Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Special Matrices

Editor-in-Chief: da Fonseca, Carlos Martins

1 Issue per year


Covered by:
WoS (ESCI)
SCOPUS
MathSciNet
zbMATH

CiteScore 2017: 0.29

SCImago Journal Rank (SJR) 2017: 0.295
Source Normalized Impact per Paper (SNIP) 2017: 0.481

Mathematical Citation Quotient (MCQ) 2017: 0.17

Open Access
Online
ISSN
2300-7451
See all formats and pricing
More options …

The expected adjacency and modularity matrices in the degree corrected stochastic block model

Dario Fasino / Francesco Tudisco
Published Online: 2018-03-07 | DOI: https://doi.org/10.1515/spma-2018-0010

Abstract

We provide explicit expressions for the eigenvalues and eigenvectors of matrices that can be written as the Hadamard product of a block partitioned matrix with constant blocks and a rank one matrix. Such matrices arise as the expected adjacency or modularity matrices in certain random graph models that are widely used as benchmarks for community detection algorithms.

Keywords: Adjacency matrix; modularity matrix; stochastic block model; inflation product

MSC 2010: 15A18; 15B99

References

  • [1] M. Bolla, Recognizing linear structure in noisy matrices. Linear Algebra Appl., 402 (2005), 228-244.Google Scholar

  • [2] M. Bolla, Penalized versions of theNewman-Girvan modularity and their relation to normalized cuts and k-means clustering. Phys. Rev. E, 84 (2011), 016108.Google Scholar

  • [3] M. Bolla, B. Bullins, S. Chaturapruek, S. Chen, and K. Friedl. Spectral properties of modularitymatrices. Linear Algebra Appl., 473 (2015), 359-376.Google Scholar

  • [4] R. A. Brualdi, The DAD theorem for arbitrary row sums. Proc. Amer. Math. Soc., 45 (1974), 189-194.Google Scholar

  • [5] K. Chaudhuri, F. Chung and A. Tsiatas, Spectral Clustering of Graphs with General Degrees in the Extended Planted Partition Model, Conference on Learning Theory (COLT), J. Mach. Learn. Res., 23 (2012), 35:1-35:23.Google Scholar

  • [6] L. Chen, Q. Yu and B. Chen, Anti-modularity and anti-community detecting in complex networks. J. Inf. Sci., 275 (2014), 293-313.Google Scholar

  • [7] F. Chung and L. Lu, Complex Graphs and Networks, volume 107 of CBMS Regional Conf. Ser. in Math. AMS, 2006.Google Scholar

  • [8] A. Coja-Oghlan and A. Lanka, Finding planted partitions in random graphswith general degree distributions. SIAM J. Discrete Math., 23 (2009/10), 1682-1714.Google Scholar

  • [9] P. Doreian, V. Batagelj and A. Ferligoj, Generalized blockmodeling, volume 25 of Structural Analysis in the Social Sciences. CUP, 2005.Google Scholar

  • [10] E. Estrada, The structure of Complex Networks. Oxford University Press, 2012.Google Scholar

  • [11] D. Fasino and F. Tudisco, A modularity based spectral method for simultaneous detection of communities and anticommunities. Linear Algebra Appl., 542 (2018), 605-623.Google Scholar

  • [12] D. Fasino and F. Tudisco, Generalized modularity matrices. Linear Algebra Appl., 502 (2016), 327-345.Google Scholar

  • [13] D. Fasino and F. Tudisco, Modularity bounds for clusters located by leading eigenvectors of the normalizedmodularitymatrix. J. Math. Inequal., 11 (2017), 701-714.Google Scholar

  • [14] D. Fasino and F. Tudisco, An algebraic analysis of the graph modularity. SIAM J. Matrix Anal. Appl., 35 (2014), 997-1018.Google Scholar

  • [15] S. Friedland, D. Hershkowitz and H. Schneider, Matrices whose powers are M-matrices or Z-matrices. Trans. Amer. Math. Soc., 300 (1987), 343-366.Google Scholar

  • [16] A. Gautier, F. Tudisco and M. Hein, The Perron-Frobenius theorem for multi-homogeneous mappings. arXiv:1801.05034, (2018)Google Scholar

  • [17] G. H. Golub and C. F. Van Loan, Matrix computations. Johns Hopkins University Press, fourth edition, 2013.Google Scholar

  • [18] P. A. Knight, The Sinkhorn-Knopp algorithm: convergence and applications. SIAM J. Matrix Anal. Appl. 30 (2008), 261-275.Google Scholar

  • [19] J. Kunegis, S. Schmidt, A. Lommatzsch, J. Lerner, E. W. De Luca and S. Albayrak, Spectral analysis of signed graphs for clustering, prediction and visualization. Proc. SIAM Int. Conf. Data Min. (2010), 559-570Google Scholar

  • [20] S.Majstorovic and D. Stevanovic, A note on graphs whose largest eigenvalues of the modularitymatrix equals zero. Electron. J. Linear Algebra, 27 (2014), 611-618.Google Scholar

  • [21] P. Mercado, F. Tudisco and M. Hein, Clustering signed networks with the geometric mean of Laplacians, Adv. Neural Inf. Process. Syst., 29 (2016), 4421-4429.Google Scholar

  • [22] M. E. J. Newman and M. Girvan, Finding and evaluating community structure in networks. Phys. Rev. E, 69 (2004), 026113.Google Scholar

  • [23] T. Qin and K. Rohe, Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel. Adv. Neural Inf. Process. Syst., 26 (2013), 3120-3128.Google Scholar

  • [24] K. Rohe, S. Chatterjee, B. Yu, Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 (2011), no. 4, 1878-1915.Google Scholar

  • [25] A. Saade, F. Krzakala and L. Zdeborová, Spectral Clustering of Graphs with the Bethe Hessian. Adv. Neural Inf. Process. Syst., 27 (2014), 406-414.Google Scholar

  • [26] S. E. Schaeffer, Graph clustering. Computer Science Review, 1 (2007), 27-64.Google Scholar

  • [27] E. E. Tyrtyshnikov, A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl., 232 (1996), 1-43.Google Scholar

  • [28] U. Von Luxburg, A tutorial on spectral clustering. Stat. Comput., 17 (2007), 395-416.Google Scholar

About the article

Received: 2017-10-20

Accepted: 2018-02-16

Published Online: 2018-03-07


Citation Information: Special Matrices, Volume 6, Issue 1, Pages 110–121, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2018-0010.

Export Citation

© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in