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.