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The Interpolation Method for Random Graphs with Prescribed Degrees

Part of: Statistical mechanics, structure of matter Graph theory

Published online by Cambridge University Press:  19 June 2015

JUSTIN SALEZ
JUSTIN SALEZ*
Affiliation:
Université Paris Diderot–LPMA, UFR de Mathématiques, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France (e-mail: [email protected])
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Abstract

We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.

MSC classification

Primary: 05C80: Random graphs
Secondary: 82-08: Computational methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 3 , May 2016 , pp. 436 - 447
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Bapat, R. B. and Raghavan, T. E. S. (1997) Nonnegative Matrices and Applications , Vol. 64 of Encyclopedia of Mathematics and its Applications, Cambridge University Press.Google Scholar
[2] Bayati, M., Gamarnik, D. and Tetali, P. (2013) Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 4080–4115.CrossRefGoogle Scholar
[3] Beineke, L. W. and Vandell, R. C. (2002) No cycling in the graphs! In The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications, Elsevier. Electron. Notes Discrete Math. 11 81–88.CrossRefGoogle Scholar
[4] Bobkov, S. and Ledoux, M. (2014) One-Dimensional Empirical Measures, Order Statistics and Kantorovich Transport Distances, book in preparation. perso.math.univ-toulouse.fr/ledoux/files/2013/11/Order.statistics.10.pdf Google Scholar
[5] Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.CrossRefGoogle Scholar
[6] Bollobás, B., Janson, S. and Riordan, O. (2007) The phase transition in inhomogeneous random graphs. Random Struct. Alg. 31 3–122.CrossRefGoogle Scholar
[7] Bollobás, B. and Riordan, O. (2011) Sparse graphs: Metrics and random models. Random Struct. Alg. 39 1–38.CrossRefGoogle Scholar
[8] Bordenave, C. (2012) Lecture Notes on Random Graphs and Probabilistic Combinatorial Optimization. www.math.univ-toulouse.fr/~bordenave/coursRG.pdf Google Scholar
[9] Borgs, C., Chayes, J. and Gamarnik, D. (2013) Convergent sequences of sparse graphs: A large deviations approach. arXiv:1302.4615 Google Scholar
[10] Borgs, C., Chayes, J., Kahn, J. and Lovász, L. (2013) Left and right convergence of graphs with bounded degree. Random Struct. Alg. 42 1–28.CrossRefGoogle Scholar
[11] de Bruijn, N. G. and Erdős, P. (1952) Some linear and some quadratic recursion formulas II. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14 152–163.CrossRefGoogle Scholar
[12] Ding, J., Sly, A. and Sun, N. (2013) Maximum independent sets on random regular graphs. arXiv:1310.4787 Google Scholar
[13] Franz, S. and Leone, M. (2003) Replica bounds for optimization problems and diluted spin systems. J. Statist. Phys. 111 535–564.CrossRefGoogle Scholar
[14] Franz, S., Leone, M. and Toninelli, F. L. (2003) Replica bounds for diluted non-Poissonian spin systems. J. Phys. A 36 10967.CrossRefGoogle Scholar
[15] Gamarnik, D. (2014) Right-convergence of sparse random graphs. Probab. Theory Rel. Fields 160 253–278.CrossRefGoogle Scholar
[16] Guerra, F. and Toninelli, F. L. (2002) The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230 71–79.CrossRefGoogle Scholar
[17] van der Hofstad, R. (2013) Random Graphs and Complex Networks, course notes. www.win.tue.nl/~rhofstad/NotesRGCN.html Google Scholar
[18] Janson, S. (2009) The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.CrossRefGoogle Scholar
[19] Janson, S. (2014) The probability that a random multigraph is simple II. J. Appl. Probab. 51A 123–137.CrossRefGoogle Scholar
[20] Panchenko, D. (2013) The Sherrington–Kirkpatrick Model, Springer Monographs in Mathematics, Springer.CrossRefGoogle Scholar
[21] Talagrand, M. (2011) Mean Field Models for Spin Glasses, Volume I: Basic Examples , Vol. 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), third series, A Series of Modern Surveys in Mathematics, Springer.Google Scholar
[22] Villani, C. (2009) Optimal Transport: Old and New , Vol. 338 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), Springer.Google Scholar
[23] Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics 1999, Vol. 267 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 239–298.Google Scholar