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 $r_{ij}^{-{\alpha }_A}({\alpha }_A\ge 0)$. 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 ${\alpha }_A\mathrm{/}d$. These observations confirm the existence of three regimes. The first one occurs in the interval ${\alpha }_A\mathrm{/}d\in [0,1]$; 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 ${\alpha }_A\mathrm{/}d=1$ that emerges in many of these properties is replaced by ${\alpha }_A\mathrm{/}d=1\mathrm{/}2$ 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 ${\alpha }_A\mathrm{/}d$ increasing from its critical value to a characteristic value ${\alpha }_A\mathrm{/}d\simeq 5$. Finally, the third regime is Boltzmannian (with $q=1$), and corresponds to short-range interactions.