Toughness in Graphs – A Survey

Bauer, Douglas C.; Broersma, Hajo; Schmeichel, E. F.

doi:10.1007/s00373-006-0649-0

Cited by 107 publications

(92 citation statements)

References 125 publications

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“…Since, for example, instead of requiring that the number of edges on average is 10, we now require that the square of the number of edges is 100 on average-we do not expect this to cause differences that are too big. Indeed, we can show that due to the fact that the new MaxEnt ensemble is concentrated around ⎯⎯ , the differences are actually rather small (see the last paragraph on pg 3 of our published paper [17]), and the new model still respects (10) with good approximation. Fig 33 shows the corrected MaxEnt distribution by edge-number (black) for the case of the nefarious network.…”

Section: Can We Eliminate Degeneracy?mentioning

confidence: 90%

“…As defined before by us also as part of this project [6], [7], the vulnerability backbone of a network is the smallest percolating/connected subgraph whose removal minimizes the size of the largest component in the graph remaining after the removal. One can show that finding the vulnerability backbone is an NP-hard problem, similar to computing the toughness measure of the graph [10]. Thus, we must work with heuristic algorithms.…”

Section: Network Vulnerability and Shatteringmentioning

confidence: 99%

See 1 more Smart Citation

Understanding the Fundamental Principles Underlying the Survival and Efficient Recovery of Multi-Scale Techno-Social Networks Following a WMD Event (A)

Toroczkai¹,

Vespignani²

2016

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The public reporting burden for this collection of information is estimated to average 1 hour per respon se. including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information . Send comments regarding this burden estimate or any other aspect of this collection of information. including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information . 1215 Jefferson Davis Highway, Sutte 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstand ing any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currentiy valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM· I. Executive summaryMotivation: Understanding the fundamental principles underlying the functional robustness of Techno-Social Networks (TSN-s) is a tall, but inevitable challenge. As the society becomes increasingly dependent on infrastructure transport networks, the implications of large-scale disruptions (e.g., due to WMD events) of these systems also become increasingly important. In our ever more connected and interdependent world, small perturbations can have far reaching effects. Large-scale events, such as cascading failures (e.g., the East coast blackout on Aug. 14, 2003) or epidemics (SARS and most recently Ebola) are typical to complex networks, and they serve as warnings about the type of behaviors one can expect in strongly interconnected systems. As a result, we demand ever-increasing predictive power and ability to evaluate the impact of potential failures. There has been, however, little to no mathematical and analytical methodology that would allow a realistic assessment of the consequences of large-scale failures in critical infrastructure TSN-s, largely due to two major aspects of these networks: i) Multivariate and heterogeneous constraints. Real-world TSNs typically operate under a wide range of constraints and limitations imposed both from the technical (physical) side and the social (usage) side. Physical constraints include geographical embeddedness; energy limitations (transportation costs, battery life); device capacity, latency and bandwidth, while usage constraints include software limitations, communication protocols and regulations. To advance empirical applications, network models will need to address these constraints since they strongly influence the response and failure modes of a network in case of WMDs. The dynamics of constrained complex networks, especially if they are directed, weighted and hierarchical are not well understood. The adaptive capabilities, including recovery of TSNs facing major destructive events has been a relatively uncharted territory. ii) Multiscale nature. Most of the TSNs are characterized by inherently complex processes involving many different scales in time a...

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Section: Can We Eliminate Degeneracy?mentioning

confidence: 90%

Section: Network Vulnerability and Shatteringmentioning

confidence: 99%

Understanding the Fundamental Principles Underlying the Survival and Efficient Recovery of Multi-Scale Techno-Social Networks Following a WMD Event (A)

Toroczkai¹,

Vespignani²

2016

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“…We obtain here sufficient and necessary conditions for an adversarial infection to become pandemic (i.e., infect all nodes) by relating existence of such strategies to the notion of toughness of the graph. The notion of graph toughness was first introduced in order to study conditions for the existence of Hamiltonian cycles in graphs, see [4]. Given an undirected graph G = (V, E) and a subset of nodes S, let |S| be the size of S and c G (S) be the number of connected components in the graph induced on V \ S obtained from G by deleting all nodes in the set S. The toughness of the graph G, where G is not the complete graph, denoted by τ (G) is defined as follows…”

Section: Pandemic Infectionsmentioning

confidence: 99%

On the Power of Adversarial Infections in Networks

Brautbar

Draief

Khanna

2013

Lecture Notes in Computer Science

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Abstract. Over the last decade we have witnessed the rapid proliferation of online networks and Internet activity. Such activity is considered as a blessing but it brings with it a large increase in risk of computer malware -malignant software that actively spreads from one computer to another. To date, the majority of existing models of malware spread use stochastic behavior, when the set of neighbors infected from the current set of infected nodes is chosen obliviously. In this work, we initiate the study of adversarial infection strategies which can decide intelligently which neighbors of infected nodes to infect next in order to maximize their spread, while maintaining a similar "signature" as the oblivious stochastic infection strategy as not to be discovered. We first establish that computing an optimal and near-optimal adversarial strategies is computational hard. We then identify necessary and sufficient conditions in terms of network structure and edge infection probabilities such that the adversarial process can infect polynomially more nodes than the stochastic process while maintaining a similar "signature" as the oblivious stochastic infection strategy. Among our results is a surprising connection between an additional structural quantity of interest in a network, the network toughness, and adversarial infections. Based on the network toughness, we characterize networks where existence of adversarial strategies that are pandemic (infect all nodes) is guaranteed, as well as efficiently computable.

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“…The best available lower bound is due to Bauer, Broersma and Veldman [3] who constructed non-Hamiltonian graphs with toughness arbitrarily close to 9 4 . Partial results related to Chvátal's conjecture have been obtained in various restricted classes of graphs (see the survey [2] for details). A number of these results concern chordal graphs.…”

Section: Introductionmentioning

confidence: 99%