Sharp Thresholds in Random Simple Temporal Graphs

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e. a temporal path). In this paper, we consider a simple model of random temporal graph, obtained by assigning to every edge of an Erd{ő}s-R{e}nyi random graph $G_{n,p}$ a uniformly random presence time in the real interval [0, 1]. It turns out that this model exhibits a surprisingly regular sequence of thresholds related to temporal reachability. In particular, we show that at p=log n/n any fixed pair of vertices can a.a.s. reach each other, at 2 log n/n at least one vertex (and in fact, any fixed node) can a.a.s. reach all others, and at 3 log n/n all the vertices can a.a.s. reach each other (i.e. the graph is temporally connected). All these thresholds are sharp. In addition, at p=4 log n/n the graph contains a spanning subgraph of minimum possible size that preserves temporal connectivity, i.e. it admits a temporal spanner with 2n-4 edges. Another contribution of this paper is to connect our model and the above results with existing ones in several other topics, including gossip theory (rumor spreading), population protocols (with sequential random scheduler), and edge-ordered graphs. In particular, our analyses can be extended to strengthen several known results in these fields.