Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs
Henry Austin, George B. Mertzios, Paul G. Spirakis
TL;DR
This work studies two natural random temporal graph models, continuous and discrete, to establish sharp thresholds for the existence of δ-temporal motifs, revealing a threshold governed by the sparsity ρ_H rather than the static ER-density. The authors develop a first- and second-moment framework augmented with automorphism counts and a domination principle to handle dependencies across motif occurrences, enabling precise thresholds for fixed motifs, cycles, and large cliques. They also analyze temporal expansion via the doubling time proxy, deriving tight upper and lower bounds in the continuous degenerate-label regime and connecting these results to known RSTG threshold phenomena. Overall, the paper provides a rigorous probabilistic toolbox for understanding when complex temporal interaction patterns emerge and how quickly reachability grows in random temporal graphs, highlighting fundamental differences from static graph thresholds and offering insight into temporal network dynamics.
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{$δ$-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.