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2,051
The second eigenvalue of the Google matrix.
, 2003
"... Abstract. We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: T , where P is an n × n rowstochastic matrix, E is a nonnegative n × n rankone rowstochastic matrix, and 0 ..."
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Cited by 90 (7 self)
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Abstract. We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: T , where P is an n × n rowstochastic matrix, E is a nonnegative n × n rankone rowstochastic matrix, and 0
The Second Eigenvalue of the Google Matrix
, 2003
"... We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: "For any matrix A = [cP + (1 c)E] , where P is an n n rowstochastic matrix, E is a nonnegative nn rankone ..."
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We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: "For any matrix A = [cP + (1 c)E] , where P is an n n rowstochastic matrix, E is a nonnegative nn rank
Reconstruction on trees: Beating the second eigenvalue
 Ann. Appl. Probab
, 2001
"... We consider a process in which information is transmitted from a given root node on a noisy dary tree network T. We start with a uniform symbol taken from an alphabet A. Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on A. The goa ..."
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Cited by 40 (13 self)
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of correct reconstruction tend to 1/A  as n → ∞? It is known that reconstruction is possible if dλ 2 2(M)> 1, where λ2(M) is the second eigenvalue of M. Moreover, in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree
A proof of Alon’s second eigenvalue conjecture
, 2003
"... A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 ..."
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Cited by 166 (1 self)
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. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1
THE SECOND EIGENVALUE OF THE FRACTIONAL p−LAPLACIAN
"... Abstract. We consider the eigenvalue problem for the fractional p−Laplacian in an open bounded, possibly disconnected set Ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue λ2(Ω) is welldefined, and ..."
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Cited by 3 (1 self)
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Abstract. We consider the eigenvalue problem for the fractional p−Laplacian in an open bounded, possibly disconnected set Ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue λ2(Ω) is well
Notes on the Second Eigenvalue of the Google Matrix
, 2003
"... If A is an n × n matrix whose n eigenvalues are ordered in terms of decreasing modules, λ1  ≥ λ2  ≥ · · · λn, it is often of interest to estimate λ2 λ1. If A is a row stochastic matrix (so λ1 = 1), one can use an old formula of R. L. Dobrushin to give a useful, explicit formula for λ ..."
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for λ2. The purpose of this note is to disseminate these known results more widely and to show how they imply, as a very special case, some recent theorems of Haveliwala and Kamvar about the second eigenvalue of the Google matrix. If A = (aij) is an n × n real matrix, A has n (counting algebraic
entropy; Zscores; Second eigenvalue
"... 1.1 Representing RNA secondary structure as planar treegraph The primary structure of a linear RNA chain molecule is the nucleotide sequence s = s1s2 …si …sL, and runs in the direction 5 ' → 3 ' terminus. L defines the number of nucleotides and si ∈ ∑ = (A, C, G, U) is the biochemical ..."
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1.1 Representing RNA secondary structure as planar treegraph The primary structure of a linear RNA chain molecule is the nucleotide sequence s = s1s2 …si …sL, and runs in the direction 5 ' → 3 ' terminus. L defines the number of nucleotides and si ∈ ∑ = (A, C, G, U) is the biochemical nucleotide at the i th position. The RNA molecule s folds upon itself relatively rapid into a twodimensional RNA secondary structure S [1]. The structure S is stabilized by the canonical WatsonCrick G≡C and A=U, and wobble G=U base pairings. (Fig. S1) A planar RNA secondary structure S is mathematically described by a set of base pairings (i, j) ∈ S connecting bases si and sj, where i < j [2]. Given (i, j) and (k, l) ∈ S, a nucleotide can base pair to at most one other nucleotide i.e., i = k ⇔ j = l. A set of ∆ ∈ Z + consecutive base pairs defines a stem for stabilizing the structure against thermal fluctuations. The number of unpaired nucleotides between
On the second eigenvalue and linear expansion of regular graphs.
 In DIMACS Serws in Dncrete Mathematics and Theoretical Computer Science
, 1993
"... Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the bestknown explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinearsized sub ..."
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Cited by 16 (2 self)
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Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the bestknown explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinear
Results 1  10
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2,051