Giant Components in Random Temporal Graphs

TL;DR

This paper establishes a precise threshold for the emergence of a giant connected component in a random temporal graph model, extending classical Erd ext{"o}s-Rényi results to temporal settings.

Contribution

It identifies a sharp connectivity threshold in a novel temporal Erd ext{"o}s-Rényi model, revealing phase transition behavior for giant component formation.

Findings

Giant component appears at p = (log n)/n

Size of largest component jumps from o(n) to n-o(n)

Threshold applies to both open and closed connectivity notions

Abstract

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 Erd\H{o}s-R\'enyi random graph model. The proposed model is obtained by randomly permuting the edges of an Erd\H{o}s-R\'enyi random graph and interpreting this permutation as an ordering of presence times. It was shown that the connectivity threshold in the Erd\H{o}s-R\'enyi 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.