Modulated Scale-free Network in the Euclidean Space

TL;DR

This paper introduces a model for growing Euclidean space networks with modulated scale-free properties, analyzing how connection probabilities based on degree and distance influence network topology and degree distributions.

Contribution

It presents a novel network growth model incorporating Euclidean distances and degree-based attachment, deriving exact link length distribution and identifying conditions for scale-free behavior.

Findings

Network is scale-free for all $\alpha > \alpha_c$

Degree distribution decays stretched exponentially for $\alpha \\leq \\alpha_c$

Link length distribution follows a power law $D(\\ell) \\sim \\ell^{\\delta}$ with exact $\\delta$

Abstract

A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its $i$-th predecessor of degree $k_i$ with a directed link of length $\ell$ using a probability proportional to $k_i \ell^{\alpha}$. Our numerical study indicates that the network is Scale-free for all values of $\alpha > \alpha_c $ and the degree distribution decays stretched exponentially for the other values of $\alpha$. The link length distribution follows a power law: $D(\ell) \sim \ell^{\delta}$ where $\delta$ is calculated exactly for the whole range of values of $\alpha$.