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

Authors Henry Austin , George B. Mertzios , Paul G. Spirakis

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LIPIcs.MFCS.2026.84.pdf
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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
Keywords
  • Random temporal graph
  • δ-temporal motif
  • sharp upper and lower bounds
  • doubling time

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Abstract

In this paper we study two natural models of random temporal graphs. In the first, the 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 discrete model, the l_e labels of each edge e are chosen uniformly at random from a set {1,2,…,T}. In both models we study the existence of 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 𝒢 = (G,λ) contains (H,P) as a δ-temporal motif if 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 sharp existence thresholds for all δ-temporal motifs, and we identify a qualitatively different behavior from the analogous static thresholds in Erdős-Rényi 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 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 sharp upper and lower bounds for the maximum doubling time in the continuous model.

Cite As Get BibTex

Henry Austin, George B. Mertzios, and Paul G. Spirakis. Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 84:1-84:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.84

Author Details

Henry Austin
  • Department of Computer Science, Durham University, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Paul G. Spirakis
  • School of Computer Science and Informatics, University of Liverpool, UK

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Spirakis, Paul G.: Supported by the EPSRC grant EP/P02002X/1.

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