Effective dimensions in networks with long-range connections

One- and two-dimensional lattices of points are connected with long-range links, whose lengths are distributed according to P(r) r-μ. By changing the decay exponent μ one can go from d-dimensional short-range networks to ∞-dimensional networks topologically similar to random graphs. An effective dimension dchem(μ) can be defined in terms of the shortest-path properties of these networks. These effective dimensions dchem are calculated here in one and two dimensions, for system sizes of up to 107 points