Abstract
We will discuss how graph based matrices are capable to find classification of the graph vertices with small within- and between-cluster discrepancies. The structural eigenvalues together with the corresponding spectral subspaces of the normalized modularity matrix are used to find a block-structure in the graph. The notions are extended to rectangular arrays of nonnegative entries and to directed graphs. We also investigate relations between spectral properties, multiway discrepancies, and degree distribution of generalized random graphs. These properties are regarded as generalized quasirandom properties, and we conjecture and partly prove that they are also equivalent for certain deterministic graph sequences, irrespective of stochastic models.
References
[1] N. Alon, J. H. Spencer, The Probabilistic Method. Wiley (2000).10.1002/0471722154Search in Google Scholar
[2] N. Alon et al., Quasi-randomness and algorithmic regularity for graphs with general degree distributions, Siam J. Comput. 39, 2336-2362 (2010).Search in Google Scholar
[3] A. L. Barabási, R. Albert, Emergence of Scaling in Random Networks, Science 286, 509-512 (1999).Search in Google Scholar
[4] P. J. Bickel, A. Chen, A nonparametric view of network models and Newman-Girvan and other modularities, Proc. Natl. Acad. Sci. USA, 106, 21068-21073 (2009).10.1073/pnas.0907096106Search in Google Scholar PubMed PubMed Central
[5] M. Bolla, Recognizing linear structure in noisy matrices, Linear Algebra Appl. 402, 228-244 (2005).10.1016/j.laa.2004.12.023Search in Google Scholar
[6] M. Bolla, Noisy random graphs and their Laplacians, Discret. Math. 308, 4221-4230 (2008).Search in Google Scholar
[7] M. Bolla, Penalized versions of the Newman–Girvan modularity and their relation to normalized cuts and k-means clustering, Phys. Rev. E 84, 016108 (2011).10.1103/PhysRevE.84.016108Search in Google Scholar PubMed
[8] M. Bolla, Spectral Clustering and Biclustering. Learning Large Graphs and Contingency Tables. Wiley (2013).10.1002/9781118650684Search in Google Scholar
[9] M. Bolla, Modularity spectra, eigen-subspaces and structure of weighted graphs, European J. Combin. 35, 105-116 (2014).10.1016/j.ejc.2013.06.019Search in Google Scholar
[10] M. Bolla, SVD, discrepancy, and regular structure of contingency tables, Discret. Appl. Math. 176, 3-11 (2014).Search in Google Scholar
[11] M. Bolla, B. Bullins, S. Chaturapruek, S. Chen, K. Friedl, Spectral properties of modularity matrices, Linear Algebra Appl. 73, 359-376 (2015).Search in Google Scholar
[12] B. Bollobás, P. Erdős, An extremal problem of graphs with diameter 2, Math. Mag. 48, 419-427 (1975).Search in Google Scholar
[13] B. Bollobás, Random Graphs, 2nd edition. Cambridge Univ. Press, Cambridge (2001).10.1017/CBO9780511814068Search in Google Scholar
[14] B. Bollobás, V. Nikiforov, Hermitian matrices and graphs: singular values and discrepancy, Discret. Math. 285, 17-32 (2004).Search in Google Scholar
[15] B. Bollobás, S. Janson, O. Riordan, The phase transition in inhomogeneous random graphs, Random Struct. Algorithms 31, 3-122 (2007).Search in Google Scholar
[16] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, K. Vesztergombi, Convergent graph sequences I: Subgraph Frequencies, metric properties, and testing, Advances in Math. 219, 1801-1851 (2008).Search in Google Scholar
[17] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, K. Vesztergombi, Convergent sequences of dense graphs II: Multiway cuts and statistical physics, Ann. Math. 176, 151-219 (2012).Search in Google Scholar
[18] S. Butler, Using discrepancy to control singular values for nonnegative matrices, Linear Algebra Appl. 419, 486-493 (2006).10.1016/j.laa.2006.05.015Search in Google Scholar
[19] F. Chung, R. Graham, R. K. Wilson, Quasi-random graphs, Combinatorica 9, 345-362 (1989).Search in Google Scholar
[20] F. Chung, R. Graham, Quasi-random graphs with given degree sequences, Random Struct. Algorithms 12, 1-19 (2008).Search in Google Scholar
[21] F. Chungm L, Lu, V. Vu, Eigenvalues of random power law graphs. Ann. Comb. 7, 21-33 (2003).Search in Google Scholar
[22] A. Coja-Oghlan, A. Lanka, Finding planted partitions in random graphs with general degree distributions, J. Discret. Math. 23, 1682-1714 (2009).Search in Google Scholar
[23] P. Erdős, A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. 5, 17-61 (1960).Search in Google Scholar
[24] P. Holland, K. B. Laskey, S. Leinhardt, Stochastic blockmodels: some first steps, Social Networks 5, 109-137 (1983).Search in Google Scholar
[25] S. Hoory, N. Linial, A. Widgerson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N. S.) 43, 439-561 (2006).Search in Google Scholar
[26] T. Kanungo et al., An efficient k-means clustering algorithm: analysis and implementation, IEEE Trans. Pattern Anal. Mach. Intell. 24, 881-892 (2002).Search in Google Scholar
[27] B. Karrer, M. E. J. Newman, Stochastic blockmodels and community structure in networks, Phys. Rev. E 83, 016107 (2011).10.1103/PhysRevE.83.016107Search in Google Scholar PubMed
[28] L. Lovász, V. T. Sós, Generalized quasirandom graphs, J. Comb. Theory B 98, 146-163 (2008).Search in Google Scholar
[29] L. Lovász, Very large graphs. In: D. Jerison et al. (Ed.), Current Developments in Mathematics, International Press, Somerville MA (2008), pp. 67-128.Search in Google Scholar
[30] F. McSherry, Spectral partitioning of random graphs. In: 42nd Annual Symposium on Foundations of Computer Science (FOCS), Las Vegas, Nevada (2001), pp. 529–537.Search in Google Scholar
[31] M. E. J. Newman, Networks. An Introduction. Oxford University Press (2010).Search in Google Scholar
[32] R.G.E. Pinch, Sequences well distributed in the square, Math. Proc. Cambridge Phil. Soc. 99, 19-22 (1986).10.1017/S0305004100063878Search in Google Scholar
[33] K. Rohe, S. Chatterjee, B. Yu, Spectral clustering and the high-dimensional stochastic blockmodel, Ann. Stat. 39, 1878-1915 (2011).Search in Google Scholar
[34] M. Simonovits, V. T. Sós, Szemerédi’s partition and quasi-randomness, Random Struct. Algorithms 2, 1-10 (1991).Search in Google Scholar
[35] E. Szemerédi, Regular partitions of graphs. In: J-C. Bermond et al. (Ed.), Colloque Inter. CNRS. No. 260, Problémes Combinatoires et Théorie Graphes (1976), pp. 399–401.Search in Google Scholar
[36] A. Thomason, Pseudo-random graphs, Ann. Discret. Math. 33, 307-331 (1987).Search in Google Scholar
[37] A. Thomason, Dense expanders and pseudo-random bipartite graphs, Discret. Math. 75, 381-386 (1989).Search in Google Scholar
[38] R. C. Thompson, The behavior of eigenvalues and singular values under perturbations of restricted rank, Linear Algebra Appl. 13, 69-78 (1976).10.1016/0024-3795(76)90044-6Search in Google Scholar
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