Are We Connected? Optimal Determination of Source–Destination Connectivity in Random Networks

This paper investigates the problem of optimally determining source-destination connectivity in random networks. Viewing the network as a random graph, we start by investigating the Erdos–Renyi (ER) graph, as well as a structured graph where, interesting, the problem appears to be open. The problem examined is that of determining whether a given pair of nodes, a source <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>, and a destination <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> are connected by a path. Assuming that at each step one edge can be tested to see if it exists or not, we determine an optimal policy that minimizes the total expected number of steps. The optimal policy has several interesting features. In order to establish the connectivity of <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>, a policy needs to check all edges on some path to see if they all exist, but to establish the disconnectivity it has to check all edges on some cut to see if none of them exists. The optimal policy has the following form. At each step, it examines the condensation multigraph formed by contracting each known connected component to a single node, and then checks an edge that is simultaneously on a shortest <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> path as well as in a minimum <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> cut. Among such edges, it chooses that which lead to the most opportunities for connection. Interestingly, for an ER graph with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> nodes, where there is an edge between two nodes with probability <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>, the optimal strategy does not depend on <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, even though the entire graph itself undergoes a sharp transition from disconnectivity to connectivity around <inline-formula> <tex-math notation="LaTeX">$p = \ln n/n$ </tex-math></inline-formula>. The policy is efficiently implementable, requiring no more than <inline-formula> <tex-math notation="LaTeX">$30\log _{2} n$ </tex-math></inline-formula> operations to determine which edge to test next. The result also extends to some more general graphs and, meanwhile, provide useful insights into the connectivity determination in random networks.

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