Scale-free network on a vertical plane

TL;DR

This paper studies the growth and properties of a scale-free network constrained to a vertical plane with a directional bias, revealing different universality classes and analytical link length distributions.

Contribution

It introduces a model of scale-free network growth with directional bias on a vertical plane and analyzes its degree and link length distributions.

Findings

Directed scale-free network for α=0 differs from isotropic networks

Degree distribution becomes stretched exponential for α<α_c

Analytical expression for link length distribution for all α

Abstract

A scale-free network is grown in the Euclidean space with a global directional bias. On a vertical plane, nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which are below it using the attachment probability: $\pi_i(t) \sim k_i(t)\ell^{\alpha}$. Our numerical results indicate that the directed scale-free network for $\alpha=0$ belongs to a different universality class compared to the isotropic scale-free network. For $\alpha < \alpha_c$ the degree distribution is stretched exponential in general which takes a pure exponential form in the limit of $\alpha \to -\infty$. The link length distribution is calculated analytically for all values of $\alpha$.