# Role of dimensionality in preferential attachment growth in the Bianconi-Barab\'asi model

## TL;DR

This paper investigates how the dimensionality of space influences the degree distribution in a geographically-aware preferential attachment model, revealing universal behaviors related to the ratio of attachment parameter to dimension.

## Contribution

It introduces a modified Bianconi-Barabási model incorporating Euclidean distances and demonstrates universal scaling laws for degree distribution and growth dynamics.

## Key findings

- Degree distribution fits a q-exponential form
- Universal behavior of parameters with respect to α_A/d
- Scaling of the growth exponent β with α_A/d

## Abstract

Scale-free networks are quite popular nowadays since many systems are well represented by such structures. In order to study these systems, several models were proposed. However, most of them do not take into account the node-to-node Euclidean distance, i.e., the geographical distance. In real networks, the distance between sites can be very relevant, e.g., those cases where it is intended to minimize costs. Within this scenario we studied the role of dimensionality $d$ in the Bianconi-Barab\'asi model with a preferential attachment growth involving Euclidean distances. The preferential attachment in this model follows the rule $\Pi_{i} \propto \eta_i k_i/r_{ij}^{\alpha_A}$ $(1 \leq i < j; \alpha_A \geq 0)$, where $\eta_i$ characterizes the fitness of the $i$-th site and is randomly chosen within the $(0,1]$ interval. We verified that the degree distribution $P(k)$ for dimensions $d=1,2,3,4$ are well fitted by $P(k) \propto e_q^{-k/\kappa}$, where $e_q^{-k/\kappa}$ is the $q$-exponential function naturally emerging within nonextensive statistical mechanics. We determine the index $q$ and $\kappa$ as functions of the quantities $\alpha_A$ and $d$, and numerically verify that both present a universal behavior with respect to the scaled variable $\alpha_A/d$. The same behavior also has been displayed by the dynamical $\beta$ exponent which characterizes the steadily growing number of links of a given site.