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

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{$δ$-temporal motifs}. Here a $δ$-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 $\mathcal{G}=(G,λ)$ contains $(H,P)$ as a $δ$-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 $δ$. We prove \textit{sharp existence thresholds} for all $δ$-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 $δ$-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.

💡 Research Summary

This paper investigates two natural random temporal‑graph models—continuous and discrete—and establishes sharp thresholds for the appearance of δ‑temporal motifs, the size of the largest δ‑temporal clique, and the maximum doubling time of reachability balls. In the continuous model each edge receives a random number of time‑labels drawn independently from a fixed discrete distribution ψ; the actual label values are sampled uniformly from the interval (0,1]. The discrete model is obtained by discretizing the interval into {1,…,T} and drawing labels uniformly from this set. Both models thus capture variability in the number of labels per edge as well as in their timing.

A δ‑temporal motif is defined as a pair (H,P) where H is a static graph and P is a partial order on its edges. A temporal graph G contains (H,P) as a δ‑motif if there exists a simple temporal subgraph isomorphic to H whose edge‑labels respect P and whose overall time span does not exceed δ. The paper introduces the notion of sparsity ρ_H = min_{H’⊆H} (|V_{H’}|/|E_{H’}|)−1, which measures the inverse density of the densest subgraph of H.

The main existence result (Theorem 10 for the continuous model, Theorem 11 for the discrete model) shows that the threshold for a fixed motif is governed by ρ_H rather than the usual edge‑density. Specifically, if δ(n) = ω(n^{‑ρ_H}) then (H,P) appears with high probability; if δ(n) = o(n^{‑ρ_H}) it does not. This leads to qualitatively new behavior: cycles of different lengths have distinct asymptotic thresholds, unlike the static Erdős–Rényi case where all finite cycles share the same threshold.

The authors then study the largest δ‑temporal clique. Let r = E