Brito, Samurai, Thiago C. Nunes, Luciano R. da Silva and Constantino Tsallis

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study the scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r(ij)(-alpha A) (alpha(A) >= 0). We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient for d = 1, 2, 3, 4 and typical values of alpha(A). Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable alpha(A)/d. These observations confirm the existence of three regimes. The first one occurs in the interval alpha(A)/d is an element of [0, 1]; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a q exponential with q constant and above unity. The critical value alpha(A)/d = 1 that emerges in many of these properties is replaced by alpha(A)/d = 1/2 for the beta exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately-long-range interactions, and reflects in an index q monotonically decreasing with alpha(A)/d increasing from its critical value to a characteristic value alpha(A)/d similar or equal to 5. Finally, the third regime is Boltzmannian-like (with q similar or equal to 1) and corresponds to short-range interactions.