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Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs

Authors George B. Mertzios , Sotiris Nikoletseas , Christoforos Raptopoulos , Paul G. Spirakis

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
Keywords
  • Simple random temporal graph
  • δ-temporal clique
  • probabilistic method

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Abstract

We consider random simple temporal graphs in which every edge of the complete graph K_n appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum δ-clique (namely a clique with edges appearing at most δ apart within [0,1]) in random instances of this model, for any constant δ. In particular, using the probabilistic method, we prove that the size of a maximum δ-clique is approximately (2 log n)/(log 1/δ) with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains Θ(n²) overlapping δ-windows, which (when viewed separately) correspond to different random instances of the Erdős-Rényi random graphs model, the size of the maximum δ-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a δ-clique is δ-o(δ) whp. We use this result to show that any polynomial time algorithm for δ-Temporal Clique is unlikely to have very large probability of success.

Cite As Get BibTex

George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Brief Announcement: On the Existence of δ-Temporal Cliques in Random Simple Temporal Graphs. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 27:1-27:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SAND.2024.27

Author Details

George B. Mertzios
  • Department of Computer Science, Durham University, UK
Sotiris Nikoletseas
  • Computer Engineering and Informatics Department, University of Patras, Greece
Christoforos Raptopoulos
  • Department of Mathematics, University of Patras, Greece
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Raptopoulos, Christoforos: Supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “2nd Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers” (Project Number: 704).
  • Spirakis, Paul G.: Supported by the EPSRC grant EP/P02002X/1.

References

  1. Eleni C. Akrida, Leszek Gasieniec, George B. Mertzios, and Paul G. Spirakis. Ephemeral networks with random availability of links: The case of fast networks. J. Par. and Distr. Comp., 87:109-120, 2016. Google Scholar
  2. Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Temporal vertex cover with a sliding time window. J. Comp. Sys. Sci., 107:108-123, 2020. Google Scholar
  3. A. Anagnostopoulos, J. Lacki, S. Lattanzi, S. Leonardi, and M. Mahdian. Community detection on evolving graphs. In Proceedings of the 30th NIPS, pages 3522-3530, 2016. Google Scholar
  4. Arnaud Casteigts, Anne-Sophie Himmel, Hendrik Molter, and Philipp Zschoche. Finding temporal paths under waiting time constraints. Algorithmica, 83(9):2754-2802, 2021. Google Scholar
  5. Arnaud Casteigts, Michael Raskin, Malte Renken, and Viktor Zamaraev. Sharp thresholds in random simple temporal graphs. In Proceedings of the 62nd FOCS, pages 319-326, 2021. Google Scholar
  6. Jessica A. Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting edges to restrict the size of an epidemic in temporal networks. J. Comp. Sys. Sci., 119:60-77, 2021. Google Scholar
  7. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. On temporal graph exploration. J. Comp. Sys. Sci., 119:1-18, 2021. Google Scholar
  8. A. Ferreira. Building a reference combinatorial model for MANETs. IEEE Network, 18(5):24-29, 2004. Google Scholar
  9. Thekla Hamm, Nina Klobas, George B. Mertzios, and Paul G. Spirakis. The complexity of temporal vertex cover in small-degree graphs. In Proceedings of the 36th AAAI, pages 10193-10201, 2022. Google Scholar
  10. Klaus Heeger, Danny Hermelin, George B. Mertzios, Hendrik Molter, Rolf Niedermeier, and Dvir Shabtay. Equitable scheduling on a single machine. In Proceedings of the 35th AAAI, pages 11818-11825, 2021. Google Scholar
  11. Anne-Sophie Himmel, Hendrik Molter, Rolf Niedermeier, and Manuel Sorge. Adapting the bron-kerbosch algorithm for enumerating maximal cliques in temporal graphs. Soc. Netw. Analysis and Mining, 7(1):35:1-35:16, 2017. Google Scholar
  12. D. Kempe, J. M. Kleinberg, and A. Kumar. Connectivity and inference problems for temporal networks. In Proceedings of the 32nd (STOC), pages 504-513, 2000. Google Scholar
  13. Nina Klobas, George B. Mertzios, Hendrik Molter, Rolf Niedermeier, and Philipp Zschoche. Interference-free walks in time: temporally disjoint paths. Auton. Agents Multi-Agent Syst., 37(1), 2023. Google Scholar
  14. Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. The complexity of computing optimum labelings for temporal connectivity. In Proceedings of the 47th MFCS, pages 62:1-62:15, 2022. Google Scholar
  15. Nina Klobas and George B. Mertzios Paul G. Spirakis. Sliding into the future: Investigating sliding windows in temporal graphs. In Proceedings of the 48th MFCS, pages 5:1-5:12, 2023. Google Scholar
  16. George B. Mertzios, Othon Michail, and Paul G. Spirakis. Temporal network optimization subject to connectivity constraints. Algorithmica, 81(4):1416-1449, 2019. Google Scholar
  17. George B. Mertzios, Hendrik Molter, Malte Renken, Paul G. Spirakis, and Philipp Zschoche. The complexity of transitively orienting temporal graphs. In Proceedings of the 46th MFCS, pages 75:1-75:18, 2021. Google Scholar
  18. George B. Mertzios, Hendrik Molter, and Viktor Zamaraev. Sliding window temporal graph coloring. J. Comp. Sys. Sci., 120:97-115, 2021. Google Scholar
  19. O. Michail and P.G. Spirakis. Elements of the theory of dynamic networks. Communications of the ACM, 61(2):72-72, January 2018. Google Scholar
  20. Othon Michail and Paul G. Spirakis. Traveling salesman problems in temporal graphs. Theor. Comp. Sci., 634:1-23, 2016. Google Scholar
  21. N. Santoro. Computing in time-varying networks. In Proceedings of the 13th SSS, page 4, 2011. Google Scholar
  22. Tiphaine Viard, Matthieu Latapy, and Clémence Magnien. Computing maximal cliques in link streams. Theor. Comp. Sci., 609:245-252, 2016. Google Scholar
  23. Feng Yu, Amotz Bar-Noy, Prithwish Basu, and Ram Ramanathan. Algorithms for channel assignment in mobile wireless networks using temporal coloring. In Proceedings of the 16th MSWiM, pages 49-58, 2013. Google Scholar
  24. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexity of finding small separators in temporal graphs. J. Comp. Sys. Sci., 107:72-92, 2020. Google Scholar
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