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Modularity and Graph Expansion

Authors Baptiste Louf, Colin McDiarmid, Fiona Skerman

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Author Details

Baptiste Louf
  • CNRS and Institut de Mathématiques de Bordeaux, France
Colin McDiarmid
  • Department of Statistics, University of Oxford, UK
Fiona Skerman
  • Department of Mathematics, Uppsala University, Sweden

Funding

  • Louf, Baptiste: Part of this work done while supported by the Wallenberg Foundation.
  • Skerman, Fiona: Partially supported by the Wallenberg AI, Autonomous Systems and Software Program WASP and Uppsala University AI4Research. Part of this work done while visiting the Simons Institute for the Theory of Computing, supported by a Simons-Berkeley Research Fellowship.

Acknowledgements

We are grateful to Prasad Tetali for helpful discussions.

Cite AsGet BibTex

Baptiste Louf, Colin McDiarmid, and Fiona Skerman. Modularity and Graph Expansion. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 78:1-78:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.78

Abstract

We relate two important notions in graph theory: expanders which are highly connected graphs, and modularity a parameter of a graph that is primarily used in community detection. More precisely, we show that a graph having modularity bounded below 1 is equivalent to it having a large subgraph which is an expander. We further show that a connected component H will be split in an optimal partition of the host graph G if and only if the relative size of H in G is greater than an expansion constant of H. This is a further exploration of the resolution limit known for modularity, and indeed recovers the bound that a connected component H in the host graph G will not be split if e(H) < √{2e(G)}.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
  • Theory of computation → Theory and algorithms for application domains
  • Mathematics of computing → Spectra of graphs
Keywords
  • edge expansion
  • modularity
  • community detection
  • resolution limit
  • conductance

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References

  1. M. Ajtai, J. Komlos, and E. Szemeredi. Deterministic simulation in logspace. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC '87, pages 132-140, New York, NY, USA, 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28410.
  2. N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986. Theory of computing (Singer Island, Fla., 1984). URL: https://doi.org/10.1007/BF02579166.
  3. P. J. Bickel and A. Chen. A nonparametric view of network models and newman-girvan and other modularities. Proceedings of the National Academy of Sciences, 106(50):21068-21073, 2009. URL: https://doi.org/10.1073/pnas.0907096106.
  4. 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, 10, 2008. URL: https://doi.org/10.1088/1742-5468/2008/10/P10008.
  5. M. Bolla, B. Bullins, S. Chaturapruek, S. Chen, and K. Friedl. Spectral properties of modularity matrices. Linear Algebra and Its Applications, 473:359-376, 2015. URL: https://doi.org/10.1016/j.laa.2014.10.039.
  6. J. Böttcher, K. P. Pruessmann, A. Taraz, and A. Würfl. Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. European Journal of Combinatorics, 31(5):1217-1227, 2010. URL: https://doi.org/10.1016/j.ejc.2009.10.010.
  7. U. Brandes, D. Delling, M. Gaertler, R. Gorke, M. Hoefer, Z. Nikoloski, and D. Wagner. On modularity clustering. Knowledge and Data Engineering, IEEE Transactions on, 20(2):172-188, 2008. URL: https://doi.org/10.1109/TKDE.2007.190689.
  8. D. Chakraborti, J. Kim, J. Kim, M. Kim, and H. Liu. Well-mixing vertices and almost expanders. Proceedings of the American Mathematical Society, 2022. URL: https://doi.org/10.1090/proc/16090.
  9. Jeff Cheeger. A lower bound for the smallest eigenvalue of the laplacian. In Proceedings of the Princeton conference in honor of Professor S. Bochner, pages 195-199, 1969. Google Scholar
  10. F. Chung. Spectral graph theory, volume 92. American Mathematical Soc. Providence, RI, 1997. URL: https://doi.org/10.1090/cbms/092.
  11. V. Cohen-Addad, A. Kosowski, F. Mallmann-Trenn, and D. Saulpic. On the power of louvain in the stochastic block model. Advances in Neural Information Processing Systems, 33, 2020. Google Scholar
  12. E. Davis and S. Sethuraman. Consistency of modularity clustering on random geometric graphs. Annals of Applied Probability, 28(4):2003-2062, 2018. URL: https://doi.org/10.1214/17-AAP1313.
  13. D. Fasino and F. Tudisco. An algebraic analysis of the graph modularity. SIAM Journal on Matrix Analysis and Applications, 35(3):997-1018, 2014. URL: https://doi.org/10.1137/130943455.
  14. S. Fortunato. Community detection in graphs. Phys. Rep., 486(3-5):75-174, 2010. URL: https://doi.org/10.1016/j.physrep.2009.11.002.
  15. S. Fortunato and M. Barthélemy. Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104(1):36-41, 2007. URL: https://doi.org/10.1073/pnas.0605965104.
  16. D. Gillman. A Chernoff bound for random walks on expander graphs. SIAM J. Comput., 27(4):1203-1220, 1998. URL: https://doi.org/10.1137/S0097539794268765.
  17. S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4):439-561, 2006. URL: https://doi.org/10.1090/S0273-0979-06-01126-8.
  18. R. Kannan, L. Lovász, and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete & Computational Geometry, 13(3):541-559, 1995. URL: https://doi.org/10.1007/BF02574061.
  19. R. Kannan, S. Vempala, and A. Vetta. On clusterings: good, bad and spectral. J. ACM, 51(3):497-515, 2004. URL: https://doi.org/10.1145/990308.990313.
  20. M. Koshelev. Modularity in planted partition model. Computational Management Science, 20(1):34, 2023. URL: https://doi.org/10.1007/s10287-023-00466-y.
  21. M. Krivelevich. Finding and using expanders in locally sparse graphs. SIAM J. Discrete Math., 32(1):611-623, 2018. URL: https://doi.org/10.1137/17M1128721.
  22. M. Krivelevich. Expanders—how to find them, and what to find in them. Surveys in Combinatorics, pages 115-142, 2019. URL: https://doi.org/10.1017/9781108649094.005.
  23. A. Lancichinetti and S. Fortunato. Limits of modularity maximization in community detection. Physical Review E, 84(6):066122, 2011. URL: https://doi.org/10.1103/physreve.84.066122.
  24. M. Lasoń and M. Sulkowska. Modularity of minor-free graphs. Journal of Graph Theory, 2022. URL: https://doi.org/10.1002/jgt.22896.
  25. 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.
  26. B. Louf and F. Skerman. Finding large expanders in graphs: from topological minors to induced subgraphs. Electronic Journal of Combinatorics, 30(1), 2023. URL: https://doi.org/10.37236/10859.
  27. S. Majstorovic and D. Stevanovic. A note on graphs whose largest eigenvalues of the modularity matrix equals zero. Electronic Journal of Linear Algebra, 27(1):256, 2014. Google Scholar
  28. C. McDiarmid and F. Skerman. Modularity of regular and treelike graphs. Journal of Complex Networks, 6(4), 2018. URL: https://doi.org/10.1093/comnet/cnx046.
  29. 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.
  30. C. McDiarmid and F. Skerman. Modularity of nearly complete graphs and bipartite graphs. arXiv preprint arXiv:2311.06875, 2023. Google Scholar
  31. K. Meeks and F. Skerman. The parameterised complexity of computing the maximum modularity of a graph. Algorithmica, 82(8):2174-2199, 2020. URL: https://doi.org/10.1007/s00453-019-00649-7.
  32. F. de Montgolfier, M. Soto, and L. Viennot. Asymptotic modularity of some graph classes. In Algorithms and Computation, pages 435-444. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25591-5_45.
  33. 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.
  34. M. A. Porter, J.-P. Onnela, and P. J. Mucha. Communities in networks. Notices Amer. Math. Soc., 56(9):1082-1097, 2009. Google Scholar
  35. L. O. Prokhorenkova, P. Prałat, and A. Raigorodskii. Modularity in several random graph models. Electronic Notes in Discrete Mathematics, 61:947-953, 2017. URL: https://doi.org/10.1016/j.endm.2017.07.058.
  36. V. A. Traag, L. Waltman, and N. J. Van Eck. From louvain to leiden: guaranteeing well-connected communities. Scientific reports, 9(1):5233, 2019. URL: https://doi.org/10.1038/s41598-019-41695-z.
  37. P. Van Mieghem, X. Ge, P. Schumm, S. Trajanovski, and H. Wang. Spectral graph analysis of modularity and assortativity. Physical Review E, 82(5):056113, 2010. URL: https://doi.org/10.1103/PhysRevE.82.056113.
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