Characterizing path-factor deleted graphs via Q-index and D-index

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

A comparative study of quantum properties associated with coherent correlations between the modes of light generated by a nondegenerate parametric down-conversion process is presented. Particularly, the degree of nonclassicality that can be quantified by logarithmic negativity and quantum discord is compared, and a striking similar tendency is observed. In the same manner, a comparison of the quadrature variance, the smallest eigenvalue of the symplectic matrix and Einstein-Podolsky-Rosen (EPR) steering is made whereby a significant similarity is witnessed. The highest degree of quantum features of the radiation is in general manifested near threshold, whereas the quantifying measure related to the smallest eigenvalue of the symplectic matrix is found to capture a better degree of quantumness. Since each criterion renders only sufficient condition, it is hoped that witnessing consistent outcome across different criteria enhances the chance for experimental verification and practical application.

A tight bound is given on the speed of convergence of Newton's method with optimal scaling for the polar decomposition of a nonsingular complex matrix. Necessary and sufficient conditions are then derived that tell when an approximation to the optimal scaling value will give better results than the unscaled Newton method. For the related matrix sign problem, it is shown that optimal scaling requires complete knowledge of the eigenvalues of the original matrix. Because this is impractical, a family of scaling methods that are optimal with respect to partial eigenvalue information is derived. This family includes optimal scaling as well as a "semioptimal" scaling method based on the dominant eigenvalues of the matrix and its inverse. Semioptimal scaling can be implemented using the power method and it gives nearly optimal performance on a set of test problems. These test problems also show that a variety of other commonly used scaling strategies, including spectral scaling, determinantal scaling, and 2-norm scaling, can result in unduly slow convergence.

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable f for which all the time-delayed autocovariance exist, we construct a Hilbert space H_{f} and a Koopman-like operator K that acts on H_{f}. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of f that is represented by the eigenvectors of K. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When f is an observable of an ergodic dynamical system which has a finite invariant measure μ,H_{f} coincides with closure in L^{2}(X,dμ) of Krylov subspace generated by f, and K coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD, and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of K from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of K-eigenfrequencies. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed.

In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.

Eigenvalues of the Laplacian matrix play an important role in characterizing structural and dynamical properties of networks. In the procedure for calculating eigenvalues of the Laplacian matrix, we need to get the Laplacian matrix that represents structures of the network. Since the actual structure of networks and the strength of links are difficult to know, it is difficult to determine elements of the Laplacian matrix. To solve this problem, our previous study proposed a concept of the network resonance method, which is for estimating eigenvalues of the scaled Laplacian matrix using resonance of oscillation dynamics on networks. This method does not need a priori information about the network structure. In this research, we investigate feasibility of the network resonance method, and show that the method can estimate eigenvalues of the scaled Laplacian matrix of the entire network through observations of oscillation dynamics even if observable nodes are restricted to a part of network.

This paper focuses on the analytical study of final consensus convergence state of multi-agent dynamical systems by using a kind of generalized linear local interaction protocols. All the agents in the fixed directed network topology are governed by double-integrator dynamics. Almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining the algebraic graph theory and the matrix theory, some necessary and sufficient conditions are derived for reaching the second-order consensus. Moreover, the finial consensus convergence states of all agents are also be analytically determined. According to the obtained results, it is found that both the linear gains and the eigenvalues of the Laplacian matrix associated with the directed network topology play key roles in reaching consensus. Finally, the effectiveness and correctness of our theoretical findings are demonstrated by some numerical examples.

This technical note investigates the positive consensus problem of multiagent systems with directed communication topologies, where all the agents have identical continuous-time positive linear dynamics. Existing works of such a problem mainly focus on the case, where networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In contrast to them, we study this problem in which the communication topologies of the multiagent system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary, and sufficient conditions on positive consensus of the directed multiagent system are derived through using linear matrix inequality techniques. A primal-dual iterative algorithm is developed for the computation of solutions. Finally, several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results.

In this paper, the consensus problem for a class of second-order multi-agent systems under aperiodically intermittent communication is investigated. First, a new sampled-hold-based intermittent control protocol is designed to achieve consensus and improve the consensus performance in the absence of continuous communication. Then, some necessary and sufficient conditions are obtained for second-order consensus based on the relationship of the control gains, the eigenvalues of the Laplacian matrix, and the intermittent communication architecture. Moreover, if the sampled-hold-based aperiodically intermittent control is degenerated into the aperiodically intermittent control or the sampled-hold-based periodically intermittent control, some simpler consensus conditions are also given. Finally, some experimental simulations are shown to verify the correctness of the theoretical analysis.

Impulsive consensus algorithms for second-order multi-agent networks with sampled information

Let G be a connected simple graph. The relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. For a tree the complete description of the eigenvector corresponding to this eigenvalue is given and some results about the multiplicity of this eigenvalue are given. 1. Laplacian matrices. Let G be a connected simple graph with vertex set V = {1, 2, ··· ,n }, edge set E and let each edge be associated with a positive number, called the weight of the edge. The above graph is called a weighted graph. An unweighted graph is just a weighted graph with each of the edges bearing weight 1. All the graphs considered are weighted and simple, unless specified otherwise;all the matrices considered are real. The weight w(i )o f av ertexi is the sum of the weights of the edges incident with it. The Laplacian matrix L related to this graph is defined as L =( lij ) , where

Analysis of distance matrices

We give an upper bound on the largest eigenvalue of the signless Laplacian matrix of a Hamiltonian graph. This bound is applied to obtain sufficient spectral conditions for the non-existence of Hamiltonian cycles. Under certain additional assumptions we provide a polynomial time decisive spectral criterion for the Hamiltonicity of a given graph with sufficiently large minimum vertex degree.

Model-based monitoring systems for machine diagnostics and control are receiving increased attention. These techniques can estimate process and machine variables from inexpensive, easy to install remote sensors based on state-space models of the machine structure between the machine variables of interest and the location of the remote sensors. Model-based monitors are most frquently based on state observer techniques. Unfortunately these observers can be ill-conditioned leading to poor peformance. The authors show in previous papers that the conditioning of the observer performance can be repesented by a single performance index, the condition number of the eigensystem of the state observer matrix, and that there exists an upper bound for the index in non-normal matrices and the bound can be determined by the structure and eigenvalues of the observer matrix. In this paper, a design methodology for synthesizing well-conditioned observers is proposed based on the upper bound of the performance index. This is in contrast to using classical pole placement techniques. From the fact that small upper bound guarantees the small value of the performance index, a well-conditioned matrix form is defined and block-by block design strategy for a well-conditioned observer matrix is presented. A complete design procedure for well-conditioned deterministic state observers is given for the single-output case, while for multi-output case only a sufficient condition is given. The design stategy is illustrated with an example that shows that a well-conditioned observer performs much better than an observer designed with traditional techniques.

Accurate stabilization for linear stochastic systems based on region pole assignment and its applications