Spectral Analysis of Random Sparse Matrices

Ando, Tomonori; Kabashima, Yoshiyuki; Takahashi, H.; Watanabe, Osamu; Yamamoto, Masaki

doi:10.1587/transfun.e94.a.1247

Cited by 3 publications

(6 citation statements)

References 9 publications

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“…where p (k) is the probability of having a node of degree k and k = k kp (k) [71]. The sum in (21) starts from k = 1 since we should not be concerned with isolated nodes. Eq.…”

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

“…where p (k) is the probability of having a node of degree k and k = k kp (k) [71]. The sum in (21) starts from k = 1 since we should not be concerned with isolated nodes. Equation ( 21) is generally solved via a population dynamics algorithm (see section 6 for details).…”

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

“…In this case, there is a pair of marginal coefficients Ω i and H i living on each node. Since in the infinite size limit the nodes can only be distinguished by their degree, following the same line of reasoning that led to (21), the joint pdf of the random variables of the type Ω i and H i in the thermodynamic limit can be written as…”

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

“…when p = c/N, with c being the constant mean degree of nodes (or equivalently, the mean number of nonzero elements per row of the corresponding adjacency matrix). In this sparse regime, Krivelevich and Sudakov [20] proved a theorem stating that for any constant c the largest eigenvalue of Erdős-Rényi graph diverges slowly with N as log N/ log log N. To ensure that the largest eigenvalue remains ∼ O (1), the nodes with very large degree must be pruned (see [21]).…”

Section: Introductionmentioning

confidence: 99%

“…In this sparse regime, Krivelevich and Sudakov [20] proved a theorem stating that for any constant c the largest eigenvalue of Erdős-Rényi graph diverges slowly with N as log N/ log log N . To ensure that the largest eigenvalue remains ∼ O(1), the nodes with very large degree must be pruned (see [21]). The characterisation of eigenvectors properties has proved to be much harder and is generally a less explored area of random matrix theory.…”

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confidence: 99%

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Top eigenpair statistics for weighted sparse graphs

Susca¹,

Vivo²,

Kühn³

2019

J. Phys. A: Math. Theor.

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We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector's components coming from nodes of degree k. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further crosschecked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks. arXiv:1903.09852v2 [cond-mat.stat-mech] 25 Oct 2019 eigenpair problem for a differential operator [8]. The top eigenpair is also relevant in signal reconstruction problems employing algorithms based on the spectral method [9]. In the context of graph theory, the eigenvectors of both adjacency and Laplacian matrices are employed to solve combinatorial optimisation problems, such as graph 3-colouring [10] and to develop clustering and cutting techniques [11][12][13]. In particular, the top eigenvector of graphs is intimately related to the "ranking" of the nodes of the network [14]. Indeed, beyond the natural notion of ranking of a node given by its degree, the relevance of a node can be estimated from how "important" its neighbours are. The vector expressing the importance of each node is exactly the top eigenvector of the network adjacency matrix. Google PageRank algorithm works in a similar way [15,16]: the PageRanks vector is indeed the top eigenvector of a large Markov transition matrix between web pages.When the matrix J is random and symmetric with i.i.d. entries, analytical results on the statistics of the top eigenpair date back to the classical work by Füredi and Komlós [17]: the largest eigenvalue of such matrices follows a Gaussian distribution with finite variance, provided that the moments of the distribution of the entries do not scale with the matrix size. This result directly relates to the largest eigenvalue of Erdős-Rényi (E-R) [18] adjacency matrices in the case when the probability p for two nodes to be connected does not scale with the matrix size N . This result has been then extended by Janson [19] in the case when p is large. However, in our analysis we will be mostly dealing with the sparse case, i.e. when p = c/N , with c being the constant mean degree of nodes (or equivalently, the mean number of nonzero elements per row of the corresponding adjacency matrix). In this sparse regime, Krivelevich and Sudakov [20] proved a theorem stating that for any constant c the largest eigenvalue of Erdős-Rényi graph diverges slowly with N as log N/ log log N . To ensure that the largest eigenvalue remains ∼ O(1), the nodes with very large degree must be prun...

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“…where p (k) is the probability of having a node of degree k and k = k kp (k) [71]. The sum in (21) starts from k = 1 since we should not be concerned with isolated nodes. Eq.…”

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

Section: Thermodynamic Limit N → ∞mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Top eigenpair statistics for weighted sparse graphs

Susca¹,

Vivo²,

Kühn³

2019

J. Phys. A: Math. Theor.

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Compressed Sensing Performance of Binary Matrices with Binary Column Correlations

Dai

Xia

2017

2017 Data Compression Conference (DCC)

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Near-optimal Binary Compressed Sensing Matrix

Lu¹,

Li²,

Kpalma³

et al. 2013

Preprint

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Compressed sensing is a promising technique that attempts to faithfully recover sparse signal with as few linear and nonadaptive measurements as possible. Its performance is largely determined by the characteristic of sensing matrix. Recently several zero-one binary sensing matrices have been deterministically constructed for their relative low complexity and competitive performance. Considering the implementation complexity, it is of great practical interest if one could further improve the sparsity of binary matrix without performance loss. Based on the study of restricted isometry property (RIP), this paper proposes the near-optimal binary sensing matrix, which guarantees nearly the best performance with as sparse distribution as possible. The proposed near-optimal binary matrix can be deterministically constructed with progressive edge-growth (PEG) algorithm. Its performance is confirmed with extensive simulations.

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Spectral Analysis of Random Sparse Matrices

Cited by 3 publications

References 9 publications

Top eigenpair statistics for weighted sparse graphs

Top eigenpair statistics for weighted sparse graphs

Compressed Sensing Performance of Binary Matrices with Binary Column Correlations

Near-optimal Binary Compressed Sensing Matrix

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