Abstract
This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a complete proof of the equivalence between normalized adjacency clustering and normalized modularity clustering. Numerical experiments show that normalized adjacency clustering can be as twice efficient as normalized modulairty clustering.
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References
Bolla, M.: Penalized versions of the Newman-Girvan modularity and their relation to normalized cuts and k-means clustering. Phys. Rev. E 84(1), 016108 (2011)
Bunch, J.R., Nielsen, C.P., Sorensen, D.C.: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31(1), 31–48 (1978)
Chitta, R., Jin, R., Jain, A.K.: Efficient kernel clustering using random Fourier features. In: 2012 IEEE 12th International Conference on Data Mining (ICDM), pp. 161–170. IEEE (2012)
Chung, F.R.: Spectral graph theory, vol. 92. American Mathematical Soc. (1997)
Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23(2), 298–305 (1973)
Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslov. Math. J. 25(4), 619–633 (1975)
Hertz, T., Bar-Hillel, A., Weinshall, D.: Boosting margin based distance functions for clustering. In: Proceedings of the Twenty-First International Conference On Machine Learning, p. 50. ACM (2004)
LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324 (1998)
Meyer, C.D.: Matrix analysis and applied linear algebra. SIAM (2000)
Newman, M.E.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. 103(23), 8577–8582 (2006)
Newman, M.E., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)
Ng, A.Y., Jordan, M.I., Weiss, Y., et al.: On spectral clustering: Analysis and an algorithm. Adv. Neural. Inf. Process. Syst. 2, 849–856 (2002)
Olson, E., Walter, M.R., Teller, S.J., Leonard, J.J.: Single-cluster spectral graph partitioning for robotics applications. In: Robotics: Science and Systems, pp. 265–272 (2005)
Pothen, A.: Graph partitioning algorithms with applications to scientific computing. In: Keyes, D.E., Sameh, A., Venkatakrishnan, V. (eds.) Parallel Numerical Algorithms. ICASE/LaRC Interdisciplinary Series in Science and Engineering, vol. 4, pp. 323–368. Springer, Dordrecht (1997). https://doi.org/10.1007/978-94-011-5412-3_12
Race, S.L., Meyer, C., Valakuzhy, K.: Determining the number of clusters via iterative consensus clustering. In: Proceedings of the SIAM Conference on Data Mining (SDM), pp. 94–102. SIAM (2013)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)
Tolliver, D.A., Miller, G.L.: Graph partitioning by spectral rounding: applications in image segmentation and clustering. In: 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR2006), vol. 1, pp. 1053–1060. IEEE (2006)
Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)
Wilkinson, J.H., Wilkinson, J.H., Wilkinson, J.H.: The algebraic eigenvalue problem, vol. 87. Clarendon Press, Oxford (1965)
Yu, L., Ding, C.: Network community discovery: solving modularity clustering via normalized cut. In: Proceedings of the Eighth Workshop on Mining and Learning with Graphs, pp. 34–36. ACM (2010)
Zhang, R., Rudnicky, A.I.: A large scale clustering scheme for kernel k-means. In: 2002 Proceedings. 16th International Conference on Pattern Recognition, vol. 4, pp. 289–292. IEEE (2002)
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Jiang, H., Meyer, C. (2023). Relations Between Adjacency and Modularity Graph Partitioning. In: Kashima, H., Ide, T., Peng, WC. (eds) Advances in Knowledge Discovery and Data Mining. PAKDD 2023. Lecture Notes in Computer Science(), vol 13936. Springer, Cham. https://doi.org/10.1007/978-3-031-33377-4_15
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DOI: https://doi.org/10.1007/978-3-031-33377-4_15
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