The rank of random graphs

Costello, Kevin P.; Vu, Van

doi:10.1002/rsa.20219

Cited by 37 publications

(63 citation statements)

References 8 publications

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“…However, if the corresponding linear function depends on each of the n variables nontrivially, the fraction of vertices of the cube lying on this hyperplane tends to zero with growing n [6]. A similar result was obtained for a quadratic polynomial with sufficiently many monomials: the fraction of vertices of the cube at which such a polynomial takes a fixed value tends to zero with growing dimension of the cube [7]. An estimate for the number of vertices of the cube lying in a half-space is given in [8].…”

Section: Problem Settingmentioning

confidence: 59%

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Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

2012

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In a space of dimension 30 we find a pair of parallel hyperplanes, uniquely determined by vertices of a unit cube lying on them, such that strictly between the hyperplanes there are no vertices of the cube, though there are integer points. A similar two-sided example is constructed in dimension 37. We consider possible locations of empty quadrics with respect to vertices of the cube, which is a particular case of a discrete optimization problem for a quadratic polynomial on the set of vertices of the cube. We demonstrate existence of a large number of pairs of parallel hyperplanes such that each pair contains a large number of points of a prescribed set.

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Section: Problem Settingmentioning

confidence: 59%

“…Many papers (see, e.g., [6][7][8][9]) give estimates for the number of vertices lying on a quadric. A hyperplane contains at most half of all vertices of the cube.…”

Section: Problem Settingmentioning

confidence: 99%

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

2012

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“…Furthermore, the rate of convergence is an improvement of the one given in [7] for c ln n/n β ≤ p(n) ≤ 1/2 with c > 0 and β ∈ (0, 1). From the proof of Theorem 1.b in Section 4, if κ n = 1 − p(n), we have as n → ∞…”

Section: Resultsmentioning

confidence: 89%

“…Costello and Vu [7] have analyzed the adjacency matrices of sparse Erdös-Rényi graphs where each entry is equal to 1 with the same probability p(n), which tends to 0 as n goes to infinity (see also Costello and Vu [8], where a generalization of [7] is considered in which each entry takes the value c ∈ C with probability p and zero with probability 1 − p, and the diagonal entries are possibly non-zero). It is proved in [7] that when c ln(n)/n ≤ p(n) ≤ 1/2, c > 1/2, then with probability 1−O((ln ln(n)) −1/4 ), the rank of the adjacency matrix equals the number of non-isolated vertices.…”

Section: Resultsmentioning

confidence: 99%

On the universality of the non-singularity of general Ginibre and Wigner random matrices

Manrique

Pérez‐Abreu

Roy

2016

Random Matrices: Theory Appl.

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We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity, which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.

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“…1 Costello and Vu have recently conjectured that, for d > 2, almost every large d-regular network is nonsingular (See [15], [16]. ).…”

Section: Bounding Excess Transmissions By the Number Of Nodesmentioning

confidence: 99%

Efficient universal recovery in broadcast networks

Courtade

Wesel

2010

2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-Consider a connected broadcast network of N nodes that all wish to recover k desired packets. Each node begins with a subset of the desired packets and broadcasts messages to its neighbors. In a previous paper we established necessary and sufficient conditions on the number of transmissions from each node required for universal recovery (in which each node recovers all k packets). However, these conditions are numerous and cumbersome. The present paper gives a series of relatively simple conditions for universal recovery that apply when the number of packets is large and the distribution of packets among the nodes is well behaved.Our first result, which applies to any fixed network topology, uses only simple cuts in the network to characterize a set of transmission strategies such that for any > 0 these strategies require at most k transmissions above the minimum required for universal recovery. For certain topologies including nonsingular d-regular d-connected networks, we explicitly construct transmission strategies that achieve universal recovery while using at most N transmissions above the minimum even when the total number of required transmissions is very large. These explicit constructions essentially resolve the problem completely for many canonical networks (e.g. cliques, rings, grids on tori, etc.).

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The rank of random graphs

Cited by 37 publications

References 8 publications

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

On the universality of the non-singularity of general Ginibre and Wigner random matrices

Efficient universal recovery in broadcast networks

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