Smallest Nonzero Eigenvalue Research Articles

Local Assortativity Affects the Synchronizability of Scale-Free Network

Synchronization is critical for system-level behavior in physical, chemical, biological, and social systems. Empirical evidence has shown that the network topology strongly impacts the synchronizability of the system, and the analysis of their relationship remains an open challenge. We know that the eigenvalue distribution determines a network's synchronizability, but analytical expressions that connect network topology and all relevant eigenvalues (e.g., the extreme values) remain elusive. Here, we accurately determine its synchronizability by proposing an analytical method to estimate the extreme eigenvalues using perturbation theory. Our analytical method exposes the role that global and local topology combine to influence synchronizability. We show that the smallest nonzero eigenvalue <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda ^{(2)}$</tex-math></inline-formula> , which determines synchronizability, is estimated by the smallest degree augmented by the inverse degree difference in the least connected nodes. From this, we can conclude that there exists a clear negative relationship between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda ^{(2)}$</tex-math></inline-formula> and the local assortativity of nodes with the smallest degree value. We validate the accuracy of our framework within the setting of a scale-free network and can be driven by commonly used ordinary differential equations (e.g., 3-D Rosler dynamics or Hindmarsh–Rose neuronal circuit). From the results, we demonstrate that the synchronizability of the network can be tuned by rewiring the connections of these particular nodes while maintaining the general degree profile of the network.

Centered PSD matrices with thin spectrum are M-matrices

We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order $n$ and rank $n-1$ with eigenvalue ratio $\lambda_{\max}/\lambda_{\min}\leq n/(n-2)$ between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices.

Spectral properties of a class of treelike networks generated by motifs with single root node

In this paper, we introduce a class of treelike networks generated by motifs with single root node. We study the spectral properties of treelike networks by studying the behaviors of the corresponding adjacency matrix and Laplacian matrix. First, we get the adjacency matrix and Laplacian matrix of this treelike network. Then, we analyze symmetry of eigenvalues, multiplicity of eigenvalue [Formula: see text] and prove that its largest eigenvalue is strictly monotonically increasing and convergent. Finally, we introduce a new method to prove that the largest eigenvalue of the Laplacian matrix is strictly monotonically increasing and convergent. Also, we obtain that its smallest nonzero eigenvalue is strictly monotonically decreasing and convergent.

The Rate of Convergence of the SOR Method in the Positive Semidefinite Case

In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form G x = b , where G is a real symmetric positive semidefinite n × n matrix. The bounds are given in terms of the condition number of G , which is the ratio κ = α / β , where α is the largest eigenvalue of G and β is the smallest nonzero eigenvalue of G . Let H denote the related iteration matrix. Then, since G has a zero eigenvalue, the spectral radius of H equals 1, and the rate of convergence is determined by the size of η , the largest eigenvalue of H whose modulus differs from 1. The bound has the form η 2 ≤ 1 − 1 / κ c , where c = 2 + log 2 n . The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.

On the optimality of sliced inverse regression in high dimensions

The central subspace of a pair of random variables $(y,\boldsymbol{x})\in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y\perp\!\!\!\!\!\perp \boldsymbol{x}|P_{\mathcal{S}}\boldsymbol{x}$. In this paper, we consider the minimax rate of estimating the central space under the multiple index model $y=f(\boldsymbol{\beta }_{1}^{\tau }\boldsymbol{x},\boldsymbol{\beta }_{2}^{\tau }\boldsymbol{x},\ldots,\boldsymbol{\beta }_{d}^{\tau }\boldsymbol{x},\epsilon )$ with at most $s$ active predictors, where $\boldsymbol{x}\sim N(0,\boldsymbol{\Sigma })$ for some class of $\boldsymbol{\Sigma }$. We first introduce a large class of models depending on the smallest nonzero eigenvalue $\lambda $ of $\operatorname{var}(\mathbb{E}[\boldsymbol{x}|y])$, over which we show that an aggregated estimator based on the SIR procedure converges at rate $d\wedge ((sd+s\log (ep/s))/(n\lambda ))$. We then show that this rate is optimal in two scenarios, the single index models and the multiple index models with fixed central dimension $d$ and fixed $\lambda $. By assuming a technical conjecture, we can show that this rate is also optimal for multiple index models with bounded dimension of the central space.

HERMES: PERSISTENT SPECTRAL GRAPH SOFTWARE.

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

Diffusion in a bistable system: The eigenvalue spectrum of the Fokker-Planck operator and Kramers' reaction rate theory.

The time-dependent solution of the Fokker-Planck equation with bistable potentials is considered in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck operator. The Fokker-Planck equation is the high friction limit of the corresponding Kramers' equation. Two different potentials are considered defined with a constant diffusion coefficient, ε, and position-dependent drift coefficients. The smallest nonzero eigenvalue of the Fokker-Planck operator, λ_{1}, provides the long-time rate coefficient for the transformation of the different species in the two stable states. A novel pseudospectral method with nonclassical polynomials is applied to this class of systems. The convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator versus the number of basis functions is studied and compared with previous results. The results are consistent with Kramers' theory, and a linear relationship between lnλ_{1} and 1/ε for sufficiently small ε values is verified. A comparison with analytic approximations to λ_{1} is provided.

Inequalities for the eigenvectors associated to extremal eigenvalues in rank one perturbations of symmetric matrices

This paper considers the eigenvectors involved in rank one perturbations of symmetric matrices. A doubly stochastic matrix based on the inner products between initial and perturbed eigenvectors is introduced to derive several relations for the latter ones. A majorization theorem used together with this doubly stochastic matrix provides the main results of the paper, which deal with the eigenvectors associated to the largest and smallest non-zero eigenvalues. Further developments are also suggested for infinitesimal perturbations using convergent power expansions of both eigenvalues and eigenvectors.

Identifying the Informational/Signal Dimension in Principal Component Analysis

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that has received the largest attention thus far and several identification methods—the so-called stopping rules—have been proposed, giving very different results in practice, and some comparative study has been carried out. Some inconsistencies in the previous studies led us to try to fix the distinction between signal from noise in PCA—and its limits—and propose a new testing method. This consists in the production of simulated data according to a predefined eigenvalues structure, including zero-eigenvalues. From random populations built according to several such structures, reduced-size samples were extracted and to them different levels of random normal noise were added. This controlled introduction of noise allows a clear distinction between expected signal and noise, the latter relegated to the non-zero eigenvalues in the samples corresponding to zero ones in the population. With this new method, we tested the performance of ten different stopping rules. Of every method, for every structure and every noise, both power (the ability to correctly identify the expected dimension) and type-I error (the detection of a dimension composed only by noise) have been measured, by counting the relative frequencies in which the smallest non-zero eigenvalue in the population was recognized as signal in the samples and that in which the largest zero-eigenvalue was recognized as noise, respectively. This way, the behaviour of the examined methods is clear and their comparison/evaluation is possible. The reported results show that both the generalization of the Bartlett’s test by Rencher and the Bootstrap method by Pillar result much better than all others: both are accounted for reasonable power, decreasing with noise, and very good type-I error. Thus, more than the others, these methods deserve being adopted.

Maximizing synchronizability of duplex networks.

We study the synchronizability of duplex networks formed by two randomly generated network layers with different patterns of interlayer node connections. According to the master stability function, we use the smallest nonzero eigenvalue and the eigenratio between the largest and the second smallest eigenvalues of supra-Laplacian matrices to characterize synchronizability on various duplexes. We find that the interlayer linking weight and linking fraction have a profound impact on synchronizability of duplex networks. The increasingly large inter-layer coupling weight is found to cause either decreasing or constant synchronizability for different classes of network dynamics. In addition, negative node degree correlation across interlayer links outperforms positive degree correlation when most interlayer links are present. The reverse is true when a few interlayer links are present. The numerical results and understanding based on these representative duplex networks are illustrative and instructive for building insights into maximizing synchronizability of more realistic multiplex networks.

Convergence Rate of Distributed ADMM Over Networks

We propose a new distributed algorithm based on alternating direction method of multipliers (ADMM) to minimize sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. Our algorithm allows for a general choice of the communication weight matrix, which is used to combine the iterates at different nodes. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate $O(1/T)$ , where $T$ is the number of iterations. We then show that when functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an $\epsilon$ -optimal solution can be computed with $O\left(\sqrt{\kappa _f} \log (1/\epsilon) \right)$ iterations, where $\kappa _f$ is the condition number of the problem. Our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes, the smallest nonzero eigenvalue, and operator norm of the communication matrix.

When Slepian Meets Fiedler: Putting a Focus on the Graph Spectrum

The study of complex systems greatly benefits from graph models and their analysis. In particular, the eigendecomposition of the graph Laplacian lets emerge properties of global organization from local interactions; e.g., the Fiedler vector has the smallest nonzero eigenvalue and plays a key role for graph clustering. Graph signal processing focuses on the analysis of signals that are attributed to the graph nodes. Again, the eigendecomposition of the graph Laplacian is important to define the graph Fourier transform and extend conventional signal-processing operations to graphs. Here, we introduce the design of Slepian graph signals by maximizing energy concentration in a predefined subgraph given a graph spectral bandlimit. We establish a novel link with classical Laplacian embedding and graph clustering, which provides a meaning to localized graph frequencies.

Synchronization of cyclic power grids: Equilibria and stability of the synchronous state.

Synchronization is essential for the proper functioning of power grids; we investigate the synchronous states and their stability for cyclic power grids. We calculate the number of stable equilibria and investigate both the linear and nonlinear stabilities of the synchronous state. The linear stability analysis shows that the stability of the state, determined by the smallest nonzero eigenvalue, is inversely proportional to the size of the network. We use the energy barrier to measure the nonlinear stability and calculate it by comparing the potential energy of the type-1 saddles with that of the stable synchronous state. We find that the energy barrier depends on the network size (N) in a more complicated fashion compared to the linear stability. In particular, when the generators and consumers are evenly distributed in an alternating way, the energy barrier decreases to a constant when N approaches infinity. For a heterogeneous distribution of generators and consumers, the energy barrier decreases with N. The more heterogeneous the distribution is, the stronger the energy barrier depends on N. Finally, we find that by comparing situations with equal line loads in cyclic and tree networks, tree networks exhibit reduced stability. This difference disappears in the limit of N→∞. This finding corroborates previous results reported in the literature and suggests that cyclic (sub)networks may be applied to enhance power transfer while maintaining stable synchronous operation.

Dynamic Feedback Synchronization of Lur'e Networks via Incremental Sector Boundedness

In this note, we generalize our results on synchronization of homogeneous Lur'e networks by static, relative state information based protocols from Zhang et al to the case that for each agent only relative measurements are available. We establish sufficient conditions under which a linear dynamic synchronization protocol exists for such networks. These conditions involve feasibility of two LMI's together with a coupling inequality, reminiscent of the well-known LMI conditions in $H_{\infty}$ control by measurement feedback. We show that, regardless of the number of agents in the network, only the three inequalities are involved. In the computation of the protocol matrices, the eigenvalues of the Laplacian matrix of the interconnection graph occur. In particular, the matrices representing the protocol depend on the smallest nonzero eigenvalue and the largest eigenvalue of the Laplacian matrix. We validate our results by means of a numerical simulation example.

Design and Topological Analysis of Complex Networks with Optimal Controllability

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

Laplacians for flow networks

We define a class of Laplacians for multicommodity, undirected flow networks, and bound their smallest nonzero eigenvalues with a generalization of the sparsest cut.

Properties of the volume operator in loop quantum gravity: I. Results

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in loop quantum gravity, which is the quantum analog of the classical volume expression for regions in three-dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator; in particular the presence of a ‘volume gap’ (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5–7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices are included, for which the presence of a volume gap is shown. This paper presents our main results; details are provided by a companion paper (Brunnemann and Rideout 2007 Properties of the volume operator in loop quantum gravity: II. Detailed presentation Class. Quantum Grav. 25 065002).

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator generating an infinite-dimensional diffusion process Ξ (t), in space . Here ν is a “natural”Ξ (t)-invariant measure on which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace such that has a distinctive character related to a “quasi-particle” picture. In particular, has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point κ1>0 giving the smallest non-zero eigenvalue of a limiting problem associated with β= 0. An immediate corollary of our result is an exponentially fast L 2-convergence to equilibrium for the process Ξ(t) for small values of β.

Modeling of Ion-Exchange Column Operation. II. Mass Transport Kinetics

Abstract A previously described method for modeling the operation of ion-exchange columns by numerical integration on a microcomputer is modified to include the effect on elution curves of the finite rate of mass transport of solute ions between the resin and the aqueous phase. This is done by means of a time constant approach. The time constant is estimated as the smallest nonzero eigenvalue of a suitably-chosen diffusion problem. Results are presented showing the effect of the size of the time constant and the salt concentration in the eluting liquid on the shapes of the elution curves.

The temperature coefficients of diatomic dissociation and recombination reactions

An improved method is presented for the solution of the relaxation equations representing the dissociation-recombination kinetics of a dilute diatomic gas. The kinetic equations are transformed to normal modes, and then solved by asymptotic expansion techniques. Numerical integration is employed through the initial transient, after which the normal-mode equations can be integrated analytically. This approach is about 10 8 times faster than conventional numerical integration, and the whole relaxation trajectory is accessible in less than 1 minute of computing time. Hence, exact solutions to nozzle-flow and similar problems now come within the bounds of feasibility. Using previously calculated T-V transition probabilities, suitably smoothed and extrapolated, calculations on the dissociation of H 2 are presented in the range 1200–12,000°K. The dissociation rate constant k d ,M is strictly proportional to the smallest nonzero eigenvalue of the relaxation matrix in this temperature range, and the Arrhenius temperature coefficient is about 2 kcal/mole less than the dissociation energy. The reasons why the Arrhenius temperature coefficient of k d ,M is not considerably smaller are examined. The induction period and the vibrational relaxation time are shown to be closely related to the second nonzero eigenvalue of the relaxation matrix, and these quantities have magnitudes and temperature coefficients in reasonable accord with experiment. Recombination results are presented for temperatures in the range 300–12,000°K. Below 5000°K, the recombination rate constant k r ,M decreases with increasing temperature with a temperature coefficient which seems to be related to the temperature coefficients of transitions in the bootleneck region. Definitive criteria, in terms of the eigenvalues of the relaxation matrix, are developed for identifying the bottleneck region. Above 5000°K, recombination rates increase again, but this may be due to imperfections in the assumed transition probabilities. At all temperatures at which k d ,M could be determined, the Rate-Quotient Law, i.e., k d ,M / k r ,M = K , the equilibrium constant, is obeyed to an accuracy of 4 or 5 decimal places.