Percolation of spatially constraint networks

We study how spatial constraints are reflected in the percolation properties of networks embedded in one-dimensional chains and two-dimensional lattices. We assume long-range connections between sites on the lattice where two sites at distance r are chosen to be linked with probability p(r) similar to r(-delta). Similar distributions have been found in spatially embedded real networks such as social and airline networks. We find that for networks embedded in two dimensions, with 2 < delta < 4, the percolation properties show new intermediate behavior different from mean field, with critical exponents that depend on d. For delta < 2, the percolation transition belongs to the universality class of percolation in Erdos-Renyi networks (mean field), while for delta > 4 it belongs to the universality class of percolation in regular lattices. For networks embedded in one dimension, we find that, for delta < 1, the percolation transition is mean field. For 1 < delta < 2, the critical exponents depend on delta, while for delta > 2 there is no percolation transition as in regular linear chains. Copyright (C) EPLA, 2011