Scaling properties of d-dimensional complex networks

TL;DR

This study investigates the scaling properties of $d$-dimensional geographically located networks with preferential attachment, revealing three regimes based on the ratio $rac{ ext{alpha}_A}{d}$, characterized by different interaction ranges and degree distribution behaviors.

Contribution

It introduces a comprehensive numerical analysis of $d$-dimensional geographically located networks, identifying distinct regimes and scaling laws based on the ratio $rac{ ext{alpha}_A}{d}$, which was not previously characterized.

Findings

Three regimes identified based on $rac{ ext{alpha}_A}{d}$ ratio.

Degree distribution follows a $q$-exponential in the first two regimes.

Critical values of $rac{ ext{alpha}_A}{d}$ are 1 and 1/2 for different properties.

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 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.