Singularity of the k-core of a random graph

Ferber, Asaf; Kwan, Matthew; Sah, Ashwin; Sawhney, Mehtaab

doi:10.48550/arxiv.2106.05719

Cited by 2 publications

(22 citation statements)

References 36 publications

(70 reference statements)

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“…Indeed, an interesting open problem appears to be the extension of the present methods to the symmetric case. In particular, it would be interesting to see if the present techniques can be used to add to the line of works on the adjacency matrices of random graphs, which have been approached by means of techniques based on local weak convergence or Littlewood-Offord techniques [9,22]. 2.6.…”

Section: Expansion Around the Equitable Solutionmentioning

confidence: 99%

The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix A with prescribed numbers of non-zero entries in each row and column over any field F. The rank formula turns out to be independent of both the field and the distribution of the non-zero matrix entries. The proofs are based on a blend of algebraic and probabilistic methods inspired by ideas from mathematical physics.MSC: 05C80, 60B20, 94B05Amin Coja-Oghlan's research received support under DFG CO 646/3. Alperen A. Ergür partially funded by Einstein Foundation, Berlin.

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Section: Expansion Around the Equitable Solutionmentioning

confidence: 99%

The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Resolving a conjecture of Vu, it was recently proved by Ferber and the last three authors [33] (see also [30]) that for constants k ≥ 3 and c > 0, the k-core of G(n, c/n) is nonsingular whp. That is to say, trimming low-degree vertices typically removes any singularity present in the graph (foreshadowing the main result of this paper, that whp the only dependencies are "tree-like" or "cycle-like").…”

Section: Introductionmentioning

confidence: 92%

“…In [33], Ferber and the last three authors leveraged this observation together with the techniques discussed in the last two subsections, to prove that the k-core of a random graph (for k ≥ 3) is nonsingular whp. Specifically, for a random n-vertex graph constrained to have minimum degree at least k, they designed a procedure to identify βn vertices of high degree without actually revealing the neighbours of these vertices.…”

Section: Rank-boostingmentioning

confidence: 99%

“…It turns out that if, at some point in the process, f has few nonzero coefficients, this essentially corresponds to A(G) having a kernel vector with small support (i.e., with few nonzero entries) 7 . Therefore, an essential part of the Littlewood-Offord-based proofs mentioned so far [26,33,47,48] is to prove that A(G) has no small-support kernel vectors.…”

Section: Rank-boostingmentioning

confidence: 99%

“…We prove such a theorem: roughly speaking, the 2-core is "right on the edge of singularity", being neither singular whp nor nonsingular whp. (In retrospect, one can see that in the case k ≥ 3, the k-core is actually "quite far from being singular" with respect to natural local dependencies, and this "wiggle room" played a crucial role in the proofs in [30,33]).…”

Section: Introductionmentioning

confidence: 99%

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The Exact Rank of Sparse Random Graphs

Glasgow¹,

Kwan²,

Sah³

et al. 2023

Preprint

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Two landmark results in combinatorial random matrix theory, due to Komlós and Costello-Tao-Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph theory, when p is a fixed constant, the biadjacency matrix of a random Erdős-Rényi bipartite graph G(n, n, p) and the adjacency matrix of an Erdős-Rényi random graph G(n, p) are both nonsingular with high probability. However, very sparse random graphs (i.e., where p is allowed to decay rapidly with n) are typically singular, due to the presence of "local" dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbour.In this paper we give a combinatorial description of the rank of a sparse random graph G(n, n, c/n) or G(n, c/n) in terms of such local dependencies, for all constants c = e (and we present some evidence that the situation is very different for c = e). This gives an essentially complete answer to a question raised by Vu at the 2014 International Congress of Mathematicians.As applications of our main theorem and its proof, we also determine the asymptotic singularity probability of the 2-core of a sparse random graph, we show that the rank of a sparse random graph is extremely well-approximated by its matching number, and we deduce a central limit theorem for the rank of G(n, c/n). Corollary 1.3. Let G ∼ G(n, c/n) for a constant c < 1 or c > e, let G ∼ G(n, c/n), and let X be the rank of the adjacency matrix of G. Then (X − EX)/ √ Var X d → N (0, 1).Actually, in an upcoming paper together with Goldschmidt and Kreačić [38], we are able to close the gap between 1 and e in Corollary 1.3. Specifically, the regime c ≤ e is rather different in nature than the regime c > e, and when G ∼ G(n, c/n) for c ≤ e, we are able to give a unified proof that the rank and matching number of G both satisfy a central limit theorem (without going through Theorem 1.2(A1)). Remark.[60] and [49] provide explicit formulas for the asymptotic values of EX and Var X, though these are a bit too complicated to describe here. It is worth remarking that the asymptotic formula for Var X is the single place in this paper where there is a material difference between the "binomial" model of Erdős-Rényi random graphs (where each edge is present with probability p independently) and the "uniform" model of Erdős-Rényi random graphs (where we choose a random subset of exactly m edges, for say m = ⌊p n 2 ⌋). Indeed, the variance of the matching number (and therefore the variance of the rank) differs by a constant factor between these two settings; see [49,60] for details. For all the other results in the paper (which are all stated for the binomial model), one can make trivial changes to the proofs to obtain exactly the same result in the uniform model.Remark. We believe that a central limit theorem does not hold for the rank of G(n, n, c/n); see Section 1.6. 1.4. Exactly characterising the rank. In this subsection we finally state our main theorem, giving an exact combinatorial ...

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Singularity of the k-core of a random graph

Cited by 2 publications

References 36 publications

The rank of sparse random matrices

The rank of sparse random matrices

The Exact Rank of Sparse Random Graphs

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