Abstract
Graphs are models of communication networks. This paper applies symbolic combinatorial techniques in order to characterize the interplay between two parameters of a random graph, namely its density (the number of edges in the graph) and its robustness to link failures. Here, robustness 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 the expected number of ways to get from s to t via two edge-disjoint paths of length ℓ in the classical random graph model G n,p by means of “symbolic” combinatorial methods. We then derive bounds on related threshold probabilities as functions of ℓ and n.
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