Expected Value Expansions in Random Subgraphs with Applications to Network Reliability

Abstract

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

Distinguished vertices in probabilistic rooted graphs

“…Then the expected number of vertices reachable from the root will be a polynomial in p that we denote EV(G; p), and this polynomial may be taken as a measure of the reliability of the rooted network. A related invariant has been studied by Colbourn [9] (called network resilience) and Siegrist [16] and also by Seignist and coworkers [2,3,17] (called pairconnected reliability). More motivation and background can © 2009 Wiley Periodicals, Inc. be found in [4].…”

Combinatorial Properties of a Rooted Graph Polynomial

“…The polynomial was introduced in [1] and [2], extended to antimatroids in [18], and applied to rooted graphs in [5,19]. A closely related polynomial, called pair connected reliability by Amin, Siegrist, and Slater in [3,4] and Siegrist in [21,22] and network resilience by Colbourn in [13], is motivated by the reliability polynomial. A similar polynomial has also been defined for (nonrooted) graphs [6,23].…”