Network reliability: Heading out on the highway

Abstract

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.

Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

“…From the earliest days of network reliability, researchers have tried to develop algorithms to efficiently compute the reliability of graphs/networks. In a recent survey paper [1], Brown et al have rediscovered the way of this research domain, by shedding light on some of the theoretical advances made in the past as well as the new directions. This scientific adventure started with the work of Moore and Shannon [2,3] and von Neumann [4], when the first foundations of the field were settled.…”

Generalized Convexity Properties and Shape Based Approximation in Networks Reliability

“…From the earliest days of network reliability, researchers have tried to develop algorithms to efficiently compute the reliability of graphs/networks. In a recent survey paper [1], Brown et al have rediscovered the way of this research domain, by putting into light some of the theoretical advances made in the past, as well as the new directions. This scientific adventure started with the work of Moore and Shannon [16,17] and von Neumann [23], when the first foundations of the field were settled.…”

Maximal Intervals of Decrease and Inflection Points for Node Reliability

“…Many other reliability models both of undirected and directed graphs, such as two-terminal reliability, K-terminal reliability and reachability (see, for example, [3]) have the same expected property. For a recent discussion of reliability in variants, see [1].…”

“…This begs the question of how pathological node reliability can be with respect to monotonicity -can there be multiple maximal intervals of decrease in [0, 1]? This questions, along with the question of whether the number of inflection points in [0, 1] for node reliability, were raised in [1,2] (it was previously known that there Figure 1: A plot of nRel(P 5 ; p) = 3p 5 − 12p 4 + 21p 3 − 16p 2 + 5p. is a graph whose node reliability has three points of inflection in the interval).…”