Annals of Probability

The rank of diluted random graphs

Charles Bordenave, Marc Lelarge, and Justin Salez

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We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs (Gn)n≥0 converging locally to a Galton–Watson tree T (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function φ of T. In the first part, we show that the adjacency operator associated with T is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on φ for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of (Gn)n≥0. Our proofs borrow ideas from analysis of algorithms, functional analysis, random matrix theory and statistical physics.

Article information

Ann. Probab., Volume 39, Number 3 (2011), 1097-1121.

First available in Project Euclid: 16 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 15A52
Secondary: 47A10: Spectrum, resolvent

Random graphs adjacency matrix random matrices local weak convergence Karp and Sipser algorithm


Bordenave, Charles; Lelarge, Marc; Salez, Justin. The rank of diluted random graphs. Ann. Probab. 39 (2011), no. 3, 1097--1121. doi:10.1214/10-AOP567.

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  • [1] Aizenman, M., Sims, R. and Warzel, S. (2006). Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Related Fields 136 363–394.
  • [2] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
  • [3] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [4] Bai, Z. D. and Silverstein, J. W. (2006). Spectral Analysis of Large Dimensional Random Matrices. Mathematics Monograph Series 2. Science Press, Beijing.
  • [5] Bauer, M. and Golinelli, O. (2000). On the kernel of tree incidence matrices. J. Integer Seq. 3 Art. 00.1.4, 1 HTML document (electronic).
  • [6] Bauer, M. and Golinelli, O. (2001). Exactly solvable model with two conductor-insulator transitions driven by impurities. Phys. Rev. Lett. 86 2621–2624.
  • [7] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
  • [8] Bhanidi, S., Evans, S. N. and Sen, A. (2009). Spectra of large random trees. Preprint.
  • [9] Bohman, T. and Frieze, A. (2010). Karp–Sipser on random graphs with a fixed degree sequence.
  • [10] Bordenave, C. and Lelarge, M. (2010). Resolvent of large random graphs. Random Structures and Algorithms 37 332–352.
  • [11] Bordenave, C., Lelarge, M. and Salez, J. (2010). Matchings on infinite graphs. Preprint.
  • [12] Costello, K. P. and Vu, V. H. (2008). The rank of random graphs. Random Structures Algorithms 33 269–285.
  • [13] Costello, K. P., Tao, T. and Vu, V. (2006). Random symmetric matrices are almost surely nonsingular. Duke Math. J. 135 395–413.
  • [14] Cvetković, D. M., Doob, M. and Sachs, H. (1995). Spectra of Graphs: Theory and Applications, 3rd ed. Johann Ambrosius Barth, Heidelberg.
  • [15] Karp, R. and Sipser, M. (1981). Maximum matchings in sparse random graphs. In Proc. of the Twenty-Second IEEE Annual Symposium on Foundations of Computer Science 364–375. IEEE Computer Soc., Los Angeles, CA.
  • [16] Khorunzhy, O., Shcherbina, M. and Vengerovsky, V. (2004). Eigenvalue distribution of large weighted random graphs. J. Math. Phys. 45 1648–1672.
  • [17] Klein, A. (1998). Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133 163–184.
  • [18] Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York.
  • [19] Zdeborová, L. and Mézard, M. (2006). The number of matchings in random graphs. J. Stat. Mech. Theory Exp. 5 P05003, 24 pp. (electronic).