Temporal connectivity of Random Geometric Graphs

TL;DR

This paper investigates the conditions under which temporal random geometric graphs become fully connected over time, revealing that higher edge density is needed for temporal connectivity compared to simple connectivity, unlike Erdős-Rényi graphs.

Contribution

It establishes a threshold for temporal connectivity in random geometric graphs and compares it with simple connectivity, highlighting differences from Erdős-Rényi models.

Findings

Temporal connectivity requires higher edge density than simple connectivity.

Thresholds for temporal connectivity differ significantly from Erdős-Rényi graphs.

Results apply to both standard and 'soft' random geometric graphs.

Abstract

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erd\H{o}s-R\'enyi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph.