On the nonexistence of uniformly optimal graphs for pair‐connected reliability

Amin, Ashok T.; Siegrist, Kyle; Slater, Peter J.

doi:10.1002/net.3230210307

Cited by 12 publications

(4 citation statements)

References 11 publications

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“…Finally, we look at a network design application. In order to select a network with good resilience, we may ask for a network to exhibit relatively high k-resilience for each admissible value of k. The specific question for k = 2 is treated in [4,6,11]. In the more general situation, we may want to consider multiple k-resilience measures on the same network.…”

Section: Network Designmentioning

confidence: 99%

Multiterminal resilience for series‐parallel networks

Farley

Colbourn

2007

Networks

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Network resilience measures the average two-terminal reliability (connectedness) of a network. Multiterminal resilience extends this measure to any k vertices; it is the average k -terminal reliability of a network. This generalizes two well-studied network connectedness measures. Calculating multiterminal resilience on general networks encompasses all-terminal reliability and thus is NP-hard. Multiterminal resilience is examined on undirected series-parallel networks, and an efficient (polynomial time) algorithm is developed for calculating the resilience for every k . Applications in mobile ad hoc and sensor networks are outlined.

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Section: Network Designmentioning

confidence: 99%

Multiterminal resilience for series‐parallel networks

Farley

Colbourn

2007

Networks

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“…Specifically, consider the class G n,m of all unlabelled (n, m) graphs. When k = 2 (pairconnected reliability) and p = 1 (the IID edge-failure model), it is shown in [4] that, for n 6 m 6 n 2 − 2, there does not exist a graph that is optimal for all values of ρ. When k = 2 and ρ = 1 (the IID vertex-failure model), it is shown in [3] that for some values of (n, m) there exists a graph that is optimal for all values of p, but for other values of (n, m) there does not exist such a graph.…”

Section: From Theorems 51 and 54 It Follows Thatmentioning

confidence: 99%

“…The all-terminal reliability is a global reliability measure. Another global performance measure is the pair-connected reliability or the resilience of the graph: the expected number of pairs of vertices that remain connected after the element failures (see for example [1]- [4], [8], [10], and the references in these papers). There have been two main lines of investigation in network reliability.…”

Section: Introductionmentioning

confidence: 99%

Expected Value Expansions in Random Subgraphs with Applications to Network Reliability

Siegrist¹

1998

Combinator. Probab. Comp.

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Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

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“…By way of contrast, the most reliable series-parallel network has a nice characterization [165]. Optimal graphs for resilience [8,9] and two-terminal reliability [26] have also been studied. In [36,168], uniformly least reliable connected graphs for all-terminal reliability are studied.…”

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confidence: 99%

Network reliability: Heading out on the highway

et al. 2020

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A variety of probabilistic notions of network reliability of graphs and digraphs have been proposed and studied since the early 1950s. Although grounded in the engineering and logistics of network design and analysis, the research also spans pure and applied mathematics, with connections to areas as diverse as combinatorics and graph theory, combinatorial enumeration, optimization, probability theory, real and complex analysis, algebraic topology, commutative algebra, the design and analysis of algorithms, and computational complexity. In this paper we describe the landscape of various notions of network reliability, the roads well traveled, and some that appear likely to lead to meaningful and important journeys.

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On the nonexistence of uniformly optimal graphs for pair‐connected reliability

Cited by 12 publications

References 11 publications

Multiterminal resilience for series‐parallel networks

Multiterminal resilience for series‐parallel networks

Expected Value Expansions in Random Subgraphs with Applications to Network Reliability

Network reliability: Heading out on the highway

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