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 ℓ - robust if s and t are connected via at least two edge-disjoint paths of length at most ℓ. We determine here the expected number of ways to get from s to t via two edge-disjoint paths in the random graph model G _n,p . We then derive bounds on related threshold probabilities pn;ℓ as functions of ℓ and n.