On the roots of all-terminal reliability polynomials

Abstract

Given a graph G in which each edge fails independently with probability q ∈ [0, 1], the all-terminal reliability of G is the probability that all vertices of G can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in q whose roots (all-terminal reliability roots) were conjectured to have modulus at most 1 by Brown and Colbourn. Royle and Sokal proved the conjecture false, finding roots of modulus larger than 1 by a slim margin. Here, we present the first nontrivial upper bound on the modulus of any all-terminal reliability root, in terms of the number of vertices of the graph. We also find all-terminal reliability roots of larger modulus than any previously known. Finally, we consider the all-terminal reliability roots of simple graphs; we present the smallest known simple graph with all-terminal reliability roots of modulus greater than 1, and we find simple graphs with all-terminal reliability roots of modulus greater than 1 that have higher edge connectivity than any previously known examples.

On the reliability roots of simplicial complexes and matroids

“…So we see that the usual techniques from real analysis (such as the Intermediate Value Theorem) that one might use to try to locate roots outside the unit disk are going to fail for matroids, as then any such root must necessarily be nonreal. From [6] and [18] we know that there are (nonreal) reliability roots of some matroids (namely cographic matroids) that lie a little bit outside the disk. These matroids have dimension 13 or higher.…”

“…This graph has a root of its reliability polynomial whose modulus is approximately 1.0017, just barely outside of the disk. There have been subsequent attempts to find roots with larger modulus, with [6] pushing the roots out as far from the origin as 1.113, but still the examples seem few and far between, and still bounded.…”

On the roots of Wiener polynomials of graphs

“…For the allterminal reliability polynomial, it is suspected that the collection of all roots is bounded in modulus (as no roots have been found outside of the disk |z −1| ≤ 1.2). Currently the best known upper bound on the modulus of the roots for all graphs of order n is linear in n as shown in [6]. For the independence polynomials, the all-terminal-reliability polynomials and the chromatic polynomials, the extremal graphs that yield roots of maximum modulus are not known.…”

Features of modeling failures of recoverable complex technical objects with a hierarchical constructive structure

“…A large number of studies are devoted to calculating and predicting the reliability, as well as creating models (including simulation ones). The paper [3] offers a method of inheritance and development of modern methods of reliability estimation based on numerical modeling, based on dynamic Bayesian networks and numerical modeling. Thus, it overcomes the limitations of the analytical method and the multilevel synthesis method, and also provides an effective tool for assessing the reliability of complex dynamic systems.…”

“…3. In addition to the previously made changes, we also set the value of τ sub2 =0.2 h (and enter the corresponding changes in the database) for the element '2' (for example).…”