Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs

TL;DR

This paper investigates the emergence of specific temporal motifs and the growth dynamics of reachability in random temporal graphs, establishing sharp thresholds and bounds that differ from static graph behaviors.

Contribution

It introduces models for random temporal graphs, proves sharp thresholds for temporal motifs, and characterizes the doubling time of reachability, revealing new phenomena distinct from static graphs.

Findings

Sharp thresholds for the existence of all δ-temporal motifs.

Different behavior from static Erdős-Rényi graph thresholds.

Bounds on the doubling time of reachability in the continuous model.

Abstract

In this paper we study two natural models of \textit{random temporal} graphs. In the first, the \textit{continuous} model, each edge $e$ is assigned $l_e$ labels, each drawn uniformly at random from $(0,1]$, where the numbers $l_e$ are independent random variables following the same discrete probability distribution. In the second, the \textit{discrete} model, the $l_e$ labels of each edge $e$ are chosen uniformly at random from a set $\{1,2,\ldots,T\}$. In both models we study the existence of \textit{$\delta$-temporal motifs}. Here a $\delta$-temporal motif consists of a pair $(H,P)$, where $H$ is a fixed static graph and $P$ is a partial order over its edges. A temporal graph $\mathcal{G}=(G,\lambda)$ contains $(H,P)$ as a $\delta$-temporal motif if $\mathcal{G}$ has a simple temporal subgraph on the edges of $H$ whose time labels are ordered according to $P$, and whose life duration is at most $\delta$. We prove \textit{sharp existence thresholds} for all $\delta$-temporal motifs, and we identify a qualitatively different behavior from the analogous static thresholds in Erdos-Renyi random graphs. Applying the same techniques, we then characterize the growth of the largest $\delta$-temporal clique in the continuous variant of our random temporal graphs model. Finally, we consider the \textit{doubling time} of the reachability ball centered on a small set of vertices of the random temporal graph as a natural proxy for temporal expansion. We prove \textit{sharp upper and lower bounds} for the maximum doubling time in the continuous model.