Inflection points for network reliability
Given a finite, undirected graph G (possibly with multiple edges), we assume that the vertices are operational, but the edges are each independently operational with probabilityp. The (all-terminal) reliability, \(\operatorname{Rel}(G,p)\), of G is the probability that the spanning subgraph of operational edges is connected. It has been conjectured that reliability functions have at most one point of inflection in (0,1). We show that the all-terminal reliability of almost every simple graph of order n has a point of inflection, and there are indeed infinite families of graphs (both simple and otherwise) with more than one point of inflection.
graph connected point of inflection all-terminal reliability