随机简单时间图中的尖锐阈值
Sharp Thresholds in Random Simple Temporal Graphs
AI总结：
本文研究随机简单时间图模型，揭示时间可达性的多组尖锐阈值，证明随机时间图中几乎总存在接近最优甚至最优的时间生成树，弥补了一般时间图生成树规模下界的相关结论。
摘要：
边仅在特定时间点出现的图被称为时间图（temporal graph，也有其他名称）。若每一对有序顶点都能通过一条按时间顺序遍历边的路径（即时间路径）相连，则称该图是时间连通的。本文研究一种简单的随机时间图模型：由Erdős-Rényi随机图$G\sim G_{n,p}$出发，对边取随机排列$\pi$，并将$\pi$中的秩解释为边的出现时间。该模型中的时间可达性呈现出出人意料的规则阈值序列。具体而言，我们证明当$p=\log n/n$时，任意固定顶点对几乎必然（a.a.s.）可互相到达；当$p=2\log n/n$时，至少有一个顶点（实际上任意固定顶点）几乎必然能到达所有其他顶点；当$p=3\log n/n$时，所有顶点几乎必然可互相到达，即图是时间连通的。此外，图一旦达到时间连通，就存在大小为$2n+o(n)$的时间生成树（temporal spanner），这接近最优，因为$2n-4$是下界。这一结果意义重大，因为时间图一般不存在大小为$O(n)$的生成树（Kempe等人，STOC 2000），甚至不存在大小为$o(n^2)$的生成树（Axiotis等人，ICALP 2016）。因此，我们的结果表明，这些工作中发现的阻碍，以及更一般而言所有不可忽略的阻碍，在统计上都是不显著的：随机时间图中几乎总是存在接近最优的生成树。上述所有阈值都是尖锐的。进一步推进时间生成树的研究，我们证明关键生成树（pivotal spanners，即大小为$2n-2$、由两棵在单个顶点处粘合的生成树构成的生成树，一棵随时间递减，另一棵随后随时间递增）在$p=4\log n/n$时几乎必然存在，该阈值也是尖锐的。最后，我们证明最优生成树（大小为$2n-4$）在$p = 4\log n/n$时也几乎必然存在。
英文摘要：
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 Erdős-Rényi random graph $G~G_{n,p}$ by considering a random permutation $π$ of the edges and interpreting the ranks in $π$ 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 $p = 4\log n/n$.
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