Long-Range Percolation Model

Intuitively, a -dimensional discrete percolation model is said to be long-range if direct flow is possible between pairs of graph vertices or graph edges which are "very distant" (Grimmett 1999). This is in contrast to the more-studied cases of bond percolation and site percolation, the standard models for which allow flow only between adjacent edges and vertices, respectively.

To make this intuition more precise, some authors describe long-range percolation to be a model in which any two elements  and  within some metric space  are connected by an edge  with some probability  where  is inversely proportional to the distance  between them (Coppersmith et al. 2002).

Besides simply extending the classical models of percolation on regular point lattices, the study of long-range percolation allows one to model a number of significant real-world scenarios for which classical discrete models are ill-adapted, e.g.,social networking.

REFERENCES

Coppersmith, D.; Gamarnik, D.; and Sviridenko, M. "The Diameter of a Long-Range Percolation Graph." In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2002.

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.