Positive Eigenvector Research Articles

Nutritional, antioxidants and protein profiling of leaves of Moringa oleifera germplasm

This study aimed to select elite germplasm based on nutritional, antioxidants and protein profiling of leaves of Moringa oleifera germplasm. Principal components analysis (PCA) using the fifteen nutritional traits of thirty-four accessions of M. oleifera indicated that 74.70 % variability was accounted for the first six principal components (PCs) with eigenvalues ≥1. The first principal component (PC1) had an eigenvalue of 3.6 and explained 23.90 % of the total variation. Nitrogen, protein, magnesium, iron, zinc, calcium and potassium had high positive eigenvectors in PC1. The second component (PC2) explained 15.27 % of the total variance with an eigen value of 2.29 and positively correlated to copper, calcium, phosphorus and total phenols. Protein content was significantly and highly correlated with nitrogen content in leaves (r = +1.0). High magnitude and significant correlation were found between FRAP and DPPH (r = +0.9). The thirty-four germplasm were investigated in this study on quality parameters at ICAR-Central Horticultural Experiment Station, Godhra, Gujarat, India and grouped in five main clusters. Distance estimates based on the fifteen qualitative traits ranged from 2734.15 to 3868.43. The cluster V had maximum mean values for nutrients (mg/100 g dw) iron (81.76),zinc (7.66), manganese (27.13), calcium (4470.58), magnesium (1068.28), sulfur (3758.68), nitrogen (4615.84), potassium (1558.51), protein (29.40 g/100 g), total phenols (900.21 mg GAE/100 g fw), DPPH (31.17 µmol (TE)/g fw) and FRAP (50.42 µmol (TE)/g fw). The germplasm CHES D_40 and CHES D_45 (cluster V) could be utilized for further exploration based on their nutritional, antioxidants and protein content and for conservation of nutritionally superior germplasm.

The two-steps eigenvector centrality in complex networks

Eigenvector centrality refers to the principal eigenvector of the adjacency matrix of a graph. The adjacency matrix can be regarded as the matrix describing the one-step walks on the graph, shortly, the one-walk matrix, and then eigenvector centrality can refer to the principal eigenvector of the one-walk matrix. In this paper, we consider the effect of combining the first and second neighbors of a given node in a network. Analogously to the one-walk matrix, we construct a three-order tensor, namely the two-steps tensor, to represent the first and second neighbor relation among nodes. We adopt the positive eigenvector of the two-steps tensor corresponding to its spectral radius as a new centrality measure. This new centrality measure is referred to as the two-steps eigenvector centrality, which extends the notion of eigenvector centrality. Experimental results show that the new centrality measure is effective, and in certain networks, it gives an additional insight regarding node importance.

Perron-Frobenius type theorem for nonnegative tubal matrices in the sense of t-product

This paper is focused on the study of Perron-Frobenius (PF) type theorem for nonnegative tubal matrices, where a tubal matrix is a matrix with every element being a tubal scalar (a vector in the usual sense). Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce irreducible tubal matrices and provide two equivalent characterizations. Then, the celebrated PF theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices, such as the existence of a positive eigenvector, can be generalized to the tubal matrix setting, while some are not. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.

Noda iteration for computing generalized tensor eigenpairs

In this paper, we propose the tensor Noda iteration (NI) and its inexact version for solving the eigenvalue problem of a particular class of tensor pairs called generalized M-tensor pairs. A generalized M-tensor pair consists of a weakly irreducible nonnegative tensor and a nonsingular M-tensor within a linear combination. It is shown that any generalized M-tensor pair admits a unique positive generalized eigenvalue with a positive eigenvector. A modified tensor Noda iteration (MTNI) is developed for extending the Noda iteration for nonnegative matrix eigenproblems. In addition, the inexact generalized tensor Noda iteration method (IGTNI) and the generalized Newton-Noda iteration method (GNNI) are also introduced for more efficient implementations and faster convergence. Under a mild assumption on the initial values, the convergence of these algorithms is guaranteed. The efficiency of these algorithms is illustrated by numerical experiments.

Principal component and cluster analyses for assessing agro-morphological diversity in rice

Forty-five rice genotypes were evaluated for determining the pattern of variation and relationship among 14 yield contributing traits. Four principal components (PCs) exhibited eigen values >1.0 and explained about 79.5 % of the total phenotypic variability. From rotated component matrix it has been observed that the highest positive eigen vector was taken by secondary branches (0.945), followed by total spikelet number (0.945), fertile spikelet number (0.889), primary branches (0.676) and harvest index (0.632) in PC1, indicating the major effects in the overall variation among the genotypes. Seven groups were formed after cluster analysis. Cluster I had lowest average for days to 50% flowering, Cluster II had highest mean value for harvest index, Cluster III had highest mean for flag leaf area, test weight, and straw and grain yield per plant, and Cluster V had highest mean value for primary branches, total spikelet number, fertile spikelet number and fertility %. So, desirable genotypes fromdifferent cluster can be selected and hybridization programme may be initiated to utilize heterosis in F1 generation and wide spectrum of recombinants in segregating generations for selection of promising segregants.

Leader-Based Consensus in Directed Networks of Homogeneous Positive Agents With Multiple Inputs

Abstract In this paper, the non-zero finite positive state consensus problem of leader-following multi-agent systems (MASs) connected over a directed network is investigated. The multi-agent network consists of homogeneous linear time-invariant (LTI) positive agents whose minimal positive state-space realization is assumed to be known. In this paper, a state-feedback hierarchical control protocol is proposed where the local controller synthesizes a singular, Lyapunov stable, and Metzler system matrix with a simple dominant eigenvalue at origin. The obtained consensus vector is the positive eigenvector associated with the zero eigenvalue of the synthesized system matrix. The controller gain matrices are obtained by solving a set of necessary and sufficient conditions derived in linear programing framework. A numerical example is given to elucidate the usefulness of the proposed results.

Monotone convergence of Newton-like iteration for a structured nonlinear eigen-problem

A structured eigen-problem Ax+F(x)=λx is studied in this paper, where in applications A∈Rn×n is an irreducible Stieltjes matrix. Under certain restrictions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of A, the Newton-like algorithm for this eigen-problem is well defined and converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method.

Locally eventually positive operator semigroups

We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive C0-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces − for instance, the square of the Laplace operator with Dirichlet boundary conditions on L2 and the Dirichlet bi-Laplacian on Lp-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.

The dimension of eigenvariety of nonnegative tensors associated with spectral radius

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar, called Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. We prove that the dimension of the above projective eigenvariety is zero, i.e. there are finitely many eigenvectors associated with the spectral radius up to a scalar. We characterize a nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs.

Identifying and correcting the defects of the Saaty analytic hierarchy/network process: A comparative study of the Saaty analytic hierarchy/network process and the Markov chain-based analytic network process

The Saaty analytic network process (Saaty-ANP) is a generalization of the analytic hierarchy process. The Markov chain-based ANP (MC-ANP) is another decision-making approach suitable for general structures. Both ANPs use a relative measurement (paired comparisons with ratio scales) to estimate tangible and intangible factors, use a stochastic matrix (SM) to solve feedback problems and obtain the same results under some conditions. The Saaty-ANP does not define the basic concepts, nor does it check the rationality of the structure, which may lead to meaningless solutions and ignore a subclass of feedback decision problems. MC-ANP separates the alternatives from the criteria and defines the attributes, criteria, criterion dominated relations (CDRs) and reasonable constraints of the CDRs by means of digraphs; it also represents CDRs as Markov chain transition diagrams and corresponding (stochastic) adjacency matrices and obtains solutions from a system of linear equations. With the MC-ANP, for the real alternative problems (Class I), the solutions are priorities of the alternatives obtained by the parametric positive left eigenvectors of the SM, and for the nominal alternative problems (Class II), the solutions are priorities of the criteria obtained by the nonnegative right eigenvector of the SM. We analyze the conditions and causes of rank reversal; note that rank reversal does not appear in Class II problems; the study offers a rank reversal ANP example (with feedback) and presents a rank-preserving method for Class I problems. We discuss the contribution of MC-ANP, how to compensate for the defects of Saaty-AHP/ANP, and present issues that need further consideration.

A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied.

Optimal portfolio allocation and asset centrality revisited

This paper revisits the relationship between eigenvector asset centrality and optimal asset allocation in a minimum variance portfolio. We show that the standard definition of eigenvector centrality is misleading when the adjacency matrix in a network can take negative values. This is, for example, the case when the network topology is induced by the correlation matrix between assets in a portfolio. To correct for this, we introduce the concept of positive and negative eigenvector centrality. Our results show that the loss function associated to the minimum variance portfolio is positively/negatively related to the positive and negative eigenvector centrality under short-selling constraints but cannot be generalized beyond that. Furthermore, in contrast to what is claimed in the related literature, this relationship does not imply any monotonic relationship between the centrality of an asset and its optimal portfolio allocation. These theoretical insights are illustrated empirically in a portfolio allocation exercise with assets from U.S. and U.K. financial markets.

Dynamic behavior for social networks with state-dependent susceptibility and antagonistic interactions

In this paper, we investigate a general nonlinear model of opinion dynamics in which both state-dependent susceptibility to persuasion and antagonistic interactions are considered. According to the existing literature and socio-psychological theories, we examine three specializations of state-dependent susceptibility, that is, stubborn positives scenario, stubborn neutrals scenario, and stubborn extremists scenario. Interactions among agents form a signed graph, in which positive and negative edges represent friendly and antagonistic interactions, respectively. We conduct a comprehensive theoretical analysis of the generalized nonlinear opinion dynamics. For stubborn positives and stubborn neutrals scenarios, the general model is well-posed if and only if the system matrix is diagonally dominant. Based on the property of limit set and the existing results, we obtain some sufficient conditions such that the states of all agents converge to specific equilibrium point or bipartite consensus. For stubborn extremists scenario, by using the Perron–Frobenius property of eventually positive matrices, we establish some sufficient conditions such that the states of all agents converge into the subspace spanned by the right positive eigenvector of an eventually positive matrix. When there exists at least one entry of the equilibrium point or right positive eigenvector which is not equal to one, the derived results can be used to describe different levels of an opinion. In addition, for stubborn extremists scenario, we also consider the cases that the system matrix forms the opposing Laplacian and repelling Laplacian matrix, respectively. Finally, we present two examples to demonstrate the effectiveness of the theoretical findings.

Spectral properties of a non-compact operator in ecology.

Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate [Formula: see text], or a stable-stage distribution [Formula: see text]. In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector [Formula: see text] corresponding to its spectral radius [Formula: see text], this [Formula: see text] is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population [Formula: see text], there is a [Formula: see text] such that [Formula: see text].

The principal eigenvector to $$\alpha $$-spectral radius of hypergraphs

For a connected hypergraph H with $$rank(H)=r$$ , let $$\mathcal {D}(H)$$ and $$\mathcal {A}(H)$$ be the diagonal tensor of degrees and the adjacency tensor of H, respectively. For $$0 \le \alpha < 1$$, the $$\alpha $$-spectral radius $$\rho _{\alpha }(H)$$ of H is defined as $$\rho _{\alpha }(H)=\max \{x^{T}(\mathcal {A}_{\alpha }x)|x \in \mathbf {R}_{+}^{n},\Vert x\Vert _{r}=1\}$$, where $$\mathcal {A}_{\alpha }(H)=\alpha \mathcal {D}(H)+(1-\alpha )\mathcal {A}(H)$$. In this paper, we present some bounds on entries of the positive unit eigenvector corresponding to the $$\alpha $$-spectral radius of connected uniform hypergraphs. Furthermore, we obtain some bounds on entries of the positive unit eigenvector corresponding to the $$\alpha $$-spectral radius of connected general hypergraphs.

A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors

A classic result of Banach states that the supreme of a multivariate homogenous polynomial is equivalent to that of its associated symmetric multilinear form over unit balls. Using the language of higher-order tensors in finite-dimensional spaces, this means that for a symmetric tensor, its largest singular value is in fact equivalent to the largest magnitude of its eigenvalues. This note strengthens Banach’s results on nonnegative higher-order tensors. It is shown that for a symmetric nonnegative irreducible tensor: (1) any singular value admitting a positive singular vector tuple must be an eigenvalue, where the singular vectors in the tuple must be equal to each other; i.e., they amount to an eigenvector; (2) when the order is odd, any nonnegative singular vector tuple corresponding to the largest singular value must also boil down to a positive eigenvector. These results give the flexibility of directly solving tensor eigenvalue problems via solving tensor singular value problems.

MULTI-DOMAIN NEYMAN-TCHUPROV OPTIMAL ALLOCATION

Abstract The eigenproblem solution of the multi-domain efficient allocation is identified as a direct generalization of the classical Neyman-Tchuprov optimal allocation in stratified SRSWOR. This is achieved through analysis of eigenvalues and eigenvectors of a suitable population-based matrix D. Such a solution is an analytical companion to NLP approaches, which are often used in applications, see, e.g. Choudhry, Rao and Hidiroglou (2012). In this paper we are interested rather in the structure of the optimal allocation vector and relative variance than in such purely numerical tools (although the eigenproblem solution provides also numerical solutions, see, e.g. Wesołowski and Wieczorkowski (2017)). The domain-wise optimal allocation and the respective optimal variance of the estimator are determined by the unique direction (defined in terms of the positive eigenvector of matrix D) in the space ℝ I , where I is the number of domains in the population.

A Rayleigh quotient method for criticality eigenvalue problems in neutron transport

The alpha- and k-effective eigenproblems describe the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. We present Rayleigh quotient methods that are applied to demonstrably primitive discretizations of the one-dimensional slab, multigroup in energy, neutron transport equation. These methods are capable of solving subcritical and supercritical alpha- and k-effective eigenvalue problems. The derived eigenvalue updates are optimal in the least squares sense and positive eigenvector updates are guaranteed from the Perron-Frobenius Theorem for primitive matrices.

On the spectral theory of positive operators and PDE applications

The strong version of the Krein-Rutman theorem requires that the positive cone of the Banach space has nonempty interior and the compact map is strongly positive, mapping nonzero points in the cone into its interior. In this paper, we first generalize this version of the Krein-Rutman theorem to the case of 'semi-strongly positive' operators; and we prove it in a totally elementary fashion. We then prove the equivalence of semi-strong positivity and irreducibility in a Banach lattice, linking the afore-mentioned result with the Krein-Rutman theorem for irreducible operators. One of the things we emphasize is to use 'upper and lower spectral radii' to characterize, in the fashion of Collatz-Wielandt formula for nonnegative irreducible matrices, the principal eigenvalue of these operators. For reducible operators, we prove that the lower spectral radius always serves as the least upper bound of the set of eigenvalues pertaining to positive eigenvectors, and the upper spectral radius the greatest lower bound of the set. Finally, we demonstrate the full power of these Krein-Rutman theorems on some PDE examples such as elliptic eigenvalue problems on non-smooth domains, and cooperative systems which may or may not be fully coupled, by using as few PDE tools as possible.

On the Perron-Frobenius Theory of $M_v-$matrices and Eventually Exponentially Nonnegative Matrices

Mv−matrix is a matrix of the form A = sI −B,&nbsp;where 0 ≤ ρ(B) ≤ s and B is an eventually nonnegative matrix. In this paper, Mv−matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an Mv−matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if index0B ≤ 1 are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or Mv−matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and Mv−matrices are presented. &nbsp;Various numerical examples are given to support our theoretical findings.