Analysis and Optimization of Robustness in Multiplex Flow Networks Against Cascading Failures

TL;DR

The paper introduces a two-layer multiplex flow-network model to study cascading failures driven by random attacks, with within-layer load redistribution and two overload conditions (layer-independent and layer-influenced). It derives a mean-field recursive framework to compute the final surviving fraction $n_{\infty}(p)$ and develops a fixed-point approach to obtain robust final states, represented by $n_{\infty}(p)=(1-p)\mathbb{P}[S_A>x^*+\beta_B y^*, S_B>y^*+\beta_A x^*]$. Crucially, it proves an optimality principle: under a total free-space constraint, allocating free space across layers proportional to mean effective loads and distributing it equally within each layer (layer-weighted equal free space) maximizes the critical attack size $p^*$ and yields the maximal final system size $n_{\infty}(p)=1-p$ for all $p\le p^*$. Numerical results confirm the analytical predictions across diverse distributions and illustrate the impact of cross-layer influence on robustness and phase-transition behavior. The framework offers design guidelines for robust, multi-commodity networks in engineering, transportation, and financial systems.

Abstract

Networked systems are susceptible to cascading failures, where the failure of an initial set of nodes propagates through the network, often leading to system-wide failures. In this work, we propose a multiplex flow network model to study robustness against cascading failures triggered by random failures. The model is inspired by systems where nodes carry or support multiple types of flows, and failures result in the redistribution of flows within the same layer rather than between layers. To represent different types of interdependencies between the layers of the multiplex network, we define two cases of failure conditions: layer-independent overload and layer-influenced overload. We provide recursive equations and their solutions to calculate the steady-state fraction of surviving nodes, validate them through a set of simulation experiments, and discuss optimal load-capacity allocation strategies. Our results demonstrate that allocating the total excess capacity to each layer proportional to the mean effective load in the layer and distributing that excess capacity equally among the nodes within the layer ensures maximum robustness. The proposed framework for different failure conditions allows us to analyze the two overload conditions presented and can be extended to explore more complex interdependent relationships.

Figures (6)

• Figure 1: Multiplex Flow Network Model. Each layer is defined on the same set of vertices but with (potentially) different edge sets, and each layer is responsible for carrying/supplying a different flow type. The highlighted node, say $v_x$, carries flow/load of $L_{x,A}$ in network $A$ and $L_{x,B}$ in network $B$; its flow/load vector is then given by $\mathbf{L}_x = [L_{x,A}, L_{x,B}]$. If $v_x$ fails, $L_{x,A}$ amount of flow of type-$A$ will be redistributed to functional nodes in network $A$ and $L_{x,B}$ amount of type-$B$ flow is redistributed to functional nodes in $B$.
• Figure 2: Two different cases for failure/overload conditions defined as a partitioning of the ($L_A, L_B$) plane into regions determining a node’s current state on each flow as functioning or failed. (a) Layer-independent Overload: Independent capacity for each layer and the node fails if capacity is exceeded on any layer. (b) Layer-influenced Overload: Failure condition depends jointly on ($L_A$, $L_B$) indicating that the ability of the node to take on extra load on any given layer is influenced by its load on other layers. Here, we use linear boundaries to define the region where the node is able to function on both layers.
• Figure 3: The resulting plots illustrating the graphical solution procedure. The initial loads in both layers are distributed uniformly with $L_{\textrm{min}} = 20$ and $\mathbb{E}[L] = 30$, while the free spaces in both layers are also distributed uniformly with $S_{\textrm{min}} = 25$ and $\mathbb{E}[S] = 50$. Additionally, $\beta_A = \beta_B = 0.25$. Subfigures (a) and (b) depict the surviving conditions for Networks $A$ and $B$, where the blue surface represents the left-hand side values of the inequalities. The gray plane (parallel to the x-y plane) corresponds to $z =1/(1-p)$ when $p = 0.25$, and the $(x,y)$ values above this plane are projected onto the x-y plane in green (horizontally striped area). Figure (c) shows the intersection of points that satisfy both inequalities, referred to as stable points, highlighted in red (dotted area). The upper-left corner of the stable points (black square), indicating the element-wise minimum, represents the resulting excess load for the given attack size at steady state, corresponding to a surviving fraction of $0.75$. As $p$ increases, the gray plane shifts upwards, shrinking the set of stable points. Eventually there will be a critical attack size $p^*$ beyond which we observe complete failure of the system indicated by the lack of feasible solutions for the stable points.
• Figure 4: Final system size for different $L_A,L_B,S_A,S_B$ distributions under random attack $p$. Analytical results are obtained by (\ref{['eq_main:solution set']}) are shown in solid lines where the averaged results from 100 independent simulations are shown with different markers. We see that theoretical results perfectly match with the simulation averages.
• Figure 5: Final system size for $L_A \sim U(10,30)$, $L_B\sim Wei(10,10.78,6)$, $S_A \sim U(10,60)$, $S_B \sim U(20,100)$ distributions under random attack $p$. Analytical results are obtained by (\ref{['eq_main:solution set']}) are shown in solid lines where the averaged results from 100 independent simulations with $N=10^5$ are shown with different markers. We see that theoretical results perfectly match with the simulation averages.

Theorems & Definitions (6)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma B.1
• Lemma B.2
• Theorem C.1