Scaling properties of d-dimensional complex networks

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 scaling properties of a wide class of \(d\)-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through \(r_{ij}^ {-\alpha_A} \;(\alpha_A \geq 0)\). 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 \(\alpha_A/d \in [0,1]\); 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 \(\alpha_A/d \simeq 5\). Finally, the third regime is Boltzmannian (with \(q=1\)), and corresponds to short-range interactions.