Limit distributions for the diameter and the shortest path hop count in random graphs with positive integer edge costs

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