A temporal graph is a graph whose edges appear only at certain points in time. In these graphs, reachability among the nodes relies on paths that traverse edges in chronological order (\emph{temporal paths}). Unlike standard paths, temporal paths are not always composable, thus the reachability relation is \emph{not transitive} and connected components do not form equivalence classes. We investigate the evolution of connected components in a simple model of random temporal graphs. In this model, a random temporal graph is obtained by permuting uniformly at random the edges of an Erd\"os-R\'enyi graph and interpreting the positions in this permutation as presence times. Phase transitions for several reachability properties were recently characterized in this model [Casteigts et al., FOCS 2021], in particular for one-to-one, one-to-all, and all-to-all reachability. The characterization of similar transitions for the existence of giant components was left open. In this paper, we develop a set of new techniques and use them to characterize the emergence of giant components in random temporal graphs. Our results imply that the growth of temporal components departs significantly from its classical analog. In particular, the largest component transitions abruptly from containing almost no vertices to almost all vertices at $p = \log n / n$, whereas in static random graphs (directed or not), a giant component of intermediate size arises first, and keeps steadily growing afterwards. This threshold holds for both \emph{open} and \emph{closed} temporal components, i.e., components that respectively allow or forbid the use of external nodes to achieve internal reachability, a distinction arising in the absence of transitivity.

翻译：时间图是一种边仅在特定时间点出现的图。在这些图中，节点间的可达性依赖于按时间顺序遍历边的路径（即时间路径）。与标准路径不同，时间路径不一定可组合，因此可达性关系不具有传递性，并且连通分量不构成等价类。我们研究了一类简单随机时间图模型中连通分量的演化过程。在该模型中，随机时间图是通过对Erdős-Rényi图的边进行均匀随机排列，并将排列中的位置解释为出现时间而获得的。最近已有研究刻画了该模型中若干可达性性质的相变过程（Casteigts 等，FOCS 2021），特别是针对一对一的、一对多的以及全部对全部的可达性。关于巨连通分量存在的类似相变刻画问题尚未解决。本文开发了一系列新技术，并利用它们刻画了随机时间图中巨连通分量的涌现过程。我们的结果表明，时间连通分量的增长显著有别于其经典对应物。特别地，当$p = \log n / n$时，最大分量从几乎不含顶点骤变为包含几乎所有顶点，而在静态随机图（无论有向或无向）中，首先会出现中等规模的巨连通分量，随后持续增长。这一阈值同时适用于开放型和封闭型时间连通分量，即分别允许或禁止利用外部节点实现内部可达性的分量——这种区分源于传递性的缺失。