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Structural Robustness and Vulnerability of Networks

Alice C. Schwarze, Jessica Jiang, Jonny Wray, Mason A. Porter

Abstract

Networks are useful descriptions of the structure of many complex systems. Unsurprisingly, it is thus important to analyze the robustness of networks in many scientific disciplines. In applications in communication, logistics, finance, ecology, biomedicine, and many other fields, researchers have studied the robustness of networks to the removal of nodes, edges, or other subnetworks to identify and characterize robust network structures. A major challenge in the study of network robustness is that researchers have reported that different and seemingly contradictory network properties are correlated with a network's robustness. Using a framework by Alderson and Doyle~\cite{Alderson2010}, we categorize several notions of network robustness and we examine these ostensible contradictions. We survey studies of network robustness with a focus on (1)~identifying robustness specifications in common use, (2)~understanding when these specifications are appropriate, and (3)~understanding the conditions under which one can expect different notions of robustness to yield similar results. With this review, we aim to give researchers an overview of the large, interdisciplinary body of work on network robustness and develop practical guidance for the design of computational experiments to study a network's robustness.

Structural Robustness and Vulnerability of Networks

Abstract

Networks are useful descriptions of the structure of many complex systems. Unsurprisingly, it is thus important to analyze the robustness of networks in many scientific disciplines. In applications in communication, logistics, finance, ecology, biomedicine, and many other fields, researchers have studied the robustness of networks to the removal of nodes, edges, or other subnetworks to identify and characterize robust network structures. A major challenge in the study of network robustness is that researchers have reported that different and seemingly contradictory network properties are correlated with a network's robustness. Using a framework by Alderson and Doyle~\cite{Alderson2010}, we categorize several notions of network robustness and we examine these ostensible contradictions. We survey studies of network robustness with a focus on (1)~identifying robustness specifications in common use, (2)~understanding when these specifications are appropriate, and (3)~understanding the conditions under which one can expect different notions of robustness to yield similar results. With this review, we aim to give researchers an overview of the large, interdisciplinary body of work on network robustness and develop practical guidance for the design of computational experiments to study a network's robustness.
Paper Structure (131 sections, 46 equations, 7 figures, 7 tables)

This paper contains 131 sections, 46 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Defining robustness
  3. Related concepts
  4. Different name, but similar concept?
  5. Resilience.
  6. Stability.
  7. Reliability.
  8. Same name, but different concept?
  9. Performance.
  10. Mechanisms of robustness, resilience, and stability.
  11. Insensitivity of a system to parameter changes.
  12. Further reviews of robustness and related concepts
  13. Simple networks, subnetworks, and matrix representations of networks
  14. Networks and simple networks
  15. Neighbors and neighborhoods.
  16. ...and 116 more sections

Figures (7)

  • Figure 1: Specifications of a robustness problem for a networked system. In each column, we categorize widely used choices for a specification of a network-robustness problem. Gray lines connect two categories for different specifications if, to the best of our knowledge, scholars have chosen specifications from this pair of categories in a study of network robustness. We omit edges between concepts in non-consecutive chapters in this visualization.
  • Figure 2: Examples of formalizations of robustness and related concepts. In panel (a), we show performance--time curves for three example systems for which we use the inverse performance drop at the time $t_{\textrm{pert}}$ of perturbation as a measure of robustness. In panel (b), we use the inverse length of the time interval between $t_{\textrm{pert}}$ and the time $t_{\textrm{rec}}$ of complete performance recovery as a measure of resilience. In panel (c), we show state--time curves for three example systems for which we use the inverse decay rate of a small displacement as a measure of stability. In panel (d), we show performance over a system parameter $r$ for three example systems for which we use the slope of the curve close to a reference value $r_{\textrm{ref}}$ of $r$ as a measure of persistence. In panel (e), we illustrate that a single system can respond differently to different perturbations, and we use the fraction of perturbations after which the system recovers a minimal performance given by $X_{\textrm{min}}$ by time $t_{\textrm{check}}$.
  • Figure 3: Subgraphs and walks on a network. We show an example network with $N = 8$ nodes and $m = 8$ edges in a black box in the bottom-left corner. In the top half of the figure, we show several examples of subgraphs, connected subgraphs, node-induced subgraphs, paths, and connected components of the example graph. In the bottom-left corner, we show several examples of walks and closed walks on the example network.
  • Figure 4: A map of a connected transportation network from the video game Mini Metro. White symbols with black borders indicate train stations. Small black symbols indicate passengers who are waiting to embark on a train. The shape of a station indicates its type. The shape of a passenger indicates their desired destination type. (Copyright © 2024 Dinosaur Polo Club. Mini Motorways and Mini Metro are trademarks of Dinosaur Polo Club. All rights reserved.)
  • Figure 5: A map of a fragmented transportation network from the video game Mini Metro. White symbols with black borders indicate train stations. Small black symbols indicate passengers who are waiting to embark on a train. The shape of a station indicates its type. The shape of a passenger indicates their desired destination type. (Copyright © 2024 Dinosaur Polo Club. Mini Motorways and Mini Metro are trademarks of Dinosaur Polo Club. All rights reserved.)
  • ...and 2 more figures