Roots of two‐terminal reliability polynomials

Abstract

Assume that the vertices of a graph G are always operational, but the edges of G are operational independently with probability p ∈ [0, 1]. For fixed vertices s and t, the two‐terminal reliability of G is the probability that the operational subgraph contains an (s, t)‐path, while the all‐terminal reliability of G is the probability that the operational subgraph contains a spanning tree. Both reliabilities are polynomials in p, and have very similar behavior in many respects. However, unlike all‐terminal reliability polynomials, little is known about the roots of two‐terminal reliability polynomials. In a variety of ways, we shall show that the nature and location of the roots of two‐terminal reliability polynomials have significantly different properties than those held by roots of the all‐terminal reliability polynomials.

Complex Network-Based Resilience Capability Assessment for a Combat System of Systems

“…There are currently several exact solution methods for the two-terminal connectivity of networks [27][28][29], and reference [30] has proven that the exact solution to the problem in networks is an NP-hard problem. In addition, the existing analytical solution methods mainly focus on equiprobable networks.…”

Network reliability: Heading out on the highway

“…We close this section with an intriguing observation concerning the roots of two‐terminal reliability polynomials [54]. Suppose we have a graph H with two identified (distinct) terminals, u and v , and let the two‐terminal reliability of H with terminals u and v be h = h ( p ).…”

Reliability polynomials of consecutive‐k‐out‐of‐n:Fsystems have unbounded roots

“…Although the reliability of a network Rel(N, p) is defined for p ∈ [0, 1], being a polynomial, it is completely characterized by its roots in the complex plane C [19]. As reliability of networks (as well as circuits, be they CMOS or quantum based) has lately taken center stage, the location of the roots of various reliability polynomials has become a highly explored topic in recent decades (see, for instance, [5][6][7]13]). The roots of all-terminal reliability polynomials were analyzed for the first time by Brown and Colbourn [5], who conjectured (in 1992) that all these roots lie inside the closed unit disk centered at z = 1.…”

“…Although the reliability of a network $$$ Rel\left(N,p\right) $$$ is defined for $$$ p\in \left[0,1\right] $$$, being a polynomial, it is completely characterized by its roots in the complex plane $$$ \mathbb{C} $$$ [19]. As reliability of networks (as well as circuits, be they CMOS or quantum based) has lately taken center stage, the location of the roots of various reliability polynomials has become a highly explored topic in recent decades (see, for instance, [5‐7, 13]).…”

“…Since 2006 the roots of two‐terminal reliability polynomials have also started to be scrutinized by Christian Tanguy, in a series of papers tackling particular networks [21‐23]. Recently, Brown and DeGagné [6] proved that the closure of two‐terminal roots contains the closed unit disks centered at $$$ z=0 $$$ and $$$ z=1 $$$, but the question if the two‐terminal roots are bounded also remains an open problem [6, 14].…”