Giant Components in Random Temporal Graphs [RANDOM 2023]

A temporal graph is a graph whose edges appear only at certain points in time. Recently, the second and the last three authors proposed a natural temporal analog of the Erdős-Rényi random graph model. The proposed model is obtained by randomly permuting the edges of an Erdős-Rényi random graph and interpreting this permutation as an ordering of presence times. It was shown that the connectivity threshold in the Erdős-Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting.

In the present paper, we identify a sharp threshold for the emergence of a giant temporally connected component. We show that at p=log n/n the size of the largest temporally connected component increases from o(n) to n-o(n). This threshold holds for both open and closed connected components, i.e. components that allow, respectively forbid, their connecting paths to use external nodes.