  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.

Combinatorics