Trade-Offs between Density and Robustness in Random Interconnection Graphs

Graphs are models of communication networks. This paper applies combinatorial and symbolic-analytic techniques in order to characterize the interplay between two parameters of a random graph: its density (the number of edges in the graph) and its robustness to link failures, where robustness here means multiple connectivity by short disjoint paths. A triple (G, s, t), where G is a graph and s, t are designated vertices, is called ` − robust if s and t are connected via at least two edge-disjoint paths of length at most `. We determine here the expected number of ways to get from s to t via two edge-disjoint paths in the random graph model Gn,p. We then derive bounds on related threshold probabilities pn,` as functions of ` and n.