Optimal Paths in Probabilistic Networks

The random graph theory was initially proposed by P. Erdős and A. Rényi in the 1950s–1960s. More recently, B. Bollobás presented the first systematic and extensive treatment of results in the theory of random graphs. Associating to each edge of a random graph a real random variable, we obtain a probabilistic network. The determination of an optimal path between two nodes in a probabilistic network was first studied by H. Frank in 1969. Since then few theoretical results have become known, even though there is a recognizable applicability of this type of network to real problems, namely, to social and telecommunication networks. The mathematical model proposed in this paper maximizes the expected value of a utility function over a directed random network, where the costs related to the arcs are real random variables following Gaussian distributions. We consider the linear, quadratic, and exponential cases, presenting a theoretical formulation based on multi-criteria models as well as the resulting algorithms and computational tests.

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