The Annals of Probability

The rank of diluted random graphs

Charles Bordenave, Marc Lelarge, and Justin Salez

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Abstract

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

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

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281733

Digital Object Identifier
doi:10.1214/10-AOP567

Mathematical Reviews number (MathSciNet)
MR2789584

Zentralblatt MATH identifier
1298.05283

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1300281733.


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