- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Block-corrected modularity for community detection

Hasti Narimanzadeh*, Takayuki Hiraoka, and Mikko Kivelä

  • *Contact author: hasti.narimanzadeh@aalto.fi
PDF

Phys. Rev. E 112, 024307 – Published 27 August, 2025

DOI: https://doi.org/10.1103/7sjf-c6jz

Abstract

Unknown node attributes in complex networks may introduce community structures that are important to distinguish from those driven by known attributes. We propose a block-corrected modularity that discounts given block structures present in the network to reveal communities masked by them. We show analytically how the proposed modularity finds the community structure driven by an unknown attribute in a simple network model. Further, we observe that the block-corrected modularity finds the underlying community structure on a number of simple synthetic network models while methods using different null models fail. We develop an efficient spectral method as well as two Louvain-inspired fine-tuning algorithms to maximize the proposed modularity and demonstrate their performance on several synthetic network models. Finally, we assess our methodology on various real-world citation networks built using the OpenAlex data by correcting for the temporal citation patterns.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. M. Girvan and M. E. Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99, 7821 (2002).
  2. J. Ji, A. Zhang, C. Liu, X. Quan, and Z. Liu, Survey: Functional module detection from protein-protein interaction networks, IEEE Trans. Knowledge Data Eng. 26, 261 (2012).
  3. A. L. Traud, P. J. Mucha, and M. A. Porter, Social structure of Facebook networks, Physica A 391, 4165 (2012).
  4. P. Guerra, W. Meira Jr, C. Cardie, and R. Kleinberg, A measure of polarization on social media networks based on community boundaries, in Proceedings of the International AAAI Conference on Web and Social Media (The Association for the Advancement of Artificial Intelligence, Cambridge, Massachusetts, 2013), Vol. 7, pp. 215–224.
  5. D. Hric, K. Kaski, and M. Kivelä, Stochastic block model reveals maps of citation patterns and their evolution in time, J. Info. 12, 757 (2018).
  6. A. Salloum, T. H. Y. Chen, and M. Kivelä, Separating polarization from noise: Comparison and normalization of structural polarization measures, Proc. ACM Hum.-Comput. Interact. 6, 1 (2022).
  7. M. E. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E 74, 036104 (2006).
  8. V. A. Traag and J. Bruggeman, Community detection in networks with positive and negative links, Phys. Rev. E 80, 036115 (2009).
  9. V. D. Blondel, J.-L. Guillaume, R. Lambiotte, and E. Lefebvre, Fast unfolding of communities in large networks, J. Stat. Mech. (2008) P10008.
  10. T. Richardson, P. J. Mucha, and M. A. Porter, Spectral tripartitioning of networks, Phys. Rev. E 80, 036111 (2009).
  11. S. White and P. Smyth, A spectral clustering approach to finding communities in graphs, in Proceedings of the 2005 SIAM International Conference on Data Mining (SIAM, Philadelphia, 2005), pp. 274–285.
  12. T. P. Peixoto, Descriptive vs. inferential community detection in networks Pitfalls, myths and half-truths (Cambridge University Press, Cambridge, 2023).
  13. B. H. Good, Y.-A. De Montjoye, and A. Clauset, Performance of modularity maximization in practical contexts, Phys. Rev. E 81, 046106 (2010).
  14. L. Speidel, T. Takaguchi, and N. Masuda, Community detection in directed acyclic graphs, Eur. Phys. J. B 88, 203 (2015).
  15. M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M. A. Porter, S. Gómez, and A. Arenas, Mathematical formulation of multilayer networks, Phys. Rev. X 3, 041022 (2013).
  16. M. Bazzi, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn, and S. D. Howison, Community detection in temporal multilayer networks, with an application to correlation networks, Multiscale Model. Sim. 14, 1 (2016).
  17. P. Expert, T. S. Evans, V. D. Blondel, and R. Lambiotte, Uncovering space-independent communities in spatial networks, Proc. Natl. Acad. Sci. USA 108, 7663 (2011).
  18. X. Liu, T. Murata, and K. Wakita, Extending modularity by capturing the similarity attraction feature in the null model, arXiv:1210.4007.
  19. A. Arenas, A. Fernandez, S. Fortunato, and S. Gomez, Motif-based communities in complex networks, J. Phys. A: Math. Theor. 41, 224001 (2008).
  20. C. Bothorel, J. D. Cruz, M. Magnani, and B. Micenkova, Clustering attributed graphs: Models, measures and methods, Netw. Sci. 3, 408 (2015).
  21. P. Chunaev, Community detection in node-attributed social networks: A survey, Comput. Sci. Rev. 37, 100286 (2020).
  22. V. Red, E. D. Kelsic, P. J. Mucha, and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks, SIAM Rev. 53, 526 (2011).
  23. C. Jia, Y. Li, M. B. Carson, X. Wang, and J. Yu, Node attribute-enhanced community detection in complex networks, Sci. Rep. 7, 2626 (2017).
  24. Y. Asim, A. Majeed, R. Ghazal, B. Raza, W. Naeem, and A. K. Malik, Community detection in networks using node attributes and modularity, Int. J. Adv. Comput. Sci. Appl. 8, 382 (2017).
  25. Y. Zhou, H. Cheng, and J. X. Yu, Graph clustering based on structural/attribute similarities, Proc. VLDB Endow. 2, 718 (2009).
  26. Z. Huang, Y. Ye, X. Li, F. Liu, and H. Chen, Joint weighted nonnegative matrix factorization for mining attributed graphs, in Advances in Knowledge Discovery and Data Mining: 21st Pacific-Asia Conference, PAKDD 2017, Jeju, South Korea, May 23–26, 2017, Proceedings, Part I 21 (Springer, Berlin, 2017), pp. 368–380.
  27. C. Zhe, A. Sun, and X. Xiao, Community detection on large complex attribute network, in Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (Association for Computing Machinery (ACM), Anchorage, Alaska, 2019), pp. 2041–2049.
  28. P. D. B. Parolo, R. K. Pan, R. Ghosh, B. A. Huberman, K. Kaski, and S. Fortunato, Attention decay in science, J. Info. 9, 734 (2015).
  29. T. P. Peixoto, Ordered community detection in directed networks, Phys. Rev. E 106, 024305 (2022).
  30. P. Erdős and A. Rényi, On random graphs I, Publ. Math. Debrecen 6, 18 (1959).
  31. P. Erdős and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).
  32. M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct. Algorithms 6, 161 (1995).
  33. B. Bollobás, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Eur. J. Combinatorics 1, 311 (1980).
  34. P. W. Holland, K. B. Laskey, and S. Leinhardt, Stochastic blockmodels: First steps, Social Netw. 5, 109 (1983).
  35. M. J. Barber, Modularity and community detection in bipartite networks, Phys. Rev. E 76, 066102 (2007).
  36. B. Karrer and M. E. Newman, Random graph models for directed acyclic networks, Phys. Rev. E 80, 046110 (2009).
  37. J. N. Pougué-Biyong and R. Lambiotte, Signedlouvain: Louvain for signed networks, arXiv:2407.19288.
  38. G. Zeno, T. La Fond, and J. Neville, Dymond: Dynamic motif-nodes network generative model, in Proceedings of the Web Conference 2021 (International World Wide Web Conference Committee (IW3C2), Ljubljana, Slovenia, 2021), pp. 718–729.
  39. S. Purohit, L. B. Holder, and G. Chin, Temporal graph generation based on a distribution of temporal motifs, in Proceedings of the 14th International Workshop on Mining and Learning with Graphs (Association for Computing Machinery (ACM), London, UK 2018), Vol. 7.
  40. B. Karrer and M. E. Newman, Stochastic blockmodels and community structure in networks, Phys. Rev. E 83, 016107 (2011).
  41. E. A. Leicht and M. E. Newman, Community structure in directed networks, Phys. Rev. Lett. 100, 118703 (2008).
  42. D. Krackhardt and R. N. Stern, Informal networks and organizational crises: An experimental simulation, Social Psychol. Q. 51 123 (1988).
  43. W. M. Rand, Objective criteria for the evaluation of clustering methods, J. Am. Stat. Assoc. 66, 846 (1971).
  44. L. Hubert and P. Arabie, Comparing partitions, J. Class. 2, 193 (1985).
  45. U. Brandes, D. Delling, M. Gaertler, R. Gorke, M. Hoefer, Z. Nikoloski, and D. Wagner, On modularity clustering, IEEE Trans. Knowledge Data Eng. 20, 172 (2007).
  46. A. Clauset, M. E. Newman, and C. Moore, Finding community structure in very large networks, Phys. Rev. E 70, 066111 (2004).
  47. M. E. Newman, Modularity and community structure in networks, Proc. Natl. Acad. Sci. USA 103, 8577 (2006).
  48. M. Newman, Spectral community detection in sparse networks, arXiv:1308.6494.
  49. J. Priem, H. Piwowar, and R. Orr, Openalex: A fully-open index of scholarly works, authors, venues, institutions, and concepts, arXiv:2205.01833.
  50. L. Peel, D. B. Larremore, and A. Clauset, The ground truth about metadata and community detection in networks, Sci. Adv. 3, e1602548 (2017).
  51. M. E. Newman and A. Clauset, Structure and inference in annotated networks, Nat. Commun. 7, 11863 (2016).
  52. D. Hric, T. P. Peixoto, and S. Fortunato, Network structure, metadata, and the prediction of missing nodes and annotations, Phys. Rev. X 6, 031038 (2016).
  53. D. Hric, R. K. Darst, and S. Fortunato, Community detection in networks: Structural communities versus ground truth, Phys. Rev. E 90, 062805 (2014).
  54. L. Mangold and C. Roth, Quantifying metadata relevance to network block structure using description length, Commun. Phys. 7, 331 (2024).
  55. https://help.openalex.org/hc/en-us/articles/24397285563671-About-the-data.
  56. 10.5281/zenodo.14867442
  57. R. Guimera, M. Sales-Pardo, and L. A. N. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70, 025101 (2004).
  58. J. Reichardt and S. Bornholdt, When are networks truly modular? Physica D 224, 20 (2006).
  59. T. P. Peixoto, Bayesian stochastic blockmodeling, Advances in Network Clustering and Blockmodeling, edited by P. Doreian, V. Batagelj, and A. Ferligoj, Chapter 11 (John Wiley & Sons, Ltd, 2019), pp. 289–332.
  60. M. Jerdee, A. Kirkley, and M. Newman, Normalized mutual information is a biased measure for classification and community detection, arXiv:2307.01282.

Outline

Information

Phys. Rev. E 112, 024307– Published 27 August, 2025
  • Received 26 February 2025
  • Revised 24 June 2025
  • Accepted 29 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation