On the roots of the node reliability polynomial
  Given a graph $G$ whose edges are perfectly reliable and whose nodes each operate independently with probability $p\in[0,1],$ the node reliability of $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the well studied \textit{all-terminal reliability}, where the nodes are perfectly reliable but the edges fail randomly. In sharp contrast to what is known about the roots of the all-terminal reliability polynomial, we show that the node reliability polynomial of any connected graph on at least three nodes has a nonreal polynomial root, the collection of real roots of all node reliability polynomials is unbounded, and the collection of complex roots of all node reliability polynomials is dense in the entire complex plane. 
Jason Brown and Lucas Mol
math.CO math.CA math.PR