[论文解读] Two phase transitions in modular multiplex networks

Yael Kfir-Cohen, Dana Ben Porath+3 more|arXiv (Cornell University)|Feb 10, 2026

Complex Network Analysis Techniques被引用 0

一句话总结

该论文在模块化多层网络中发现存在两次显著的相变：模块断连与全局崩溃，且相变类型（连续、突变或混合序）取决于模块间连通性范围，有分析阈值支撑。

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.

研究动机与目标

• 理解由密集模块构成、模块间连接稀疏的模块化网络的鲁棒性
• 研究两种模块化多层模型（二维晶格和随机规则RR）中的渗透引发的相变，并与单层模块化网络进行比较
• 推导模块断连的解析阈值并描述跨模型级联的性质

提出的方法

• 模型(a)：具有与四个最近邻模块相连的模块化多层晶格；节点具有模块内和模块间连接，功能依赖于两层中的巨型分量是否存在
• 模型(b)：模块化网络的每个模块连接到D个其他模块（随机规则拓扑），含两层且在每个节点内存在相互依赖
• 只有当节点在两层中均属于巨型分量才是功能性节点（多层渗透）
• 通过比较跨模块连边失败未达到局部巨型分量的概率，用G以及参数Q、D、k_intra，推导空间（模块断连）阈值p_c^{sp}的解析结果
• 利用非模块化多层晶格和RR网络的已知结果来验证p_c^{sp}（式2–4）并与仿真比较
• 评估相变的性质（连续与突变）并测量临界指数β ≈ 0.506以及级联动力学ζ ≈ 0.506

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

实验结果

研究问题

• RQ1 在模块化多层网络中，是什么触发模块彼此断连的第一道相变？
• RQ2 模块间连通性的范围/拓扑（短程晶格与长程RR）如何影响第二道相变的序与性质？
• RQ3 是否能推导空间（模块断连）相变的解析阈值并将其与模块内连通性及模块间拓扑联系起来？
• RQ4 模块化多层网络是否呈现混合序相变，及其对应的临界指数与级联系统动力学？
• RQ5 结果在相变行为与阈值方面，与单层模块化网络和ER多层网络相比有何差异？

主要发现

• 存在两次显著的相变：模块断连与模块内的全局崩溃
• 基于晶格的多层网络呈现一个连续相变与一个突变（混合序）相变，而RR多层呈现两个突变相变
• 上部相变（模块断连）受模块间连通性范围影响（短程与长程）
• 下部相变（模块崩溃）类似ER分支阈值p_c^{ER} ≈ 2.4554/k_intra，在单层情形为连续，在多层情形为突变
• RR模型的空间相变呈现混合序行为，测得β ≈ 0.506，级联系统持续时间指数ζ ≈ 0.506，指示近临界态的长寿命动力学
• 解析表达式（式2–4）通过模块度Q和互连度D预测p_c^{sp}，与仿真一致（图4）
• 在固定k_inter时增大模块规模ζ，则上部相变被消除，保留标准ER样的相变，显示模块间拓扑的作用

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