Percolation in networks with long-range connections

Two-dimensional lattices of points are connected with long-range links, whose lengths are distributed according to P(r)∼r-α. By changing the decay exponent α one can go from d-dimensional short-range networks to ∞-dimensional networks topologically similar to random graphs. Percolation on these networks is numerically studied for systems of up to 107 sites. The shortest-path, fractal and chemical dimensions are determined at the critical threshold, as a function of the decay exponent α.