Role of dimensionality in complex networks: Connection with nonextensive statistics

TL;DR

This paper explores how the dimensionality of geographically-located networks influences their degree distributions, revealing universal behaviors linked to nonextensive statistics and the ratio of attachment decay to dimension.

Contribution

It introduces a model of $d$-dimensional networks with distance-based preferential attachment and demonstrates universal $q$-exponential degree distribution behaviors depending on $rac{ ext{distance decay exponent}}{ ext{dimension}}$.

Findings

Degree distributions follow $q$-exponentials with universal dependence on $rac{ ext{alpha}_A}{d}$.

As $rac{ ext{alpha}_A}{d}$ increases, the distribution approaches the exponential limit ($q=1$).

Results verified numerically for dimensions 1 through 4.

Abstract

Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form $P(k) \propto e_q^{-k/\kappa}$, where the $q$-exponential form $e_q^z \equiv [1+(1-q)z]^{\frac{1}{1-q}}$ optimizes the nonadditive entropy $S_q$ (which, for $q\to 1$, recovers the Boltzmann-Gibbs entropy). We introduce and study here $d$-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through $r_{ij}^{-\alpha_A} \; (\alpha_A \ge 0)$. Revealing the connection with $q$-statistics, we numerically verify (for $d$ =1, 2, 3 and 4) that the $q$-exponential degree distributions exhibit, for both $q$ and $\kappa$, universal dependences on the ratio $\alpha_A/d$. Moreover, the $q=1$ limit is rapidly achieved by increasing $\alpha_A/d$ to infinity.