  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.

Statistical Mechanics