Mobile Geometric Graphs: Detection, Coverage and Percolation

TL;DR

This paper analyzes a dynamic geometric graph model where nodes move randomly, studying detection, coverage, and percolation times with precise asymptotic results using advanced stochastic geometry techniques.

Contribution

It provides the first detailed asymptotic analysis of detection, coverage, and percolation times in a dynamic Brownian motion-based geometric graph model.

Findings

Derived precise asymptotics for detection times

Established asymptotics for coverage times

Analyzed percolation times in the dynamic model

Abstract

We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R^d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes if their distance is at most r. We study three features in this model: detection (the time until a target point---fixed or moving---is within distance r from some node of the graph), coverage (the time until all points inside a finite box are detected by the graph), and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these features by combining ideas from stochastic geometry, coupling and multi-scale analysis.