iop.org/article/10.1088/1367-2630/aeb4e4},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Two phase transitions in modular multiplex networks
AU - Yael Kfir-Cohen
AU - Dana Vaknin Ben Porath
AU - Bnaya Gross
AU - Sergey V Buldyrev
AU - Shlomo Havlin
PY - 2026
DA - 2026/10/15
JO - New Journal of Physics
DO - 10.1088/1367-2630/aeb4e4
UR - https://iopscience.iop.org/article/10.1088/1367-2630/aeb4e4
AB - 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.
SN - 1367-2630
ER -