  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. We
study analytic properties of the node reliability on the interval $[0,1]$
including monotonicity, concavity, and fixed points. Our results show a stark
contrast between this model of network robustness and models that arise from
coherent set systems (including all-terminal, two-terminal and K-terminal
reliability).
