Document Open Access Logo

Modularity of Preferential Attachment Graphs

Authors Katarzyna Rybarczyk , Małgorzata Sulkowska

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2026.76.pdf
  • Filesize: 0.73 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Stochastic processes
Keywords
  • Modularity
  • preferential attachment model
  • edge expansion

Metrics

Abstract

We study a preferential attachment model G_n^h. The graph G_n^h is generated from a finite initial graph by adding new vertices one at a time. Each new vertex connects to h ≥ 1 already existing vertices, and these are chosen with probability proportional to their current degrees. We are particularly interested in the community structure of G_n^h, which is expressed in terms of the so-called modularity. We prove that the modularity of G_n^h is, with high probability, upper bounded by a function that tends to 0 as h tends to infinity. This resolves a conjecture of Prokhorenkova, Prałat, and Raigorodskii from 2016.
As a byproduct, we obtain novel concentration results (which are interesting in their own right) for the volume and edge density parameters of vertex subsets of G_n^h. The key ingredient here is the definition of a function μ, which serves as a natural measure for vertex subsets, and is proportional to the average size of their volumes. This extends previous results on the topic by Frieze, Pérez-Giménez, Prałat, and Reiniger from 2019.

Cite As Get BibTex

Katarzyna Rybarczyk and Małgorzata Sulkowska. Modularity of Preferential Attachment Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 76:1-76:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.76

Author Details

Katarzyna Rybarczyk
  • Adam Mickiewicz University, Poznań, Poland
Małgorzata Sulkowska
  • Department of Fundamentals of Computer Science, Wrocław University of Science and Technology, Poland

Funding

This research was funded in whole or in part by National Science Centre, Poland, grant OPUS-25 no 2023/49/B/ST6/02517. For the purpose of Open Access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

References

  1. K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19(3):357-367, 1967. URL: https://doi.org/10.2748/tmj/1178243286.
  2. J.P. Bagrow. Communities and bottlenecks: Trees and treelike networks have high modularity. Physical Review E, 85:066118, 2012. URL: https://doi.org/10.1103/PhysRevE.85.066118.
  3. A.L. Barabási and R. Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999. URL: https://doi.org/10.1126/science.286.5439.509.
  4. P. Bennett and A. Dudek. A gentle introduction to the differential equation method and dynamic concentration. Discrete Mathematics, 345(12):113071, 2022. URL: https://doi.org/10.1016/j.disc.2022.113071.
  5. V.D. Blondel, J.L. Guillaume, R. Lambiotte, and E. Lefebvre. Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008(10):P10008, 2008. URL: https://doi.org/10.1088/1742-5468/2008/10/P10008.
  6. B. Bollobás and O. Riordan. Handbook of Graphs and Networks: From the Genome to the Internet. Wiley-VCH, 2003. Pages 1-34. URL: https://doi.org/10.1002/3527602755.
  7. B. Bollobás and O. Riordan. The diameter of a scale-free random graph. Combinatorica, 24:5-34, 2004. URL: https://doi.org/10.1007/s00493-004-0002-2.
  8. B. Bollobás, O. Riordan, J. Spencer, and G. Tusnády. The degree sequence of a scale‐free random graph process. Random Structures & Algorithms, 18(3):279-290, April 2001. URL: https://doi.org/10.1002/rsa.1009.
  9. J. Chellig, N. Fountoulakis, and F. Skerman. The modularity of random graphs on the hyperbolic plane. Journal of Complex Networks, 10(1):cnab051, 2021. URL: https://doi.org/10.1093/comnet/cnab051.
  10. T.N. Dinh and M.T. Thai. Finding community structure with performance guarantees in scale-free networks. In 2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing, pages 888-891, 2011. URL: https://doi.org/10.1109/PASSAT/SocialCom.2011.185.
  11. D. Freedman. On tail probabilities for martingales. The Annals of Probability, 3(1):100-118, 1975. URL: https://doi.org/10.1214/aop/1176996452.
  12. A. Frieze, X. Pérez-Giménez, P. Prałat, and B. Reiniger. Perfect matchings and Hamiltonian cycles in the preferential attachment model. Random Structures & Algorithms, 54(2):258-288, 2019. URL: https://doi.org/10.1002/rsa.20778.
  13. G. Gilad and R. Sharan. From Leiden to Tel-Aviv University (TAU): exploring clustering solutions via a genetic algorithm. PNAS Nexus, 2(6):pgad180, 2023. URL: https://doi.org/10.1093/pnasnexus/pgad180.
  14. W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):13-30, 1963. URL: https://doi.org/10.1080/01621459.1963.10500830.
  15. R. van der Hofstad. Random Graphs and Complex Networks. Vol.1. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781316779422.
  16. S. Janson, T. Łuczak, and A. Ruciński. Random Graphs. John Wiley & Sons, Inc., 2000. URL: https://doi.org/10.1002/9781118032718.
  17. R. P. Boas Jr. and J. W. Wrench Jr. Partial sums of the harmonic series. The American Mathematical Monthly, 78(8):864-870, 1971. URL: https://doi.org/10.2307/2316476.
  18. B. Kamiński, V. Poulin, P. Prałat, P. Szufel, and F. Théberge. Clustering via hypergraph modularity. Plos One, 14:e0224307, February 2019. URL: https://doi.org/10.1371/journal.pone.0224307.
  19. B. Kamiński, P. Prałat, and F.Théberge. Mining Complex Networks. Chapman and Hall/CRC, 2021. URL: https://doi.org/10.1201/9781003218869.
  20. M. Lasoń and M. Sulkowska. Modularity of minor-free graphs. Journal of Graph Theory, 102(4):728-736, 2023. URL: https://doi.org/10.1002/jgt.22896.
  21. L. Lichev and D. Mitsche. On the modularity of 3-regular random graphs and random graphs with given degree sequences. Random Structures & Algorithms, 61(4):754-802, 2022. URL: https://doi.org/10.1002/rsa.21080.
  22. Hosam M. Mahmoud, R. T. Smythe, and J. Szymański. On the structure of random plane-oriented recursive trees and their branches. Random Structures & Algorithms, 4(2):151-176, 1993. URL: https://doi.org/10.1002/rsa.3240040204.
  23. C. McDiarmid, K. Rybarczyk, F. Skerman, and M. Sulkowska. Note on edge expansion and modularity in preferential attachment graphs, 2025. Preprint. Google Scholar
  24. C. McDiarmid and F. Skerman. Modularity in random regular graphs and lattices. Electronic Notes in Discrete Mathematics, 43:431-437, 2013. URL: https://doi.org/10.1016/j.endm.2013.07.063.
  25. C. McDiarmid and F. Skerman. Modularity of regular and treelike graphs. Journal of Complex Networks, 6(4):596-619, 2018. URL: https://doi.org/10.1093/comnet/cnx046.
  26. C. McDiarmid and F. Skerman. Modularity of Erdős-Rényi random graphs. Random Structures & Algorithms, 57(1):211-243, 2020. URL: https://doi.org/10.1002/rsa.20910.
  27. M. Mihail, C. Papadimitriou, and A. Saberi. On certain connectivity properties of the internet topology. Journal of Computer and System Sciences, 72:239-251, 2006. URL: https://doi.org/10.1016/j.jcss.2005.06.009.
  28. M.E.J. Newman. Networks. Oxford University Press, 2018. URL: https://doi.org/10.1093/oso/9780198805090.001.0001.
  29. M.E.J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69(2):026113, 2004. URL: https://doi.org/10.1103/physreve.69.026113.
  30. L. Prokhorenkova, A. Raigorodskii, and P. Prałat. Modularity of complex networks models. Internet Mathematics, 2017. URL: https://doi.org/10.24166/im.12.2017.
  31. H. Robbins. A remark on Stirling’s formula. The American Mathematical Monthly, 62(1):26-29, 1955. URL: https://doi.org/10.2307/2308012.
  32. K. Rybarczyk. Modularity of random intersection graphs. In M. Bloznelis, P. Drungilas, B. Kamiński, P. Prałat, M. Šileikis, F. Théberge, and R. Vaicekauskas, editors, Modelling and Mining Networks, pages 30-44, Cham, 2025. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-92898-7_3.
  33. K. Rybarczyk and M. Sulkowska. Modularity of preferential attachment graphs, 2025. URL: https://doi.org/10.48550/arXiv.2501.06771.
  34. K. Rybarczyk and M. Sulkowska. New bounds on the modularity of G(n,p), 2025. URL: https://doi.org/10.48550/arXiv.2504.16254.
  35. J. Szymański. On a nonuniform random recursive tree. In A. Barlotti, M. Biliotti, A. Cossu, G. Korchmaros, and G. Tallini, editors, Annals of Discrete Mathematics (33), volume 144 of North-Holland Mathematics Studies, pages 297-306. North-Holland, 1987. URL: https://doi.org/10.1016/S0304-0208(08)73062-7.
  36. V.A. Traag, L. Waltman, and N.J. van Eck. From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9(5233), 2019. URL: https://doi.org/10.1038/s41598-019-41695-z.
  37. L. Warnke. On the method of typical bounded differences. Combinatorics, Probability & Computing, 25:269-299, 2022. URL: https://doi.org/10.1017/S0963548315000103.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact