The diameter of a long range percolation graph

Abstract

One can model a social network as a long-range percolation model on a graph {0,1,... ,N}2. The edges (x,y) of this graph are selected with probability ≈ β/||x -y||s if ||x -y|| > 1, and with probability 1 if ||x -y|| = 1, for some parameters β, α > 0. That is, people are more likely to be acquainted with their neighbors than with people at large distance. This model was introduced by Benjamini and Berger [2] and it resembles a model considered by Weinberg in [6], [7]. We are interested in how small (probabilistically) is the diameter of this graph as a function of β and s, thus relating to the famous Milgram's experiment which led to the "six degrees of separation" concept. Extending the work by Benjamini and Berger, we consider a d-dimensional version of this question on a node set {0,1,..., N}d and obtain upper and lower bounds on the expected diameter of this graph. Specifically, we show that the expected diameter experiences phase transitions at values s = d and s = 2d. We compare the algorithmic implication of our work to the ones of Kleinberg, [6].