Factorization Theory in Diameter Constrained Reliability

In classical network reliability, the system under study is a network with perfect nodes and imperfect links that fail randomly and independently. The typical problem in the area is to find the probability that the resulting random graph is connected(or more generally, that a given subset K of nodes belong to the same connected component), called reliability. Although (and because) the exact reliability computation belongs to the class of NP-Hard problems, the literature offers many methods for this purpose, given the importance of the models. This paper deals with diameter-constrained reliability, a similar metric defined in 2001, inspired by delay-sensitive applications in telecommunications, where terminals are required to be connected by d hops or less, for a given positive integer d called diameter. Factorization theory is one of the fundamental tools of classical network reliability, and today it is a mature area. However, its extension to the diameter-constrained context requires at least the recognition of irrelevant links, which is an open problem. In this paper, irrelevant links are efficiently determined in the most basic and, by far, used case, the source-terminal one, where vertical bar K vertical bar = 2, thus providing a first step towards a Factorization theory in diameter constrained reliability. We also discuss the metric value in some special graphs, and we propose a specific Factoring algorithm. The paper is closed with a discussion of trends for future work.