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

2026-02-02

AI Summary

This study investigates the existence thresholds of δ-temporal motifs and the doubling time of temporal reachability expansion in random temporal graphs. Under both continuous and discrete temporal labeling models, and assuming a fixed underlying static graph structure together with a prescribed partial order on its edges, the work analyzes the conditions under which δ-temporal motifs emerge. By employing probabilistic methods, random graph theory, and structural analysis of partial orders, it establishes for the first time that all δ-temporal motifs exhibit sharp existence thresholds, revealing a fundamental distinction from the classical thresholds in static Erdős–Rényi graphs. Furthermore, the paper characterizes the asymptotic growth of the largest δ-temporal cliques and provides tight upper and lower bounds on the doubling time of temporal reachability balls in the continuous model.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• temporal motifs
• random temporal graphs
• sharp thresholds
• doubling time
• temporal cliques

Innovation

• temporal motifs
• sharp thresholds
• random temporal graphs
• doubling time
• temporal cliques