Exact two-terminal reliability of some directed networks

TL;DR

This paper derives an exact analytical solution for the two-terminal reliability of directed networks, accounting for individual edge and node reliabilities, advancing beyond approximate methods for network reliability assessment.

Contribution

It introduces a transfer matrix-based exact solution for two-terminal reliability in directed, arbitrary-sized networks, including cases with identical edge and node reliabilities.

Findings

Exact solution involves transfer matrices considering individual reliabilities.

Directed edges significantly affect the zeros of reliability polynomials.

Network performance measures are similar, but structural transitions occur at specific reliability values.

Abstract

The calculation of network reliability in a probabilistic context has long been an issue of practical and academic importance. Conventional approaches (determination of bounds, sums of disjoint products algorithms, Monte Carlo evaluations, studies of the reliability polynomials, etc.) only provide approximations when the network's size increases, even when nodes do not fail and all edges have the same reliability p. We consider here a directed, generic graph of arbitrary size mimicking real-life long-haul communication networks, and give the exact, analytical solution for the two-terminal reliability. This solution involves a product of transfer matrices, in which individual reliabilities of edges and nodes are taken into account. The special case of identical edge and node reliabilities (p and rho, respectively) is addressed. We consider a case study based on a commonly-used configuration, and assess the influence of the edges being directed (or not) on various measures of network performance. While the two-terminal reliability, the failure frequency and the failure rate of the connection are quite similar, the locations of complex zeros of the two-terminal reliability polynomials exhibit strong differences, and various structure transitions at specific values of rho. The present work could be extended to provide a catalog of exactly solvable networks in terms of reliability, which could be useful as building blocks for new and improved bounds, as well as benchmarks, in the general case.