Temporal connectivity of random geometric graphs

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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 Erdős–Rényi 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.

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