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Scale-free cascading failures: Generalized approach for all simple, connected graphs

Agnieszka Janicka, Fiona Sloothaak, Maria Vlasiou

TL;DR

This work characterizes cascade sequences of failures in the asymptotic regime and yields a universal theorem applicable to all simple, connected graphs, revealing that when a cascade leads to network disconnections, the total failure size exhibits a scale-free tail inherited from the input characteristics.

Abstract

Cascading failures, wherein the failure of one component triggers subsequent failures in complex interconnected systems, pose a significant risk of disruptions and emerge across various domains. Understanding and mitigating the risk of such failures is crucial to minimize their impact and ensure the resilience of these systems. In multiple applications, the failure processes exhibit scale-free behavior in terms of their total failure sizes. Various models have been developed to explain the origin of this scale-free behavior. A recent study proposed a novel hypothesis, suggesting that scale-free failure sizes might be inherited from scale-free input characteristics in power networks. However, the scope of this study excluded certain network topologies. Here, motivated by power networks, we strengthen this hypothesis by generalizing to a broader range of graph topologies where this behavior is manifested. Our approach yields a universal theorem applicable to all simple, connected graphs, revealing that when a cascade leads to network disconnections, the total failure size exhibits a scale-free tail inherited from the input characteristics. We do so by characterizing cascade sequences of failures in the asymptotic regime.

Scale-free cascading failures: Generalized approach for all simple, connected graphs

TL;DR

This work characterizes cascade sequences of failures in the asymptotic regime and yields a universal theorem applicable to all simple, connected graphs, revealing that when a cascade leads to network disconnections, the total failure size exhibits a scale-free tail inherited from the input characteristics.

Abstract

Cascading failures, wherein the failure of one component triggers subsequent failures in complex interconnected systems, pose a significant risk of disruptions and emerge across various domains. Understanding and mitigating the risk of such failures is crucial to minimize their impact and ensure the resilience of these systems. In multiple applications, the failure processes exhibit scale-free behavior in terms of their total failure sizes. Various models have been developed to explain the origin of this scale-free behavior. A recent study proposed a novel hypothesis, suggesting that scale-free failure sizes might be inherited from scale-free input characteristics in power networks. However, the scope of this study excluded certain network topologies. Here, motivated by power networks, we strengthen this hypothesis by generalizing to a broader range of graph topologies where this behavior is manifested. Our approach yields a universal theorem applicable to all simple, connected graphs, revealing that when a cascade leads to network disconnections, the total failure size exhibits a scale-free tail inherited from the input characteristics. We do so by characterizing cascade sequences of failures in the asymptotic regime.
Paper Structure (21 sections, 17 theorems, 160 equations, 2 figures, 1 table)

This paper contains 21 sections, 17 theorems, 160 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and notation
  3. Main results
  4. Main theorem and roadmap of the proof
  5. Proof of Theorem \ref{['mainThm']}
  6. Proofs and supporting results
  7. Proof of Lemma \ref{['AltOptSol']}
  8. Proof of Proposition \ref{['propProjection']}
  9. Proof of Lemma \ref{['firstColumnAgeneral']}
  10. Proof of Proposition \ref{['propositionKKT']}
  11. Proof of Lemma \ref{['cascadeEpsilon']}
  12. Proof of Lemma \ref{['asymptoticS']}
  13. Uniqueness of flow exceedance
  14. Impact of the tie-breaking rule
  15. Conclusions
  16. ...and 6 more sections

Key Result

Lemma 1

Given a graph $\mathcal{G}$ and $\gamma\in \Gamma$, it is possible to adjust the orientation of edges in the graph so that $Vd \geq 0$ for all $\varepsilon\geq 0$ sufficiently small, where $d = e_1 + \varepsilon \gamma$.

Figures (2)

  • Figure 1: Example of a graph with a positive probability of a non-unique cascade sequence. Specifically, if node 1 has the largest demand, then the failure of edge (7) leads to the subsequent failure of edges (11) and (10). At this stage of the cascade, edges (5), (6), and (9) are the maximizers of the flow exceedance.
  • Figure 2: Example of a graph where non-unique relative flow exceedance occurs with positive probability.

Theorems & Definitions (36)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 2: Tail of the total cascade size
  • Lemma 3: Lemma IV.1 of Nesti_2020
  • Lemma 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Lemma 8
  • ...and 26 more