A long-range percolation model is a discrete percolation model in which 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 distinct elements 
and 
within some finite or countable metric space 
are independently connected by an edge 
with some probability 
that decays as a power of the
distance 
between them. In a common model, 
for non-nearest neighbors, with
and 
,
while nearest neighbors are connected with probability
1 (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.
