Scaling properties of d-dimensional complex networks.

Abstract

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}^{-α_{A}} (α_{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 α_{A}. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable α_{A}/d. These observations confirm the exist- ence of three regimes. The first one occurs in the interval α_{A}/d∈[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 α_{A}/d=1 that emerges in many of these properties is replaced by α_{A}/d=1/2 for the β 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 α_{A}/d increasing from its critical value to a characteristic value α_{A}/d≃5. Finally, the third regime is Boltzmannian-like (with q≃1) and corresponds to short-range interactions.