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 from an Erd\H{o}s-R\'enyi random graph $G~G_{n,p}$ by considering a random permutation $\pi$ of the edges and interpreting the ranks in $\pi$ as presence times. Temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. 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 vertex) 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. Furthermore, the graph admits a temporal spanner of size $2n+o(n)$ as soon as it becomes temporally connected, which is nearly optimal as $2n-4$ is a lower bound. This result is significant because temporal graphs do not admit spanners of size $O(n)$ in general (Kempe et al, STOC 2000). In fact, they do not even admit spanners of size $o(n^2)$ (Axiotis et al, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, all non-negligible obstructions, must be statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners further, we show that pivotal spanners -- i.e., spanners of size $2n-2$ made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently) -- exist a.a.s. at $4\log n/n$, this threshold being also sharp. Finally, we show that optimal spanners (of size $2n-4$) also exist a.a.s. at $p = 4\log n/n$.

翻译：图的边仅在特定时间点出现被称为时态图（另有其他称谓）。若图中每一对有序顶点均存在一条按时间顺序遍历边的路径（即时态路径），则称该图为时态连通的。本文考虑随机时态图的简单模型，该模型由Erdős-Rényi随机图$G~G_{n,p}$经随机排列$\pi$赋予边出现时间（即$\pi$中的秩次）而构建。此模型中的时态可达性展现出惊人的规律性阈值序列。特别地，我们证明：当$p=\log n/n$时，任意固定顶点对几乎必然可相互到达；当$2\log n/n$时，至少一个顶点（事实上任意固定顶点）几乎必然能到达所有其他顶点；当$3\log n/n$时，所有顶点几乎必然可相互到达，即图成为时态连通。此外，图一旦变得时态连通，便存在大小为$2n+o(n)$的时态稀疏子图，该值接近最优下界$2n-4$。此结果意义重大，因为一般时态图不存在$O(n)$规模的稀疏子图（Kempe等，STOC 2000），甚至不存在$o(n^2)$规模的稀疏子图（Axiotis等，ICALP 2016）。因此，我们的结果表明这些文献中发现的障碍（以及所有非平凡障碍）在统计上必定不显著：随机时态图中始终存在近乎最优的稀疏子图。上述所有阈值均为尖锐的。进一步研究时态稀疏子图后，我们证明：枢轴稀疏子图（即由两棵生成树在单个顶点处拼接而成的大小为$2n-2$的稀疏子图，一棵时间递减，另一棵时间递增）在$4\log n/n$处几乎必然存在，且该阈值同样尖锐。最后，我们证明最优稀疏子图（大小为$2n-4$）在$p=4\log n/n$处亦几乎必然存在。