The diameter of a long range percolation graph.

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 approximate to beta/\\x - y\\(s) if \\x - y\\ > 1, and with probability \\x - y\\(s) for some parameters beta, s > 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 Kleinberg in [6], [7]. We are interested in how small (probabilistically) is the diameter of this graph as a function of 3 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, 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].