Reliability polynomials and link importance in networks

The reliability polynomial is a graph invariant which is of interest where graphs are used as models of systems such as communication networks, computer networks, and transportation networks. This paper examines the use of reliability polynomials to rank the edges in a graph in terms of overall importance to graph reliability. For a given edge e in the graph G, G-e and G*e denote the graph with the link deleted and contracted (respectively); p (0<p<1) is the common edge-reliability. The relative importance of two edges in the graph can be compared by computing the reliability polynomials of G-e and G*e for each of the 2 edges. If the comparisons are made via the reliability polynomials as functions of p with 0<p<1, one edge can be judged as more important than the other uniformly for all values of p. This is meaningful from the standpoint of network design, because it implies that relative importance of edges can be compared without prior knowledge of what the common edge-reliability is. Existing computational algorithms for deriving reliability polynomials make the approach feasible. The authors show the ranking of the edges obtained via reliability polynomial considerations to be more informative than rankings obtained by other combinatoric features of the networks. This partial ordering of the edges in the graph according to reliability importance is compared to other combinatoric measures of reliability based on considerations of minimal-size pathsets and cutsets. The examples demonstrate that the reliability polynomials often provide a more precise tool for ranking edges according to importance