The probability distribution of the diameter of a network model consisting of a complete digraph with positive integer random edge costs is considered. Edge costs are chosen independently for each node pair according to common probability distributions for each edge direction. Bounds and some sharp limit results for the diameter distribution in the limit as the number of nodes tends to infinity are derived. In addition, limit bounds are determined for the distribution of the number of hops in a shortest path between two arbitrary nodes in the graph. Numerical examples are presented to illustrate these results.>

The probability distribution of the shortest path cost from a source node to an arbitrary destination node is considered for a random network model consisting of a complete digraph with positive integer random edge costs. Edge costs are chosen according to a common probability distribution for each direction. For this model, the joint distribution of the number of nodes which have a given sequence of shortest path costs from an arbitrary source node is determined explicitly. An expression is then obtained for the distribution of the shortest path cost between two arbitrary nodes using this joint distribution. The main result is the derivation of tight bounds and a sharp limit result for the distribution of the shortest path cost as the number of nodes tends to infinity. Numerical examples are presented to illustrate these results.>