Roots of Two-Terminal Reliability

TL;DR

This paper investigates the roots of two-terminal reliability polynomials in graphs, revealing that their properties differ significantly from those of all-terminal reliability roots, with implications for understanding network robustness.

Contribution

The study provides new insights into the nature and location of roots of two-terminal reliability polynomials, contrasting them with all-terminal reliability roots.

Findings

Two-terminal reliability roots have distinct properties from all-terminal roots.

The roots' location and nature differ significantly from those of all-terminal reliability.

The paper advances understanding of polynomial roots in network reliability models.

Abstract

Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ are operational independently with probability $p \in[0,1]$. For fixed vertices $s$ and $t$, the \emph{two-terminal reliability} of $G$ is the probability that the operational subgraph contains an $(s,t)$-path, while the \emph{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 behaviour in many respects. However, unlike all-terminal reliability, little is known about the roots of two-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.