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Heavy tails in dynamic flow networks: Universal explanation of their emergence

Agnieszka Janicka, Fiona Sloothaak, Maria Vlasiou, Bert Zwart

TL;DR

This work provides a universal, analytically tractable framework for overload cascades in multi-commodity flow networks, showing that heavy-tailed cascade costs inherit their Pareto tails from Pareto-tailed external inputs. Central to the theory are the catastrophe principle and a transformation linking input tail exponent $\alpha$ to output tail exponent $\alpha/\delta$ under $\delta$-scale-invariant, SRC cascade costs and cascade probabilities. The authors instantiate the framework with concrete flow/production/cascade-cost choices, proving satisfiability across centralized and decentralized settings and deriving tail behavior for power grids, traffic networks, and processing systems. The results unify previously domain-specific explanations and offer a broad, robust explanation for scale-free disruptions, with practical implications for capacity planning and targeted resilience interventions.

Abstract

Overload-induced cascading failures can cause extreme disruptions in a wide range of networked systems, such as power grids, transportation networks, or financial systems. Empirical studies across domains report that the size of such disruptions often follows a Pareto- or heavy-tailed distribution. While many models reproduce this scaling behavior, they are either tailored to specific domains or based on simplified mechanisms that overlook key aspects of overload cascading behavior. Hence, a general understanding of the mechanisms driving scale-free behavior in these settings remains incomplete. In this paper, we develop a universal and analytically tractable model of overload cascading failures on flow networks, offering a new perspective on how Pareto-tailed disruptions emerge across networks. Our framework shows, under mild assumptions, that heavy-tailed disruptions can arise naturally from Pareto-tailed external inputs, and it establishes a transformation law linking the input and output tail exponents. We further identify broad conditions under which the resulting cascade cost exhibits a heavy-tailed distribution and show that the mechanism is robust across several domains, including power transmission, traffic networks, and processing systems. Our results provide a unified explanation for the emergence of scale-free failures in overload-driven systems and connect previously disparate, application-specific models under a unified framework.

Heavy tails in dynamic flow networks: Universal explanation of their emergence

TL;DR

This work provides a universal, analytically tractable framework for overload cascades in multi-commodity flow networks, showing that heavy-tailed cascade costs inherit their Pareto tails from Pareto-tailed external inputs. Central to the theory are the catastrophe principle and a transformation linking input tail exponent to output tail exponent under -scale-invariant, SRC cascade costs and cascade probabilities. The authors instantiate the framework with concrete flow/production/cascade-cost choices, proving satisfiability across centralized and decentralized settings and deriving tail behavior for power grids, traffic networks, and processing systems. The results unify previously domain-specific explanations and offer a broad, robust explanation for scale-free disruptions, with practical implications for capacity planning and targeted resilience interventions.

Abstract

Overload-induced cascading failures can cause extreme disruptions in a wide range of networked systems, such as power grids, transportation networks, or financial systems. Empirical studies across domains report that the size of such disruptions often follows a Pareto- or heavy-tailed distribution. While many models reproduce this scaling behavior, they are either tailored to specific domains or based on simplified mechanisms that overlook key aspects of overload cascading behavior. Hence, a general understanding of the mechanisms driving scale-free behavior in these settings remains incomplete. In this paper, we develop a universal and analytically tractable model of overload cascading failures on flow networks, offering a new perspective on how Pareto-tailed disruptions emerge across networks. Our framework shows, under mild assumptions, that heavy-tailed disruptions can arise naturally from Pareto-tailed external inputs, and it establishes a transformation law linking the input and output tail exponents. We further identify broad conditions under which the resulting cascade cost exhibits a heavy-tailed distribution and show that the mechanism is robust across several domains, including power transmission, traffic networks, and processing systems. Our results provide a unified explanation for the emergence of scale-free failures in overload-driven systems and connect previously disparate, application-specific models under a unified framework.
Paper Structure (30 sections, 15 theorems, 166 equations, 2 figures, 2 tables)

This paper contains 30 sections, 15 theorems, 166 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Multi-commodity cascade model of flow network
  3. Notational conventions
  4. Flow network architecture
  5. Temporal dynamics
  6. Planning and operational phase
  7. Emergency phase
  8. Emergence of Pareto-tailed cascade cost
  9. Frameworks for cascading flow models
  10. Flow matrices and cost functions
  11. Source matrices
  12. Cascade cost functions
  13. Satisfiability results
  14. Application examples
  15. Power transmission systems
  16. ...and 15 more sections

Key Result

Proposition 1

Suppose that Assumption assumption holds for some $\delta>0$. Fix $\varepsilon>0$, and let $\overrightarrow{\mkern-4mu X }_{\mkern-7mu\max}$ be the largest weight amoung all vertices, i.e., $\overrightarrow{\mkern-4mu X }_{\mkern-7mu\max} = \max_{1\leq i\leq |\mathcal{V}|}\{\overrightarrow{\mkern-4m where $\alpha>0$ is the tail parameter of $\overrightarrow{\mkern-4mu X }\mkern-2mu$ as given in de

Figures (2)

  • Figure 1: Example of a small processing network. Here, red, black, and blue vertices correspond to the origin, processing, and destination vertices, respectively.
  • Figure 2: Optimal flow $\bm{F}^*(\bm{U}(\eta))$ on the $K_4$ graph for different values of parameter $\eta$. In each case, one unit of flow needs to be transferred to vertex 1 (green). Depending on the value of $\eta$, the flow originates from vertex 2 and/or 3 (orange). Negative flows represent movement opposite to the direction of the edge.

Theorems & Definitions (32)

  • Proposition 1: Catastrophe principle
  • Theorem 2: Tail of the cascade cost
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Lemma 5
  • proof
  • proof
  • Lemma 6
  • proof
  • ...and 22 more