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Distributed Triangle and Simplex Enumeration in Hypergraphs

Authors Duncan Adamson , Will Rosenbaum , Paul G. Spirakis

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LIPIcs.DISC.2026.33.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Hypergraphs
  • Mathematics of computing → Graph enumeration
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Extremal graph theory
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Information theory
Keywords
  • Triangle Enumeration
  • Simplex Enumeration
  • Hypergraph
  • Distributed Algorithms
  • CONGEST

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Abstract

In the last decade, subgraph detection and enumeration have emerged as central problems in distributed graph algorithms. This is largely due to the problems' theoretical challenges and practical applications. In this paper, we initiate the systematic study of distributed sub-hypergraph enumeration in hypergraphs. To this end, we (1) introduce several computational models for hypergraphs that generalize the CONGEST model for graphs and evaluate their relative computational power, (2) devise algorithms for distributed triangle and simplex enumeration in our computational models and prove their optimality in two such models by showing matching lower bounds, (3) introduce classes of sparse and "everywhere sparse" hypergraphs and describe efficient distributed algorithms for triangle and simplex enumeration in these classes, and (4) describe general techniques that we believe to be useful for designing efficient algorithms in our hypergraph models.

Cite As Get BibTex

Duncan Adamson, Will Rosenbaum, and Paul G. Spirakis. Distributed Triangle and Simplex Enumeration in Hypergraphs. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.33

Author Details

Duncan Adamson
  • School of Computer Science, University of St Andrews, UK
Will Rosenbaum
  • School of Computer Science and Informatics, University of Liverpool, UK
Paul G. Spirakis
  • School of Computer Science and Informatics, University of Liverpool, UK

References

  1. Duncan Adamson, Magnús M Halldórsson, and Alexandre Nolin. Distributed coloring of hypergraphs. In International Colloquium on Structural Information and Communication Complexity, pages 89-111. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32733-9_5.
  2. Duncan Adamson, Will Rosenbaum, and Paul G Spirakis. Distributed weak independent sets in hypergraphs: Upper and lower bounds. In International Symposium on Algorithmics of Wireless Networks, pages 1-16. Springer, 2025. URL: https://doi.org/10.1007/978-3-032-09120-8_1.
  3. Duncan Adamson, Will Rosenbaum, and Paul G. Spirakis. Distributed triangle and simplex enumeration in hypergraphs, 2026. URL: https://arxiv.org/abs/2602.17834.
  4. Alessia Antelmi, Gennaro Cordasco, Mirko Polato, Vittorio Scarano, Carmine Spagnuolo, and Dingqi Yang. A Survey on Hypergraph Representation Learning. ACM Computing Surveys, 56(1):1-38, January 2024. URL: https://doi.org/10.1145/3605776.
  5. Luigi Atzori, Antonio Iera, and Giacomo Morabito. The Internet of Things: A survey. Computer Networks, 54(15):2787-2805, October 2010. URL: https://doi.org/10.1016/j.comnet.2010.05.010.
  6. Baruch Awerbuch, Oded Goldreich, Ronen Vainish, and David Peleg. A trade-off between information and communication in broadcast protocols. Journal of The Acm, 37(2):238-256, April 1990. URL: https://doi.org/10.1145/77600.77618.
  7. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Distributed maximal matching and maximal independent set on hypergraphs. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2632-2676. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH100.
  8. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using nash-williams decomposition. In Proceedings of the Twenty-Seventh ACM Symposium on Principles of Distributed Computing, pages 25-34, Toronto Canada, August 2008. ACM. URL: https://doi.org/10.1145/1400751.1400757.
  9. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Synthesis Lectures on Distributed Computing Theory, 4(1):1-171, July 2013. URL: https://doi.org/10.2200/S00520ED1V01Y201307DCT011.
  10. Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean-Gabriel Young, and Giovanni Petri. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874:1-92, 2020. URL: https://doi.org/10.1016/j.physrep.2020.05.004.
  11. Ran Ben Basat, Guy Even, Ken-ichi Kawarabayashi, and Gregory Schwartzman. Optimal Distributed Covering Algorithms. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, pages 104-106, Toronto ON Canada, July 2019. ACM. URL: https://doi.org/10.1145/3293611.3331577.
  12. Austin R. Benson, Rediet Abebe, Michael T. Schaub, Ali Jadbabaie, and Jon Kleinberg. Simplicial closure and higher-order link prediction. Proceedings of the National Academy of Sciences, 115(48), November 2018. URL: https://doi.org/10.1073/pnas.1800683115.
  13. Austin R. Benson, David F. Gleich, and Jure Leskovec. Higher-order organization of complex networks. Science, 353(6295):163-166, July 2016. URL: https://doi.org/10.1126/science.aad9029.
  14. Theophilus Benson, Aditya Akella, and David A. Maltz. Network traffic characteristics of data centers in the wild. In Proceedings of the 10th ACM SIGCOMM Conference on Internet Measurement, pages 267-280, Melbourne Australia, November 2010. ACM. URL: https://doi.org/10.1145/1879141.1879175.
  15. Alain Bretto. Hypergraph Theory: An Introduction. Mathematical Engineering. Springer International Publishing, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-319-00080-0.
  16. Keren Censor-Hillel. Distributed subgraph finding: Progress and challenges, 2025. URL: https://arxiv.org/abs/2203.06597.
  17. Amit Chakrabarti and Themistoklis Haris. Counting Simplices in Hypergraph Streams. LIPIcs, Volume 244, ESA 2022, 244:32:1-32:19, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.32.
  18. Yi-Jun Chang, Seth Pettie, Thatchaphol Saranurak, and Hengjie Zhang. Near-optimal Distributed Triangle Enumeration via Expander Decompositions. Journal of the ACM, 68(3):1-36, June 2021. URL: https://doi.org/10.1145/3446330.
  19. Danny Dolev, Christoph Lenzen, and Shir Peled. "Tri, Tri Again": Finding Triangles and Small Subgraphs in a Distributed Setting. In David Hutchison, Takeo Kanade, Josef Kittler, Jon M. Kleinberg, Friedemann Mattern, John C. Mitchell, Moni Naor, Oscar Nierstrasz, C. Pandu Rangan, Bernhard Steffen, Madhu Sudan, Demetri Terzopoulos, Doug Tygar, Moshe Y. Vardi, Gerhard Weikum, and Marcos K. Aguilera, editors, Distributed Computing, volume 7611, pages 195-209. Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-33651-5_14.
  20. Mohsen Ghaffari, Fabian Kuhn, and Hsin-Hao Su. Distributed MST and Routing in Almost Mixing Time. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 131-140, Washington DC USA, July 2017. ACM. URL: https://doi.org/10.1145/3087801.3087827.
  21. Mohsen Ghaffari and Jason Li. New Distributed Algorithms in Almost Mixing Time via Transformations from Parallel Algorithms. LIPIcs, Volume 121, DISC 2018, 121:31:1-31:16, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.31.
  22. Mark S. Granovetter. The Strength of Weak Ties. American Journal of Sociology, 78(6):1360-1380, 1973. URL: https://www.jstor.org/stable/2776392.
  23. Jin Huang, Rui Zhang, and Jeffrey Xu Yu. Scalable Hypergraph Learning and Processing. In 2015 IEEE International Conference on Data Mining, pages 775-780, Atlantic City, NJ, USA, November 2015. IEEE. URL: https://doi.org/10.1109/ICDM.2015.33.
  24. Taisuke Izumi and François Le Gall. Triangle Finding and Listing in CONGEST Networks. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 381-389, Washington DC USA, July 2017. ACM. URL: https://doi.org/10.1145/3087801.3087811.
  25. Madhav Jha, C. Seshadhri, and Ali Pinar. A Space-Efficient Streaming Algorithm for Estimating Transitivity and Triangle Counts Using the Birthday Paradox. ACM Transactions on Knowledge Discovery from Data, 9(3):1-21, April 2015. URL: https://doi.org/10.1145/2700395.
  26. Stasys Jukna. Extremal Combinatorics: With Applications in Computer Science. Texts in Theoretical Computer Science. An EATCS Series. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-17364-6.
  27. Howard J. Karloff, Siddharth Suri, and Sergei Vassilvitskii. A Model of Computation for MapReduce. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 938-948, 2010. URL: https://doi.org/10.1137/1.9781611973075.76.
  28. M. Kivela, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter. Multilayer networks. Journal of Complex Networks, 2(3):203-271, September 2014. URL: https://doi.org/10.1093/comnet/cnu016.
  29. Mohsen Koohi Esfahani, Peter Kilpatrick, and Hans Vandierendonck. LOTUS: Locality optimizing triangle counting. In Proceedings of the 27th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, pages 219-233, Seoul Republic of Korea, April 2022. ACM. URL: https://doi.org/10.1145/3503221.3508402.
  30. Fabian Kuhn and Chaodong Zheng. Efficient distributed computation of mis and generalized mis in linear hypergraphs. arXiv preprint, 2018. URL: https://arxiv.org/abs/1805.03357.
  31. Shay Kutten, Danupon Nanongkai, Gopal Pandurangan, and Peter Robinson. Distributed Symmetry Breaking in Hypergraphs. In Fabian Kuhn, editor, Distributed Computing, volume 8784, pages 469-483. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014. URL: https://doi.org/10.1007/978-3-662-45174-8_32.
  32. Timothy LaRock and Renaud Lambiotte. Encapsulation structure and dynamics in hypergraphs. Journal of Physics: Complexity, 4(4):045007, December 2023. URL: https://doi.org/10.1088/2632-072X/ad0b39.
  33. Geon Lee, Fanchen Bu, Tina Eliassi-Rad, and Kijung Shin. A Survey on Hypergraph Mining: Patterns, Tools, and Generators, February 2025. URL: https://doi.org/10.48550/arXiv.2401.08878.
  34. Christoph Lenzen. Optimal deterministic routing and sorting on the congested clique. In ACM Symposium on Principles of Distributed Computing, PODC '13, Montreal, QC, Canada, July 22-24, 2013, pages 42-50, 2013. URL: https://doi.org/10.1145/2484239.2501983.
  35. N. Linial. Locality in Distributed Graph Algorithms. SIAM Journal on Computing, 21(1):193-201, February 1992. URL: https://doi.org/10.1137/0221015.
  36. Zvi Lotker, Boaz Patt-Shamir, Elan Pavlov, and David Peleg. Minimum-Weight Spanning Tree Construction in O(log log n) Communication Rounds. SIAM Journal on Computing, 35(1):120-131, January 2005. URL: https://doi.org/10.1137/S0097539704441848.
  37. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: Simple building blocks of complex networks. Science, 298(5594):824-827, 2002. URL: https://doi.org/10.1126/science.298.5594.824.
  38. C. St.J. A. Nash-Williams. Decomposition of finite graphs into forests. Journal of the London Mathematical Society, s1-39(1):12-12, 1964. URL: https://doi.org/10.1112/jlms/s1-39.1.12.
  39. M. E. J. Newman. The Structure and Function of Complex Networks. SIAM Review, 45(2):167-256, January 2003. URL: https://doi.org/10.1137/S003614450342480.
  40. Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. On the Distributed Complexity of Large-Scale Graph Computations. ACM Transactions on Parallel Computing, 8(2):1-28, June 2021. URL: https://doi.org/10.1145/3460900.
  41. David Peleg. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, 2000. URL: https://doi.org/10.1137/1.9780898719772.
  42. Larry L. Peterson and Bruce S. Davie. Computer Networks, Fifth Edition: A Systems Approach. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 5 edition, 2011. Google Scholar
  43. Igor Rivin. Counting cycles and finite dimensional Lp norms. Advances in Applied Mathematics, 29(4):647-662, November 2002. URL: https://doi.org/10.1016/S0196-8858(02)00037-4.
  44. Alistair Sinclair and Mark Jerrum. Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computation, 82(1):93-133, July 1989. URL: https://doi.org/10.1016/0890-5401(89)90067-9.
  45. Konstantinos Sotiropoulos and Charalampos E. Tsourakakis. Triangle-aware Spectral Sparsifiers and Community Detection. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 1501-1509, Virtual Event Singapore, August 2021. ACM. URL: https://doi.org/10.1145/3447548.3467260.
  46. Charalampos E. Tsourakakis. Counting triangles in real-world networks using projections. Knowledge and Information Systems, 26(3):501-520, March 2011. URL: https://doi.org/10.1007/s10115-010-0291-2.
  47. Duncan J Watts and Steven H Strogatz. Collective dynamics of `small-world' networks. Nature, 393(6684), June 1998. Google Scholar
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