Spectral Radius Of Matrix Research Articles

The spectral radius of signless Laplacian matrix and sum-connectivity index of graphs

The sum-connectivity index of a graph G is defined as the sum of weights over all edges uv of G, where du and dv are the degrees of the vertices u and v in G, respectively. The sum-connectivity index is one of the most important indices in chemical and mathematical fields. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The sum-connectivity index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the sum-connectivity index of a graph G and the spectral radius of the matrix Q(G). We prove that for some connected graphs with n vertices and m edges,

The $\mathcal{H}_2$-optimal Control Problem of CSVIU Systems: Discounted, Counterdiscounted, and Long-Run Solutions

The paper deals with stochastic control problems associated with $H_2$ performance indices such as energy or power norms or energy measurements. The control applies to a class of systems for which a stochastic process conveys the underlying uncertainties, known as control and state variation increase uncertainty (CSVIU). These indices allow various emphases, focusing on the transient behavior with the discounted norm to stricter conditions on stability and steady-state mean-square error and convergence rate, using the optimal overtaking criterion. The long-run average power control stands as a midpoint in this respect. A critical advance regards the explicit form of the optimal control law, expressed in two forms. One takes a perturbed affine Riccati-like form of feedback solution; the other comes from a generalized normal equation that arises from the nondifferentiable local optimal problem. They are equivalent, but the former allows optimality and stability, and the latter develops into a search method to attain the optimal law. A detectability notion and a Riccati solution grant stochastic stability from the behavior of the norms. The energy overtaking criterion requires a further constraint on a matrix spectral radius. With these findings, the paper revisits the rising of the inaction solution, a prominent feature of CSVIU models to deal with the uncertainty inherent to poorly known models. Besides the optimal solution, it provides the tools to pursue it.

On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G) 2 d(u)+d-(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have ( 2 || ----n----- ||{ 2 (n − 1), if n ≥ 6, -q(G-)- ≤ | 16-, if n = 5, H (G ) || 5 |( 3, if n = 4, and the bounds are best possible.

Containment Control of Asynchronous Discrete-Time General Linear Multiagent Systems With Arbitrary Network Topology

In this contribution, we propose and investigate the containment control issue for general linear multiagent systems (MASs) under the asynchronous setting, where the network topology is not subjected to any structural restrictions and the roles of the leaders and the followers are entirely determined by the network topology. It is assumed that the interaction time instants of each agent, at which this agent interacts with its neighbors, are independent of the other agents' and can be unevenly distributed. An asynchronous distributed algorithm is proposed to implement the control strategy of linear MASs. The non-negative matrix theory and the composition of binary relations are utilized to handle the asynchronous containment control issue. It is shown that the leaders in each closed and strongly connected component of the network topology will reach a common state and the followers will gradually enter the dynamic convex hull constructed by the leaders. Moreover, it is also proved that the system matrix can be strictly unstable, and the upper bound of the system matrix's spectral radius is explicitly stated. Finally, two simulation examples are also provided to verify the efficacy of our theoretical results.

Analog circuit soft fault diagnosis utilizing matrix perturbation analysis

This paper proposes a novel approach of analog circuit soft fault diagnosis utilizing matrix perturbation analysis. This method establishes an output response square matrix whose elements will change if circuits fail. Fault can be diagnosed via comparing the difference between the fault-free output matrix and the faulty. According to matrix theory, matrix spectral radius and perturbation matrix m1 norm are utilized to describe the difference. Differing from artificial intelligence algorithms, it is all completely unnecessary to train samples, and can be applied to more complex circuit diagnostics with fewer test nodes. Fault diagnosis, fault location and parameter identification can be realized by quadratic curve fitting in single fault mode. Experiments confirm the feasibility and correctness of this method.

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

SummaryThis paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.

Analysis of voltage stability uncertainty using stochastic response surface method related to wind farm correlation

Wind speed follows the Weibull probability distribution and wind power can have a significant influence on power system voltage stability. In order to research the influence of wind plant correlation on power system voltage stability, in this paper, the stochastic response surface method (SRSM) is applied to voltage stability analysis to establish the polynomial relationship between the random input and the output response. The Kendall rank correlation coefficient is selected to measure the correlation between wind farms, and the joint probability distribution of wind farms is calculated by Copula function. A dynamic system that includes system node voltages is established. The composite matrix spectral radius of the dynamic system is used as the output of the SRSM, whereas the wind speed is used as the input based on wind farm correlation. The proposed method is compared with the traditional Monte Carlo (MC) method, and the effectiveness and accuracy of the proposed approach is verified using the IEEE 24-bus system and the EPRI 36-bus system. The simulation results also indicate that the consideration of wind farm correlation can more accurately reflect the system stability.

Global stability of a discrete tuberculosis model

In this paper,a discrete tuberculosis model is investigated.By means of calculating the next generation matrix's spectral radius,we derive the reproduction number R_o of the model.The solutions of the model are bounded and positive,which can be verified through the relation theory of the difference equation.It is proved that R_0=1 is a threshold to determine the disease extincation or persistence.The disease-free equilibrium is global asymptotically stable when the reproduction number R_0 1.The endemic equilibrium is global asymptotically stable when the reproduction number R_0 1.

Further results on the spectral radius of matrices and graphs

In this paper, we obtain some results on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these results to signless Laplacian matrices to obtain some known results. We also apply these results to distance signless Laplacian matrix of a graph, and obtain some sharp bounds on the distance signless Laplacian spectral radius.

Study on power system small‐disturbance uncertainty stability considering wind power

AbstractIn order to study in depth the influence of the uncertainty factors on power system stability and to obtain the probability of instability under different levels, an evaluation method of power system stability based on monomial cubature rules (MCRs) is proposed. In this paper, the wind speed and the matrix spectral radius are considered as the uncertainty parameter and the output response, respectively. The relationship between the wind speed and the matrix spectral radius is found with MCRs to assess power system dynamic stability, and linear regression is used to solve chaotic polynomial coefficients. The comparison of the calculation results of the MCRs and Monte Carlo simulations on the IEEE 30‐bus system indicates that the MCRs require a smaller number of computations and give high accuracy. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Maximizing Sum Rates in Cognitive Radio Networks: Convex Relaxation and Global Optimization Algorithms

A key challenge in wireless cognitive radio networks is to maximize the total throughput also known as the sum rates of all the users while avoiding the interference of unlicensed band secondary users from overwhelming the licensed band primary users. We study the weighted sum rate maximization problem with both power budget and interference temperature constraints in a cognitive radio network. This problem is nonconvex and generally hard to solve. We propose a reformulation-relaxation technique that leverages nonnegative matrix theory to first obtain a relaxed problem with nonnegative matrix spectral radius constraints. A useful upper bound on the sum rates is then obtained by solving a convex optimization problem over a closed bounded convex set. It also enables the sum-rate optimality to be quantified analytically through the spectrum of specially-crafted nonnegative matrices. Furthermore, we obtain polynomial-time verifiable sufficient conditions that can identify polynomial-time solvable problem instances, which can be solved by a fixed-point algorithm. As a by-product, an interesting optimality equivalence between the nonconvex sum rate problem and the convex max-min rate problem is established. In the general case, we propose a global optimization algorithm by utilizing our convex relaxation and branch-and-bound to compute an ε-optimal solution. Our technique exploits the nonnegativity of the physical quantities, e.g., channel parameters, powers and rates, that enables key tools in nonnegative matrix theory such as the (linear and nonlinear) Perron-Frobenius theorem, quasi-invertibility, Friedland-Karlin inequalities to be employed naturally. Numerical results are presented to show that our proposed algorithms are theoretically sound and have relatively fast convergence time even for large-scale problems

Signal Processing Interpolation and Problem Conditioning Educational Workbench

This work presents a newer and more complete version of an educational tool to be used in signal processing interpolation-related subjects. Besides the consolidation of acquired theoretical knowledge, the tool allows now its users to apply three error geometry patterns, test minimum or maximum dimension signal reconstruction algorithms and problem conditioning through the analysis of the matrix spectral radius or the condition number: these new features gives the possibility to alter the problem definitions to the desired goal before the reconstruction begins. The time unit that measures the algorithms performance is ¶(nlogn) thus independent from the machine’s architecture. A video of the developed tool can be seen in: https://www.dropbox.com/s/t6yiiuy31ramxse/FILME_1.avi.

Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets

The problem of construction of Barabanov norms for analysis ofproperties of the joint (generalized) spectral radius ofmatrix sets has been discussed in a number of publications. In[18, 21] the method of Barabanovnorms was the key instrument in disproving the Lagarias-WangFiniteness Conjecture. The related constructions wereessentially based on the study of the geometrical properties ofthe unit balls of some specific Barabanov norms. In thiscontext the situation when one fails to find among currentpublications any detailed analysis of the geometricalproperties of the unit balls of Barabanov norms looks a bitparadoxical. Partially this is explained by the fact thatBarabanov norms are defined nonconstructively, by an implicitprocedure. So, even in simplest cases it is very difficult tovisualize the shape of their unit balls. The present work maybe treated as the first step to make up this deficiency. In thepaper an iteration procedure is considered that allows to buildnumerically Barabanov norms for the irreducible matrix sets andsimultaneously to compute the joint spectral radius of thesesets.

Some Observations on the TLM Numerical Solution of the Laplace Equation

This paper describes progress on the TLM modelling of the Laplace equation, in particular, how the rate of convergence is influenced by the choice of scattering parameter for a particular discretisation. The hypothesis that optimum convergence is achieved when the real and imaginary parts for the lowest harmonic in a Fourier solution cancel appears to be upheld. The Fourier solution for the problem has been advanced by a better understanding of the nature of the initial excitation. The relationship between the form of the initial condition used in this and many other numerical solutions of the Laplace equation and oscillatory behavior in the results is given a firmer theoretical basis. A correlation between TLM numerical results and those obtained from matrix spectral radius calculations has confirmed much previous work.

Effects of Hebbian learning on the dynamics and structure of random networks with inhibitory and excitatory neurons

The aim of the present paper is to study the effects of Hebbian learning in random recurrent neural networks with biological connectivity, i.e. sparse connections and separate populations of excitatory and inhibitory neurons. We furthermore consider that the neuron dynamics may occur at a (shorter) time scale than synaptic plasticity and consider the possibility of learning rules with passive forgetting. We show that the application of such Hebbian learning leads to drastic changes in the network dynamics and structure. In particular, the learning rule contracts the norm of the weight matrix and yields a rapid decay of the dynamics complexity and entropy. In other words, the network is rewired by Hebbian learning into a new synaptic structure that emerges with learning on the basis of the correlations that progressively build up between neurons. We also observe that, within this emerging structure, the strongest synapses organize as a small-world network. The second effect of the decay of the weight matrix spectral radius consists in a rapid contraction of the spectral radius of the Jacobian matrix. This drives the system through the “edge of chaos” where sensitivity to the input pattern is maximal. Taken together, this scenario is remarkably predicted by theoretical arguments derived from dynamical systems and graph theory.

Convergence rate of Gibbs sampler and its application

Based on the convergence rate defined by the Pearson– χ 2 distance, this paper discusses properties of different Gibbs sampling schemes. Under a set of regularity conditions, it is proved in this paper that the rate of convergence on systematic scan Gibbs samplers is the norm of a forward operator. We also discuss that the collapsed Gibbs sampler has a faster convergence rate than the systematic scan Gibbs sampler as proposed by Liu et al. Based on the definition of convergence rate of the Pearson– χ 2 distance, this paper proved this result quantitatively. According to Theorem 2, we also proved that the convergence rate defined with the spectral radius of matrix by Robert and Shau is equivalent to the corresponding radius of the forward operator.

Characterization of Continuous, Four-Coefficient Scaling Functions via Matrix Spectral Radius

We characterize the existence of continuous solutions of a four-coefficient dilation equation in terms of the usual spectral radius of a matrix. The criteria for the existence of such a solution can be very quickly examined. As a result we give an affirmative answer to a conjecture raised by Colella and Heil in 1992. Moreover, using our criteria we find the smoothest compactly supported four-coefficient orthogonal scaling function and thus the smoothest compactly supported orthonormal wavelet generated by this scaling function.

Neutral evolution of mutational robustness.

We introduce and analyze a general model of a population evolving over a network of selectively neutral genotypes. We show that the population's limit distribution on the neutral network is solely determined by the network topology and given by the principal eigenvector of the network's adjacency matrix. Moreover, the average number of neutral mutant neighbors per individual is given by the matrix spectral radius. These results quantify the extent to which populations evolve mutational robustness-the insensitivity of the phenotype to mutations-and thus reduce genetic load. Because the average neutrality is independent of evolutionary parameters-such as mutation rate, population size, and selective advantage-one can infer global statistics of neutral network topology by using simple population data available from in vitro or in vivo evolution. Populations evolving on neutral networks of RNA secondary structures show excellent agreement with our theoretical predictions.

Separation properties forMW-fractals

Corresponding to the irreducible 0–1 matrix (aij)n×n, take similitude contraction mappingsϕij for eachaij=1, inaij=1, in Rd with ratio 0<rij<1. There are unique nonempty compact setsF1,…,Fn satisfying for each1≤i≤n, Fi. We prove that open set condition holds if and only ifFi is ans-set for some1≤i≤n, wheres is such that the spectral radius of matrix (rij3)n x n is 1.

Further results on the stability of discrete-time matrix polynomials

This paper concentrates on the stability of discrete-time systems. The stability of the system is deduced from the roots of the determinant of a matrix polynomial of nth order. New and less restrictive conditions than those published previously are derived to ensure that roots of the determinant of the matrix polynomial are located inside the unit circle. The conditions are given in terms of the spectral radius of matrix constructed from the coefficient matrices. Examples are given for illustration.