2013
DOI: 10.37236/3576
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Spectra of Edge-Independent Random Graphs

Abstract: Let G be a random graph on the vertex set {1, 2, . . . , n} such that edges in G are determined by independent random indicator variables, while the probability pij for {i, j} being an edge in G is not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of G are recently studied by Oliveira and Chung-Radcliffe. Let A be the adjacency matrix of G,Ā = E(A), and ∆ be the maximum expected degree of G. Oliveira first proved that almost surely A −Ā = O( √ ∆ ln n) provided ∆ ≥ C l… Show more

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Cited by 39 publications
(50 citation statements)
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“…We note that while P = XX T is not observed, existing spectral norm bounds (e.g., Oliveira, 2009;Lu and Peng, 2013) establish that if A ∼ Bernoulli(P), the spectral norm of A − P is comparatively small. As a result, we regard A as a noisy version of P, and we begin our inference procedures with a spectral decomposition of A.…”
Section: Notation and Definitionsmentioning
confidence: 61%
“…We note that while P = XX T is not observed, existing spectral norm bounds (e.g., Oliveira, 2009;Lu and Peng, 2013) establish that if A ∼ Bernoulli(P), the spectral norm of A − P is comparatively small. As a result, we regard A as a noisy version of P, and we begin our inference procedures with a spectral decomposition of A.…”
Section: Notation and Definitionsmentioning
confidence: 61%
“…By assumption, for i = 1, 2, the maximum expected degree of G i , ∆, satisfies ∆ ≫ log 4 (n), hence A i − P 2 = O( √ ∆) with probability 1 − o(1) by [33]. The assumption σ r (O) ≥ c∆ implies that σ r (Ô) ≥ C∆ with probability 1 − o(1) by Weyl's inequality, so Σ −1 2 = O(1/∆).…”
Section: Theorem 43mentioning
confidence: 99%
“…First, consider the following expression. [25]). Since L(P) = Θ(1) and the non-zero eigenvalues of L(P) are all of order Θ(1), this implies, by the Davis-Kahan theorem, that the eigenspace spanned by the d largest eigenvalues of L(A) is "close" to that spanned by the d largest eigenvalues of L(P).…”
Section: Proofs Sketch For Theorem and Theorem 32mentioning
confidence: 99%
“…Lemma B.1 ( [25,32]). Let A ∼ Bernoulli(P), i.e., A is a symmetric matrix whose upper triangular entries are independent Bernoulli random variables with P[a ij = 1] = p ij .…”
Section: B2 Proof Of Eq (31)mentioning
confidence: 99%