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DOI: 10.4230/LIPIcs.MFCS.2026.84

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

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

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

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.

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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)

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@InProceedings{austin_et_al:LIPIcs.MFCS.2026.84,
  author =	{Austin, Henry and Mertzios, George B. and Spirakis, Paul G.},
  title =	{{Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{84:1--84:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.84},
  URN =		{urn:nbn:de:0030-drops-274660},
  doi =		{10.4230/LIPIcs.MFCS.2026.84},
  annote =	{Keywords: Random temporal graph, \delta-temporal motif, sharp upper and lower bounds, doubling time}
}

Document

DOI: 10.4230/LIPIcs.DISC.2025.10

Amnesiac Flooding: Easy to Break, Hard to Escape

Authors: Henry Austin, Maximilien Gadouleau, George B. Mertzios, and Amitabh Trehan

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract

Broadcast is a central problem in distributed computing. Recently, Hussak and Trehan [PODC'19/ STACS'20/DC'23] proposed a stateless broadcasting protocol (Amnesiac Flooding), which was surprisingly proven to terminate in asymptotically optimal time (linear in the diameter of the network). However, it remains unclear: (i) Are there other stateless terminating broadcast algorithms with the desirable properties of Amnesiac Flooding, (ii) How robust is Amnesiac Flooding with respect to faults? In this paper we make progress on both of these fronts. Under a reasonable restriction (obliviousness to message content) additional to the fault-free synchronous model, we prove that Amnesiac Flooding is the only strictly stateless deterministic protocol that can achieve terminating broadcast. We achieve this by identifying four natural properties of a terminating broadcast protocol that Amnesiac Flooding uniquely satisfies. In contrast, we prove that even minor relaxations of any of these four criteria allow the construction of other terminating broadcast protocols. On the other hand, we prove that Amnesiac Flooding can become non-terminating or non-broadcasting, even if we allow just one node to drop a single message on a single edge in a single round. As a tool for proving this, we focus on the set of all configurations of transmissions between nodes in the network, and obtain a dichotomy characterizing the configurations, starting from which, Amnesiac Flooding terminates. Additionally, we characterise the structure of sets of Byzantine agents capable of forcing non-termination or non-broadcast of the protocol on arbitrary networks.

Cite as

Henry Austin, Maximilien Gadouleau, George B. Mertzios, and Amitabh Trehan. Amnesiac Flooding: Easy to Break, Hard to Escape. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 10:1-10:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{austin_et_al:LIPIcs.DISC.2025.10,
  author =	{Austin, Henry and Gadouleau, Maximilien and Mertzios, George B. and Trehan, Amitabh},
  title =	{{Amnesiac Flooding: Easy to Break, Hard to Escape}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{10:1--10:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.10},
  URN =		{urn:nbn:de:0030-drops-248273},
  doi =		{10.4230/LIPIcs.DISC.2025.10},
  annote =	{Keywords: Amnesiac flooding, Terminating protocol, Algorithm state, Stateless protocol, Flooding algorithm, Network algorithms, Graph theory, Termination, Communication, Broadcast}
}

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Documents authored by Austin, Henry

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

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Amnesiac Flooding: Easy to Break, Hard to Escape

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