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Empirical Study of Graph Spectra and Their Limitations

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Complex Networks & Their Applications XII (COMPLEX NETWORKS 2023)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1141))

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Abstract

We examine the sensitivity of community-structured graph spectra to graph size, block size and inter-block edge probability. We use the Planted Partition Model because of its transparency. While this generative model may seem simplistic, it allows us to isolate the effects of graph and block size, edge probabilities and, consequently, vertex degree distribution on spectra. These sensitivities to key graph characteristics also generalize beyond Planted Partition Model graphs, because they are based on graph structure. Notably, our results show that eigenvalues converge to those of a complete graph, with increases in graph size or inter-block edge probability. Such convergence severely limits the use of spectral techniques.

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References

  1. Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002). https://doi.org/10.1103/RevModPhys.74.47

    Article  MathSciNet  Google Scholar 

  2. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  3. Broido, A.D., Clauset, A.: Scale-free networks are rare. Nature Commun. 10(1), 1017 (2019)

    Google Scholar 

  4. Bruneau, P., Parisot, O., Otjacques, B.: A heuristic for the automatic parametrization of the spectral clustering algorithm. In: 2014 22nd International Conference on Pattern Recognition, pp. 1313–1318 (2014). https://doi.org/10.1109/ICPR.2014.235

  5. Chen, J., Lu, J., Zhan, C., Chen, G.: Laplacian Spectra and Synchronization Processes on Complex Networks, pp. 81–113. Springer US, Boston, MA (2012). https://doi.org/10.1007/978-1-4614-0754-6_4. URL https://doi.org/10.1007/978-1-4614-0754-6_4

  6. Chung, F.R.K.: Spectral graph theory. American Mathematical Soc. (1997)

    Google Scholar 

  7. Coja-Oghlan, A., Goerdt, A., Lanka, A.: Spectral partitioning of random graphs with given expected degrees. In: Navarro, G., Bertossi, L., Kohayakawa, Y. (eds.) Fourth IFIP International Conference on Theoretical Computer Science- TCS 2006, pp. 271–282. Springer, US, Boston, MA (2006)

    Chapter  Google Scholar 

  8. Condon, A., Karp, R.: Algorithms for graph partitioning on the planted partition model. Random Struct. Algorithms 18(2), 116–140 (2001). https://doi.org/10.1002/1098-2418(200103)18:2<116::AID-RSA1001>3.0.CO;2-2

  9. Erdös, P., Rényi, A.: On random graphs I. Publicationes Mathematicae Debrecen 6, 290–297 (1959)

    Article  MathSciNet  Google Scholar 

  10. Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010). https://doi.org/10.1016/j.physrep.2009.11.002

    Article  MathSciNet  Google Scholar 

  11. Fortunato, S., Hric, D.: Community detection in networks: A user guide. arXiv (2016)

    Google Scholar 

  12. Gan, L., Wan, X., Ma, Y., Lev, B.: Efficiency evaluation for urban industrial metabolism through the methodologies of emergy analysis and dynamic network stochastic block model. Sustainable Cities and Society, p. 104396 (2023)

    Google Scholar 

  13. Gilbert, E.: Random graphs. Ann. Math. Statist. 30(4), 1141–1144 (1959). https://doi.org/10.1214/aoms/1177706098.

  14. Hagberg, A., Schult, D., Swart, P.: Exploring Network Structure, Dynamics, and Function using NetworkX. In: G. Varoquaux, T. Vaught, J. Millman (eds.) Proceedings of the 7th Python in Science Conference, pp. 11–15. Pasadena, CA USA (2008)

    Google Scholar 

  15. Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. arXiv 78(4), 046110 (2008). https://doi.org/10.1103/PhysRevE.78.046110

  16. Lee, C., Wilkinson, D.J.: A review of stochastic block models and extensions for graph clustering. Appl. Netw. Sci. 4(1), 1–50 (2019)

    Article  Google Scholar 

  17. Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics 6(1), 29–123 (2009). https://doi.org/10.1080/15427951.2009.10129177

    Article  MathSciNet  Google Scholar 

  18. Lutzeyer, J.F., Walden, A.T.: Comparing Graph Spectra of Adjacency and Laplacian Matrices. arXiv e-prints arXiv:1712.03769 (2017). https://doi.org/10.48550/arXiv.1712.03769

  19. von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  20. Newman, M.E.J., Strogatz, S., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026,118 (2001). https://doi.org/10.1103/PhysRevE.64.026118.

  21. Priebe, C.E., et al.: On a two-truths phenomenon in spectral graph clustering. Proc. Natl. Acad. Sci. 116(13), 5995–6000 (2019)

    Article  MathSciNet  Google Scholar 

  22. Rao Nadakuditi, R., Newman, M.E.J.: Graph spectra and the detectability of community structure in networks. arXiv e-prints arXiv:1205.1813 (2012). https://doi.org/10.48550/arXiv.1205.1813

  23. Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878–1915 (2011). https://doi.org/10.1214/11-AOS887.

  24. Spielman, D.A.: Spectral graph theory and its applications. In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07), pp. 29–38 (2007). https://doi.org/10.1109/FOCS.2007.56

  25. documentaton page (author unknown), O.: Planted partition model. https://networkx.org/documentation/stable/reference/generated/networkx.generators.community.planted_partition_graph.html

  26. documentaton page (author unknown), O.: Stochastic block model. https://networkx.org/documentation/stable/reference/generated/networkx.generators.community.stochastic_block_model.html

  27. Zhan, C., Chen, G., Yeung, L.F.: On the distributions of Laplacian eigenvalues versus node degrees in complex networks. Physica A: Statistical Mechanics and its Applications 389(8), 1779–1788 (2010). https://doi.org/10.1016/j.physa.2009.12.005. URL https://www.sciencedirect.com/science/article/pii/S0378437109010012

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Acknowledgments

– The work of P.M. was funded by a MITACS grant (grant# IT33832).

– The authors thank Prof. Valery A. Kalyagin of the Laboratory of Algorithms and Technologies for Networks Analysis in Nizhny Novogorod, Russia, for his comments on the early stages of this work.

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Authors and Affiliations

  1. University of Toronto, Toronto, ON, Canada

    Pierre Miasnikof & Yuri Lawryshyn

  2. Queen Mary University of London, London, UK

    Alexander Y. Shestopaloff

  3. Memorial University of Newfoundland, St, John’s, NL, Canada

    Alexander Y. Shestopaloff

  4. University of Western Ontario, London, ON, Canada

    Cristián Bravo

Authors
  1. Pierre Miasnikof
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  2. Alexander Y. Shestopaloff
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  3. Cristián Bravo
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  4. Yuri Lawryshyn
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Corresponding author

Correspondence to Pierre Miasnikof .

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Editors and Affiliations

  1. University of Burgundy, Dijon Cedex, France

    Hocine Cherifi

  2. Thomas J. Watson College of Engineering and Applied Sciences, Binghamton University, Binghamton, NY, USA

    Luis M. Rocha

  3. IUT Lumière - Université Lyon 2, University of Lyon, Bron, France

    Chantal Cherifi

  4. Department of Economics, Yildiz Technical University, Istanbul, Türkiye

    Murat Donduran

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Miasnikof, P., Shestopaloff, A.Y., Bravo, C., Lawryshyn, Y. (2024). Empirical Study of Graph Spectra and Their Limitations. In: Cherifi, H., Rocha, L.M., Cherifi, C., Donduran, M. (eds) Complex Networks & Their Applications XII. COMPLEX NETWORKS 2023. Studies in Computational Intelligence, vol 1141. Springer, Cham. https://doi.org/10.1007/978-3-031-53468-3_25

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  • DOI: https://doi.org/10.1007/978-3-031-53468-3_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-53467-6

  • Online ISBN: 978-3-031-53468-3

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