A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

Abstract

We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is Θ(log (n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log (n)/log (log (n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β⋆log (n), where β⋆ ≈ 2.4626, there exists a connected Erds–Rnyi random graph that is embedded in our random graph, with high probability.