On the Spectra of General Random Graphs

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.

[1]  Mustapha Aouchiche,et al.  A survey of automated conjectures in spectral graph theory , 2009 .

[2]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[3]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[4]  Uriel Feige,et al.  Spectral techniques applied to sparse random graphs , 2005, Random Struct. Algorithms.

[5]  Benjamin Recht,et al.  A Simpler Approach to Matrix Completion , 2009, J. Mach. Learn. Res..

[6]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[7]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[8]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[9]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Klas Markström,et al.  Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales , 2008, Random Struct. Algorithms.

[11]  David Gross,et al.  Recovering Low-Rank Matrices From Few Coefficients in Any Basis , 2009, IEEE Transactions on Information Theory.

[12]  Amin Coja-Oghlan.  On the Laplacian Eigenvalues of Gn, p , 2007, Comb. Probab. Comput..

[13]  E. Lieb.  Convex trace functions and the Wigner-Yanase-Dyson conjecture , 1973 .

[14]  Fan Chung Graham,et al.  The Spectra of Random Graphs with Given Expected Degrees , 2004, Internet Math..

[15]  Rudolf Ahlswede,et al.  Strong converse for identification via quantum channels , 2000, IEEE Trans. Inf. Theory.

[16]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[17]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[18]  C. Tracy,et al.  Introduction to Random Matrices , 1992, hep-th/9210073.

[19]  Michael Doob,et al.  Spectra of graphs , 1980 .

[20]  F. Chung,et al.  Eigenvalues of Random Power law Graphs , 2003 .

[21]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[22]  R. Oliveira.  Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges , 2009, 0911.0600.

[23]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.