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The diameter of a long range percolation graph. (English) Zbl 1055.60095
Proceedings of the thirteenth annual ACM-SIAM symposium on discrete algorithms, San Francisco, CA, USA, January 6–8, 2002. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-513-X/pbk). 329-337 (2002).
Summary: 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,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 I. Benjamini and N. Berger [Random Struct. Algorithms 9, 102–111 (2001; Zbl 0983.60099)] and it resembles a model considered by J. Kleinberg [“The small-world phenomenon: An algorithmic perspective” (Cornell Comput. Sci. Techn. Rep. 99–1776, 1999) and Nature 406 (2000)]. 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 in his above, first quoted paper.
60K35Interacting random processes; statistical mechanics type models; percolation theory