Preferential attachment in growing spatial networks

TL;DR

This paper analyzes how spatial geometry influences the degree distribution and condensation phenomena in growing networks with preferential attachment, revealing effects of curvature and space type on network structure.

Contribution

It introduces a model of growing spatial networks with distance-weighted preferential attachment and explores how curvature affects degree distribution and condensation.

Findings

Degree distribution resembles the fitness model with space-dependent fitness

Curvature singularities can induce Bose-Einstein condensation

Hyperbolic spaces exhibit transient condensation phenomena

Abstract

We obtain the degree distribution for a class of growing network models on flat and curved spaces. These models evolve by preferential attachment weighted by a function of the distance between nodes. The degree distribution of these models is similar to the one of the fitness model of Bianconi and Barabasi, with a fitness distribution dependent on the metric and the density of nodes. We show that curvature singularities in these spaces can give rise to asymptotic Bose-Einstein condensation, but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature. We provide numerical results for spaces of constant curvature (sphere, flat and hyperbolic space) and we discuss the conditions for the breakdown of this approach and the critical points of the transition to distance-dominated attachment. Finally we discuss the distribution of link lengths.