Exact solutions for the two- and all-terminal reliabilities of a simple ladder network

The exact calculation of network reliability in a probabilistic context has been a long-standing issue of practical importance, but a di cult one, even for planar graphs, with perfect nodes and with edges of identical reliability p. Many approaches (determination of bounds, sums of disjoint products algorithms, Monte Carlo evaluations, studies of the reliability polynomials, etc.) can only provide approximations when the network’s size increases. We consider here a ladder graph of arbitrary size corresponding to real-life network con gurations, and give the exact, analytical solutions for the alland two-terminal reliabilities. These solutions use 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 , respectively | is solved. We show that the zeros of the two-terminal reliability polynomial exhibit structures which di er substantially for seemingly similar networks, and we compare the sensitivity of various edges. We discuss how the present work may be further extended to lead to a catalog of exactly solvable networks in terms of reliability, which could be useful as elementary bricks for a new and improved set of bounds or benchmarks in the general case.