The expected number of pairs of connected nodes: Pair-connected reliability

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

doi:10.1016/0895-7177(93)90245-t

Cited by 9 publications

(5 citation statements)

References 14 publications

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“…The expectation of NCP has been called resilience in some works [19], in the same context as in this chapter. It has been explored in a few other works [20,21], always in the setting discussed here. So, formally, the resilience Res of our model is…”

Section: The Resilience Metricmentioning

confidence: 99%

Network Reliability, Performability Metrics, Rare Events and Standard Monte Carlo

Rubino

2022

Advances in Modeling and Simulation

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In this chapter, we consider static models in network reliability, that cover a huge family of applications, going way beyond the case of networks of any kind. The analysis of these models is in general #P-complete, and Monte Carlo remains the only effective approach. We underline the interest in moving from the typical binary world where components and systems are either up or down, to a multi-variate one, where the up state is decomposed into several performance levels. This is also called a performability view of the system. The chapter then proposes a different view of Monte Carlo procedures, where instead of trying to reduce the variance of the estimators, we focus on their time complexities. This view allows a first straightforward way of exploring these metrics. The chapter focuses on the resilience, which is the expected number of pairs of nodes that are connected by at least one path in the model. We discuss the ability of the mentioned approach for quickly estimating this metric, together with variations of it. We also discuss another side effect of the sampling technique proposed in the text, the possibility of easily computing the sensitivities of these metrics with respect to the individual reliabilities of the components. We show that this can be done without a significant overhead of the procedure that estimates the resilience metric alone.

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Section: The Resilience Metricmentioning

confidence: 99%

Network Reliability, Performability Metrics, Rare Events and Standard Monte Carlo

Rubino

2022

Advances in Modeling and Simulation

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“…From the work of Jackson [18] and Thomassen [28], it is known that the closure of the collection of real chromatic roots of all graphs is {0, 1} ∪ [32/27, ∞). In [5] it is shown that the closure of the real roots of independence polynomials of graphs is (−∞, 0], and in [3] it is proven that the closure of the collection of all real roots of all-terminal reliability polynomials is {0}∪ [1,2]. Since the coefficients of the Wiener polynomial of any connected graph are all positive, any real Wiener root lies in (−∞, 0].…”

Section: Density Of Real Wiener Rootsmentioning

confidence: 99%

“…Wiener polynomials of trees will be of particular interest to us, so we mention that the Wiener polynomial of a tree also arises in the context of network reliability. Given a graph G in which each edge is operational with probability p, the resilience or pair-connected reliability of G is the expected number of pairs of vertices of G that can communicate [1,11]. In particular, in a tree T , the probability that any pair of vertices u and v can communicate is simply p d (u,v) .…”

Section: Introductionmentioning

confidence: 99%

On the roots of Wiener polynomials of graphs

Brown

Mol

Oellermann

2018

Discrete Mathematics

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The Wiener polynomial of a connected graph G is defined as W (G; x) = x d(u,v) , where d(u, v) denotes the distance between u and v, and the sum is taken over all unordered pairs of distinct vertices of G. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order n is n 2 − 1, the maximum modulus among all roots of Wiener polynomials of trees of order n grows linearly in n. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely (−∞, 0], while in the case of trees, it contains (−∞, −1]. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.MSC 2010: 05C31, 05C12

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“…Since the specific vertices that are to communicate are rarely known at the time of network design, all-terminal reliability can be used to ensure that every subset of vertices can communicate among themselves. However, as discussed in [5,10], for many applications all-terminal reliability is too restrictive. It may well happen that most desired communications can proceed despite network disconnection.…”

Section: Definitions and Backgroundmentioning

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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The expected number of pairs of connected nodes: Pair-connected reliability

Cited by 9 publications

References 14 publications

Network Reliability, Performability Metrics, Rare Events and Standard Monte Carlo

Network Reliability, Performability Metrics, Rare Events and Standard Monte Carlo

On the roots of Wiener polynomials of graphs

Multiterminal resilience for series‐parallel networks

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