Giant Components in Random Temporal Graphs

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.