Network transformations and bounding network reliability

Abstract

Three transformations on networks that reduce the all-terminal network reliability (probability of connectedness) of a network are shown not to increase any coefficient in one form of the reliability polynomial of the network. These transformations yield efficiently computable lower bounds on each coefficient of the reliability polynomial. A further transformation due to Lomonosov is shown not to decrease any coefficient in the reliability polynomial, leading to an efficiently computable upper bound on each coefficient. The resulting bounds on coefficients can, in turn, be used to obtain a substantial improvement on the Ball-Provan strategy for computing lower and upper bounds on the all-terminal reliability. 0 7993 John Wiley & Sons, Inc. NETWORKS AND RELIABILITYA simple model of reliability in communications networks is a multigraph with node set V and edge multiset E; an edge e is an unordered pair of two distinct nodes, and multiple (parallel) edges between the same two nodes are permitted. Loops are omitted without further comment.Each vertex is always present, but edges operate independently, each with the same probability p . The all-terminal reliability Rel(G,p) of such a multigraph G is the probability that G is connected when each of its edges operates independently with probability p.When G contains no multiple edges, it is a simple graph or network.All-terminal reliability by no means captures all of the ingredients of the observed reliability of a network, *On leave from University of Waterloo, Canada.ton leave from University of Saskatchewan, Canada.but has been widely studied as it serves to indicate the suitability of a network topology for supporting network communication.The computation of all-terminal reliability appears to be a computationally difficult problem; Provan and Ball [16] showed that it is #P-complete, and Vertigan [24] showed that it remains #P-complete even for planar networks. For this reason, significant efforts have been made to obtain lower and upper bounds on allterminal reliability that are efficiently computable. We explore this problem further here.Related to the analysis problem of determining (or bounding) all-terminal reliability is the synfhesis problem: Given a number n of nodes and a number m of edges, what is the most (least) reliable connected multigraph (simple graph) with n nodes and m edges? Synthesis and analysis questions have for the most part been treated separately, using different types of techniques. However, it is important to observe that the least (most) reliable network on n nodes and m edges

On a poset of trees

“…Let x, y be two vertices of some graph G and let G be the graph obtained from G by erasing the edges between y and N (y)\(N (x)∪{x}) and adding the edges between x and N (y)\(N (x)∪{x}). This transformation has many nice properties: it increases the spectral radius and decreases the number of spanning trees [1,11]. This transformation can be applied to any graph, but if we consider it as a transformation on trees we have to make a restriction on x and y, namely they should have distance at most 2 in order to obtain a connected G as a result; in this case G will be also a tree.…”

Undirected simple connected graphs with minimum number of spanning trees

“…Furthermore, it is known that if shift(G, v, w) = G, then G is a threshold graph [1,3,4,11]. These are the graphs H = H(n; d 1 , d 2 , .…”

“…It was shown in [1,3,4] that every simple connected graph G can be transformed into a threshold graph H using a series of shift(G, v, w) transformations. Consequently:…”

Non-Stanley bounds for network reliability

“…This corollary is of use in bounding individual terms in the H-vector of a graph knowing only the number of spanning trees. One striking example is obtained by examining the H-vector of the 1979 Arpanet (see [8]). That network has 272,816,563,831 trees, and the best lower and upper bounds found by network transformations for/-/11 are 13,884,089,682 and 185,684,104,966 respectively.…”

“…Second, one can employ transformations that have a predictable effect on the H-vector; when such transformations reduce the graph to a simply analyzed structure (such as a series-parallel graph), one can extract bounds on the original H-vector from that of the reduced graph. This approach is taken, for example, in [8] and [4]. The third method is to develop inequalities among the terms of the H-vector that hold for any cographic matroid.…”