Average shortest path length of graphs of diameter 3

A network topology with low average shortest path length (ASPL) provides efficient data transmission while the number of nodes and the number of links incident to each node are often limited due to physical constraints. In this paper, we consider the construction of low ASPL graphs under these constraints by using stochastic local search (SLS) algorithms. Since the ASPL cannot be calculated efficiently, the ASPL is not suitable for the evaluation function of SLS algorithms. We first derive an equality and bounds for the ASPL of graphs of diameter 3. On the basis of the simplest upper bound of the ASPL, we propose to use 3Δ + 2□ as the evaluation function for graphs of diameter 3 where Δ and □ denote the number of triangles and squares in a graph, respectively. We show that the proposed evaluation function can be evaluated in O(1) time as the number of nodes and the maximum degree tend to infinity by using some data tables. By using the simulated annealing with the proposed evaluation function, we construct low ASPL regular graphs of diameter 3 with 10 000 nodes.