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Cascade of failures in coupled network systems with multiple support-dependent relations
Jia Shao, Sergey V. Buldyrev, Shlomo Havlin, H. Eugene Stanley
TL;DR
The paper studies how cascades unfold in two coupled networks with randomly assigned multiple support relations, addressing models restricted to one-to-one dependencies. It combines analytical derivation with numerical tests and finds agreement for coupled ER and scale-free networks, including a simple ER giant-component law that approaches single-network percolation with many supports.
Problem
The paper examines cascade robustness when nodes in coupled networks may have multiple support-dependent relations rather than only one-to-one dependencies.
Method
It develops a theoretical framework for randomly built support-dependent relations and tests the resulting cascade theory numerically on coupled ER and scale-free networks.
Results
For coupled ER networks, the giant components follow a simple percolation law that becomes the single-network law in the large-support limit, while simulations agree well with the theory.
Takeaways & Limitations
The model provides a framework for understanding coupled systems with complex dependence-support relations, including real-life network systems.
Abstract
from arXiv · showhide
We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependent relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support node in the other network. If both networks A and B are Erdős-Rényi networks, A and B, with (i) sizes $N^A$ and $N^B$, (ii) average degrees $a$ and $b$, and (iii) $c^{AB}_0N^B$ support links from network A to B and $c^{BA}_0N^B$ support links from network B to A, we find that under random attack with removal of fractions $(1-R^A)N^A$ and $(1-R^B)N^B$ nodes respectively, the percolating giant components of both networks at the end of the cascading failures, $μ^A_\infty$ and $μ^B_\infty$, are given by the percolation laws $μ^A_\infty = R^A [1-\exp{({-c^{BA}_0μ^B_\infty})}] [1-\exp{({-aμ^A_\infty})}]$ and $μ^B_\infty = R^B [1-\exp{({-c^{AB}_0μ^A_\infty})}] [1-\exp{({-bμ^B_\infty})}]$. In the limit of $c^{BA}_0 \to \infty$ and $c^{AB}_0 \to \infty$, both networks become independent, and the giant components are equivalent to a random attack on a single Erdős-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.
I. INTRODUCTION
The paper addresses cascading failures in coupled networks whose nodes may have multiple support relations, extending prior one-to-one dependency models. It develops a theoretical framework and finds agreement with simulations for coupled ER and scale-free networks.
- Motivation: Previous coupled-network studies assumed that each node depends on exactly one node in the other network.The paper identifies this one-to-one dependency condition as a limitation for modeling real systems.
- Model motivation: Multiple support relations allow a node to function when at least one of several support nodes in the other network remains connected.The same rule is applied in both directions between networks A and B.
- Real-world context: Power-grid and communication-network interdependence illustrates how failures can propagate across support relations and contribute to a blackout.The passage describes damage to Italy’s communication control system after power-station failures, followed by further grid fragmentation.
- Motivation: Coupled-network failures can recurse between networks, producing cascades that differ from single-network behavior.Failures in one network can disable dependent nodes in the other, whose failures feed back recursively.
- Contribution: The framework models random support-dependent relationships and agrees well with numerical simulations for coupled Erdős-Rényi and scale-free networks.It extends earlier one-to-one relations to multiple Poissonian dependent-support relations.
- Main result: For coupled ER networks, the stable-state giant components follow a simple law that becomes single-network random percolation when support links are sufficiently numerous.The coupled system generally has a first-order percolation transition, while the large-support limit yields a second-order transition.
II. THE MODEL
The model couples two networks through randomly assigned, directed support relations while requiring every functioning node to have at least one support node and intra-network giant-component connectivity. Random attacks initiate a recursive failure process that stops at a stable state when no further failures occur.
- Network structure: Two networks A and B have specified sizes and degree distributions, with intra-links connecting nodes within each network.The model permits general degree distributions P^A(k) and P^B(k).
- Support relations: Support inter-links are random, directed, and connect support nodes in one network to dependent nodes in the other.The initial inter-links are distributed randomly in both directions between A and B.
- Functioning rule: Each node may have zero, one, or several support nodes, but functioning requires at least one support node in the other network.Nodes lacking support inter-links or connection to their network’s giant component are treated as nonfunctioning.
- Initial attack: Random attacks remove fractions 1−R^A and 1−R^B of nodes from networks A and B, respectively.The attack fractions may differ between the two networks.
- Cascade process: At each cascade stage, nodes without support or giant-component connectivity fail, causing further failures in both networks.The process is illustrated using N^A = N^B = 7 with one randomly removed node in each network.
- Stable state: The cascade continues until no further node failure occurs in either network, defining the stable state.The attack order is chosen without loss of generality, with network A attacked before network B.
III. ANALYTICAL SOLUTION
The analysis tracks how support-dependent failures propagate recursively between two coupled networks until a stable state is reached. It derives stage-dependent giant-component equations and specializes them to coupled Erdős-Rényi networks.
- Stable state: The stable state is reached when both networks retain connected giant components whose nodes each have at least one support node in the other network.The cascade ends when no further failures occur under the connectivity and support requirements.
- Cascade dynamics: The cascade alternates analysis of networks A and B, updating their remaining support links at each stage.The nth stage in each network is treated as an updated version of its first-stage cascade, with support-link averages decreasing as the cascade progresses.
- Cascade dynamics: Random support-link assignment gives Poisson inter-link degree distributions, including nodes with zero initial support that are treated as failed.The model accounts for these unsupported nodes in addition to nodes removed by the initial random attack.
- Analytical framework: Generating functions describe the underlying network degree distributions and the giant components after effective random removals at each cascade stage.Because support-dependent relations are assumed uncorrelated with network properties, the support-link distributions remain Poisson with updated average degrees.
- Erdős-Rényi solution: For coupled Erdős-Rényi networks, the final giant-component fractions obey coupled percolation laws involving attack survival, support connectivity, and intra-network connectivity.The resulting equations are simple and relate to random percolation in a single Erdős-Rényi network.
- Erdős-Rényi solution: The coupled Erdős-Rényi systems have critical thresholds of RA, RB, and support-link parameters below which nonzero mutually connected giant components may not exist.The coupled-network transition is characterized as first order rather than the known second-order transition of ordinary percolation.
IV. NUMERICAL SIMULATIONS
Numerical simulations support the theoretical predictions for coupled Erdős–Rényi and scale-free networks across cascade stages and parameter settings. Finite support-link densities produce abrupt collapse, while large support densities approach single-network random-percolation behavior.
- Cascade fluctuations: Random realizations near the critical threshold split between convergence to non-zero giant components and complete fragmentation.At later cascade stages, deviations increase because of fluctuations in the realized giant-component fraction.
- Critical behavior: Increasing mutual support makes the coupled networks behave increasingly like independent networks.The critical retained fraction approaches 1/a, and the stable giant components approach the corresponding single-network behavior as support-link density grows.
V. CONCLUSIONS AND DISCUSSIONS
The paper extends cascade-of-failure analysis to coupled networks with random support-dependent relations, finding agreement with simulations on coupled ER and SF systems. Coupled networks generally exhibit first-order percolation transitions, becoming second-order only with many support links, and the framework can inform real-world coupled systems.
- The model extends prior work by considering random support-dependent relations between two coupled network systems.
- The theory agrees excellently with numerical simulations for coupled Erdős-Rényi and scale-free network systems.
- For coupled ER networks, the giant components follow a simple percolation law that approaches the single-network law as support links become numerous.
- Only the large-support limit yields a second-order percolation transition; the general coupled-network case shows a first-order phase transition.
- The model is intended to improve understanding of real-life coupled network systems.
- The framework can be extended to study robustness under non-random percolation, complementing related work on interdependent networks.