Skip to main content
SpringerLink
Log in

Multilayer Hypergraph Clustering Using the Aggregate Similarity Matrix

  • Conference paper
  • First Online:
Algorithms and Models for the Web Graph (WAW 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13894))

Included in the following conference series:

  • 352 Accesses

Abstract

We consider the community recovery problem on a multilayer variant of the hypergraph stochastic block model (HSBM). Each layer is associated with an independent realization of a d-uniform HSBM on N vertices. Given the similarity matrix containing the aggregated number of hyperedges incident to each pair of vertices, the goal is to obtain a partition of the N vertices into disjoint communities. In this work, we investigate a semidefinite programming (SDP) approach and obtain information–theoretic conditions on the model parameters that guarantee exact recovery both in the assortative and the disassortative cases.

Supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters, and the French government through the RISE Academy of UCA\(^\text {JEDI}\) Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-0001.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Source code: https://github.com/kalaluusua/Hypergraph-clustering.git.

References

  1. Abbé, E.: Community detection and stochastic block models: recent developments. J. Mach. Learn. Res. 18, 1–86 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Ahn, K., Lee, K., Suh, C.: Hypergraph spectral clustering in the weighted stochastic block model. IEEE J. Sel. Top. Sign. Process. 12(5), 959–974 (2018)

    Article  Google Scholar 

  3. Alaluusua, K., Leskelä, L.: Consistent Bayesian community recovery in multilayer networks. In: IEEE International Symposium on Information Theory (ISIT), pp. 2726–2731 (2022)

    Google Scholar 

  4. Angelini, M.C., Caltagirone, F., Krzakala, F., Zdeborová, L.: Spectral detection on sparse hypergraphs. In: Annual Allerton Conference on Communication, Control, and Computing (2015)

    Google Scholar 

  5. Avrachenkov, K., Dreveton, M.: Statistical Analysis of Networks. Now Publishers Inc, Delft (2022)

    Book  Google Scholar 

  6. Avrachenkov, K., Dreveton, M., Leskelä, L.: Community recovery in non-binary and temporal stochastic block models (2022). https://arxiv.org/abs/2008.04790

  7. Bergman, E., Leskelä, L.: Connectivity of random hypergraphs with a given hyperedge size distribution (2022). https://arxiv.org/abs/2207.04799

  8. Brusa, L., Matias, C.: Model-based clustering in simple hypergraphs through a stochastic blockmodel (2022). https://arxiv.org/abs/2210.05983

  9. Chien, I., Lin, C.Y., Wang, I.H.: Community detection in hypergraphs: optimal statistical limit and efficient algorithms. In: International Conference on Artificial Intelligence and Statistics (AISTATS) (2018)

    Google Scholar 

  10. Chien, I.E., Lin, C.Y., Wang, I.H.: On the minimax misclassification ratio of hypergraph community detection. IEEE Trans. Inf. Theor. 65(12), 8095–8118 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cole, S., Zhu, Y.: Exact recovery in the hypergraph stochastic block model: a spectral algorithm. Linear Algebra Appl. 593, 45–73 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gaudio, J., Joshi, N.: Community detection in the hypergraph SBM: optimal recovery given the similarity matrix (2022). https://arxiv.org/abs/2208.12227

  13. Ghoshdastidar, D., Dukkipati, A.: Consistency of spectral partitioning of uniform hypergraphs under planted partition model. In: Advances in Neural Information Processing Systems (NeurIPS) (2014)

    Google Scholar 

  14. Ghoshdastidar, D., Dukkipati, A.: A provable generalized tensor spectral method for uniform hypergraph partitioning. In: International Conference on Machine Learning (ICML) (2015)

    Google Scholar 

  15. Ghoshdastidar, D., Dukkipati, A.: Spectral clustering using multilinear SVD: analysis, approximations and applications. In: AAAI Conference on Artificial Intelligence (2015)

    Google Scholar 

  16. Ghoshdastidar, D., Dukkipati, A.: Consistency of spectral hypergraph partitioning under planted partition model. Ann. Stat. 45(1), 289–315 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gösgens, M.M., Tikhonov, A., Prokhorenkova, L.: Systematic analysis of cluster similarity indices: how to validate validation measures. In: International Conference on Machine Learning (ICML) (2021)

    Google Scholar 

  18. Guerrero-Sosa, J.D., Menéndez-Domínguez, V.H., Castellanos-Bolaños, M.E., Curi-Quintal, L.F.: Analysis of internal and external academic collaboration in an institution through graph theory. Vietnam J. Comput. Sci. 7(04), 391–415 (2020)

    Article  Google Scholar 

  19. Hajek, B., Wu, Y., Xu, J.: Achieving exact cluster recovery threshold via semidefinite programming. IEEE Trans. Inf. Theor. 62(5), 2788–2797 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193–218 (1985)

    Article  MATH  Google Scholar 

  21. Kamiński, B., Poulin, V., Prałat, P., Szufel, P., Théberge, F.: Clustering via hypergraph modularity. PLoS ONE 14(11), 1–15 (2019)

    Article  Google Scholar 

  22. Kamiński, B., Prałat, P., Théberge, F.: Community detection algorithm using hypergraph modularity. In: International Conference on Complex Networks and their Applications (2021)

    Google Scholar 

  23. Kamiński, B., Prałat, P., Théberge, F.: Hypergraph artificial benchmark for community detection (h-ABCD) (2022). https://arxiv.org/abs/2210.15009

  24. Ke, Z.T., Shi, F., Xia, D.: Community detection for hypergraph networks via regularized tensor power iteration (2020). https://arxiv.org/abs/1909.06503

  25. Kim, C., Bandeira, A.S., Goemans, M.X.: Community detection in hypergraphs, spiked tensor models, and sum-of-squares. In: International Conference on Sampling Theory and Applications (SampTA) (2017)

    Google Scholar 

  26. Kim, C., Bandeira, A.S., Goemans, M.X.: Stochastic block model for hypergraphs: statistical limits and a semidefinite programming approach (2018). https://arxiv.org/abs/1807.02884

  27. Kivelä, M., Arenas, A., Barthelemy, M., Gleeson, J.P., Moreno, Y., Porter, M.A.: Multilayer networks. Journal of ComplexJournal of ComplexJournal of ComplexJournal of ComplexJournal of ComplexJ. Complex 2(3), 203–271 (2014)

    Google Scholar 

  28. Kumar, T., Vaidyanathan, S., Ananthapadmanabhan, H., Parthasarathy, S., Ravindran, B.: A new measure of modularity in hypergraphs: theoretical insights and implications for effective clustering. In: International Conference on Complex Networks and their Applications (2019)

    Google Scholar 

  29. Kumar, T., Vaidyanathan, S., Ananthapadmanabhan, H., Parthasarathy, S., Ravindran, B.: Hypergraph clustering by iteratively reweighted modularity maximization. Appl. Netw. Sci. 5(1), 1–22 (2020). https://doi.org/10.1007/s41109-020-00300-3

    Article  Google Scholar 

  30. Lee, J., Kim, D., Chung, H.W.: Robust hypergraph clustering via convex relaxation of truncated MLE. IEEE J. Sel. Areas Inf. Theor. 1(3), 613–631 (2020)

    Article  Google Scholar 

  31. Lei, J., Chen, K., Lynch, B.: Consistent community detection in multi-layer network data. Biometrika 107(1), 61–73 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lesieur, T., Miolane, L., Lelarge, M., Krzakala, F., Zdeborová, L.: Statistical and computational phase transitions in spiked tensor estimation. In: 2017 IEEE International Symposium on Information Theory (ISIT), pp. 511–515. IEEE (2017)

    Google Scholar 

  33. Pal, S., Zhu, Y.: Community detection in the sparse hypergraph stochastic block model. Random Struct. Algorithms 59, 407–463 (2021)

    Article  MathSciNet  Google Scholar 

  34. Pensky, M., Zhang, T.: Spectral clustering in the dynamic stochastic block model. Electr. J. Stat. 13(1), 678–709 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Stephan, L., Zhu, Y.: Sparse random hypergraphs: Non-backtracking spectra and community detection (2022). https://arxiv.org/abs/2203.07346

  36. Zhang, Q., Tan, V.Y.F.: Exact recovery in the general hypergraph stochastic block model. IEEE Trans. Inf. Theor. 69(1), 453–471 (2023)

    Article  MathSciNet  Google Scholar 

  37. Zhen, Y., Wang, J.: Community detection in general hypergraph via graph embedding. J. Am. Stat. Assoc. 1–10 (2022)

    Google Scholar 

  38. Zhou, D., Huang, J., Schölkopf, B.: Learning with hypergraphs: clustering, classification, and embedding. In: Advances in Neural Information Processing Systems (NeurIPS) (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aalto University, Espoo, Finland

    Kalle Alaluusua & Lasse Leskelä

  2. INRIA, Sophia Antipolis, Valbonne, France

    Konstantin Avrachenkov & B. R. Vinay Kumar

Authors
  1. Kalle Alaluusua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantin Avrachenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. R. Vinay Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lasse Leskelä
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kalle Alaluusua .

Editor information

Editors and Affiliations

  1. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    Megan Dewar

  2. Toronto Metropolitan University, Toronto, ON, Canada

    Paweł Prałat

  3. SGH Warsaw School of Economics, Warsaw, Poland

    Przemysław Szufel

  4. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    François Théberge

  5. SGH Warsaw School of Economics, Warsaw, Poland

    Małgorzata Wrzosek

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Alaluusua, K., Avrachenkov, K., Kumar, B.R.V., Leskelä, L. (2023). Multilayer Hypergraph Clustering Using the Aggregate Similarity Matrix. In: Dewar, M., Prałat, P., Szufel, P., Théberge, F., Wrzosek, M. (eds) Algorithms and Models for the Web Graph. WAW 2023. Lecture Notes in Computer Science, vol 13894. Springer, Cham. https://doi.org/10.1007/978-3-031-32296-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-32296-9_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-32295-2

  • Online ISBN: 978-3-031-32296-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation