On the Reliability Roots of Simplicial Complexes and Matroids

TL;DR

This paper extends the concept of network reliability from graphs to simplicial complexes and matroids, analyzing the location of their reliability roots within the complex plane, especially inside the unit disk.

Contribution

It generalizes reliability polynomials to simplicial complexes and matroids and studies the conditions under which their roots lie inside the unit disk.

Findings

Roots of reliability polynomials for small complexes are inside the unit disk.

Some graphs have roots outside the unit disk, but such cases are rare.

The paper provides insights into the algebraic properties of reliability in higher structures.

Abstract

Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ fail independently with probability $q \in[0,1]$. The \emph{all-terminal reliability} of $G$ is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in $q$, and it was conjectured \cite{BC1} that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when, for small simplicial complexes and matroids, the roots fall inside the closed unit disk.