Geometric scale-free random graphs on mobile vertices: broadcast and percolation times

TL;DR

This paper investigates how information spreads rapidly in mobile geometric scale-free random graphs, showing that in certain regimes, broadcast and percolation occur in poly-logarithmic or stretched exponential times.

Contribution

It introduces a model of mobile geometric scale-free graphs with Brownian motion and analyzes the speed of information propagation and percolation in these dynamic networks.

Findings

Information broadcast occurs in poly-logarithmic time on tori.

Information reaches the infinite component in stretched exponential time on Euclidean space.

Results apply to mobile versions of age-dependent and soft Boolean models.

Abstract

We study the phenomenon of information propagation on mobile geometric scale-free random graphs, where vertices instantaneously pass on information to all other vertices in the same connected component. The graphs we consider are constructed on a Poisson point process of intensity $\lambda>0$, and the vertices move over time as simple Brownian motions on either $\mathbb{R}^d$ or the $d$-dimensional torus of volume $n$, while edges are randomly drawn depending on the locations of the vertices, as well as their a priori assigned marks. This includes mobile versions of the age-dependent random connection model and the soft Boolean model. We show that in the ultrasmall regime of these random graphs, information is broadcast to all vertices on a torus of volume $n$ in poly-logarithmic time and that on $\mathbb{R}^d$, the information will reach the infinite component before time $t$ with stretched exponentially high probability, for any $\lambda>0$.