Eigenvectors Of Laplacian Research Articles

Prediction of partial synchronization in delay-coupled nonlinear oscillators, with application to Hindmarsh–Rose neurons

The full synchronization of coupled nonlinear oscillators has been widely studied. In this paper we investigate conditions for which partial synchronization of time-delayed diffusively coupled systems arises. The coupling configuration of the systems is described by a directed graph. As a novel quantitative result we first give necessary and sufficient conditions for the presence of forward invariant sets characterized by partially synchronous motion. These conditions can easily be checked from the eigenvalues and eigenvectors of the graph Laplacian. Second, we perform stability analysis of the synchronized equilibria in a (gain,delay) parameter space. For this analysis the coupled nonlinear systems are linearized around the synchronized equilibria and then the resulting characteristic function is factorized. By such a factorization, it is shown that the relation between the behaviour of different agents at the zero of the characteristic function depends on the structure of the eigenvectors of the weighted Laplacian matrix. By determining the structure of the solutions in the unstable manifold, combined with the characterization of invariant sets, we predict which partially synchronous regimes occur and estimate the corresponding coupling gain and delay values. We apply the obtained results to networks of coupled Hindmarsh–Rose neurons and verify the occurrence of the expected partially synchronous regimes by using a numerical simulation. We also make a comparison with an existing approach based on Lyapunov functionals.

Spectral clustering with epidemic diffusion

Spectral clustering is widely used to partition graphs into distinct modules or communities. Existing methods for spectral clustering use the eigenvalues and eigenvectors of the graph Laplacian, an operator that is closely associated with random walks on graphs. We propose a spectral partitioning method that exploits the properties of epidemic diffusion. An epidemic is a dynamic process that, unlike the random walk, simultaneously transitions to all the neighbors of a given node. We show that the replicator, an operator describing epidemic diffusion, is equivalent to the symmetric normalized Laplacian of a reweighted graph with edges reweighted by the eigenvector centralities of their incident nodes. Thus, more weight is given to edges connecting more central nodes. We describe a method that partitions the nodes based on the componentwise ratio of the replicator's second eigenvector to the first and compare its performance to traditional spectral clustering techniques on synthetic graphs with known community structure. We demonstrate that the replicator gives preference to dense, clique-like structures, enabling it to more effectively discover communities that may be obscured by dense intercommunity linking.

A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.

An MBO Scheme on Graphs for Classification and Image Processing

In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090--1118], Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg--Landau functional for solving problems in data classification. Here, we propose an adaptation of the classic numerical Merriman--Bence--Osher (MBO) scheme for minimizing graph-based diffuse interface functionals, like those originally proposed by Bertozzi and Flenner. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. Various computational examples are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. The results show that our method is multiple times more efficient than o...

Hydrogen bond network topology in liquid water and methanol: a graph theory approach

Networks are increasingly recognized as important building blocks of various systems in nature and society. Water is known to possess an extended hydrogen bond network, in which the individual bonds are broken in the sub-picosecond range and still the network structure remains intact. We investigated and compared the topological properties of liquid water and methanol at various temperatures using concepts derived within the framework of graph and network theory (neighbour number and cycle size distribution, the distribution of local cyclic and local bonding coefficients, Laplacian spectra of the network, inverse participation ratio distribution of the eigenvalues and average localization distribution of a node) and compared them to small world and Erdős-Rényi random networks. Various characteristic properties (e.g. the local cyclic and bonding coefficients) of the network in liquid water could be reproduced by small world and/or Erdős-Rényi networks, but the ring size distribution of water is unique and none of the studied graph models could describe it. Using the inverse participation ratio of the Laplacian eigenvectors we characterized the network inhomogeneities found in water and showed that similar phenomena can be observed in Erdős-Rényi and small world graphs. We demonstrated that the topological properties of the hydrogen bond network found in liquid water systematically change with the temperature and that increasing temperature leads to a broader ring size distribution. We applied the studied topological indices to the network of water molecules with four hydrogen bonds, and showed that at low temperature (250 K) these molecules form a percolated or nearly-percolated network, while at ambient or high temperatures only small clusters of four-hydrogen bonded water molecules exist.

Oscillations of networks: the role of soft nodes

To describe the flow of a miscible quantity on a network, we consider the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. The structure of the graph influences strongly the dynamics. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations using a basis of eigenvectors of the graph Laplacian. These lead us to introduce the notion of soft nodes. We give sufficient conditions for their existence in general graphs. They can cause several effects as we show on small graphs, for example the ineffectiveness of damping applied to them. Soft nodes may be of critical importance for complex physical networks and engineering networks like power grids.

Renormalized square of white noise quantum time shift

Quantum time shifts are a very special class of continuous time random walk type additive processes on Lie algebras whose exponentiation gives rise to quantum Markov processes whose infinitesimal generators possess a total set of unitary eigen-operators: such a class of quantum Markov gen- erators was not known in the literature. This property generalizes the known fact that trigonometric exponentials are eigenvectors of the usual classical Laplacian and was first noticed in connection with the quantum Laplacian in (5). In that case the eigen-operators turn out to be the Weyl operators, which are known to be non-commutative extensions of the trigonometric ex- ponentials. A generalization of these results, from the Weyl algebra to the harmonic oscillator algebra was obtained in (6). In the present paper we extend the above results to the renormalized square of white noise algebra. This provides among other things, a quantum extension of the Markov gen- erators of the Meixner processes. To our knowledge such an extension was not previously known.

Nodal domain partition and the number of communities in networks**This work is supported by the National Natural Science Foundation of China(no. 10971137), the National Basic Research Program (973) of China (no. 2006CB805900),and a grant of Science and Technology Commission of Shanghai Municipality (STCSM,no. 09XD1402500).

It is difficult to detect and evaluate the number of communities in complex networks,especially when the situation involves an ambiguous boundary between the inner- andinter-community densities. In this paper, discrete nodal domain theory is used to provide acriterion to determine how many communities a network has and how to partition thesecommunities by means of topological structure and geometric characterization. Bycapturing the signs of the Laplacian eigenvectors, we separate the network into severalreasonable clusters. The method leads to a fast and effective algorithm with application toa variety of real network data sets.

Efficient Decentralized Approximation via Selective Gossip

Recently, gossip algorithms have received much attention from the wireless sensor network community due to their simplicity, scalability and robustness. Motivated by applications such as compression and distributed transform coding, we propose a new gossip algorithm called Selective Gossip. Unlike traditional randomized gossip which computes the average of scalar values, we run gossip algorithms in parallel on the elements of a vector. The goal is to compute only the entries which are above a defined threshold in magnitude, i.e., significant entries. Nodes adaptively approximate the significant entries while abstaining from calculating the insignificant ones. Consequently, network lifetime and bandwidth are preserved. We show that with the proposed algorithm nodes reach consensus on the values of the significant entries and on the indices of insignificant ones. We illustrate the performance of our algorithm with a field estimation application. For regular topologies, selective gossip computes an approximation of the field using the wavelet transform. For irregular network topologies, we construct an orthonormal transform basis using eigenvectors of the graph Laplacian. Using two real sensor network datasets we show substantial communication savings over randomized gossip. We also propose a decentralized adaptive threshold mechanism such that nodes estimate the threshold while approximating the entries of the vector for computing the best <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -term approximation of the data.

Resistance distances and the Kirchhoff index in Cayley graphs

In this paper, closed-form formulae for the Kirchhoff index and resistance distances of the Cayley graphs over finite abelian groups are derived in terms of Laplacian eigenvalues and eigenvectors, respectively. In particular, formulae for the Kirchhoff index of the hexagonal torus network, the multidimensional torus and the t -dimensional cube are given, respectively. Formulae for the Kirchhoff index and resistance distances of the complete multipartite graph are obtained.

Estimation of arrival times from seismic waves: a manifold-based approach*

We propose a new method to analyse seismic time-series and estimate the arrival times of seismic waves. Our approach combines two ingredients: the time-series are first lifted into a high-dimensional space using time-delay embedding; the resulting phase space is then parametrized using a non-linear method based on the eigenvectors of the graph Laplacian. We validate our approach using a data set of seismic events that occurred in Idaho, Montana, Wyoming and Utah between 2005 and 2006. Our approach outperforms methods based on singular-spectrum analysis, wavelet analysis and short-term average/long-term average (STA/LTA).

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.

Perturbation analysis of complete synchronization in networks of phase oscillators

The behavior of weakly coupled self-sustained oscillators can often be well described by phase equations. Here we use the paradigm of Kuramoto phase oscillators which are coupled in a network to calculate first- and second-order corrections to the frequency of the fully synchronized state for nonidentical oscillators. The topology of the underlying coupling network is reflected in the eigenvalues and eigenvectors of the network Laplacian which influence the synchronization frequency in a particular way. They characterize the importance of nodes in a network and the relations between them. Expected values for the synchronization frequency are obtained for oscillators with quenched random frequencies on a class of scale-free random networks and for a Erdös-Rényi random network. We briefly discuss an application of the perturbation theory in the second order to network structural analysis.

Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks

The complete knowledge of Laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them; however, determining analytically Laplacian eigenvalues and eigenvectors is a theoretical challenge. In this paper, we study the Laplacian spectra and their corresponding eigenvectors of a class of deterministically growing treelike networks. The two interesting quantities are determined through the recurrence relations derived from the structure of the networks. Beginning from the rigorous relations one can obtain the complete eigenvalues and eigenvectors for the networks of arbitrary size. The analytical method opens the way to analytically compute the eigenvalues and eigenvectors of some other deterministic networks, making it possible to accurately calculate their spectral characteristics.

Eigenvectors of the discrete Laplacian on regular graphs—a statistical approach

In an attempt to characterize the structure of eigenvectors of random regular graphs, we investigate the correlations between the components of the eigenvectors associated with different vertices. In addition, we provide numerical observations, suggesting that the eigenvectors follow a Gaussian distribution. Following this assumption, we reconstruct some properties of the nodal structure which were observed in numerical simulations, but were not explained so far (Dakel et al 2007 APPROX-RANDOM pp 436–48).

Rewiring networks for synchronization

We study the synchronization of identical oscillators diffusively coupled through a network and examine how adding, removing, and moving single edges affects the ability of the network to synchronize. We present algorithms which use methods based on node degrees and based on spectral properties of the network Laplacian for choosing edges that most impact synchronization. We show that rewiring based on the network Laplacian eigenvectors is more effective at enabling synchronization than methods based on node degree for many standard network models. We find an algebraic relationship between the eigenstructure before and after adding an edge and describe an efficient algorithm for computing Laplacian eigenvalues and eigenvectors that uses the network or its complement depending on which is more sparse.

Coarse analysis of collective motion with different communication mechanisms

We study the effects of a signalling constraint on an individual-based model of self-organizing group formation using a coarse analysis framework. This involves using an automated data-driven technique which defines a diffusion process on the graph of a sample dataset formed from a representative stationary simulation. The eigenvectors of the graph Laplacian are used to construct ‘diffusion-map’ coordinates which provide a geometrically meaningful low-dimensional representation of the dataset. We show that, for the parameter regime studied, the second principal eigenvector provides a sufficient representation of the dataset and use it as a coarse observable. This allows the computation of coarse bifurcation diagrams, which are used to compare the effects of the signalling constraint on the population-level behavior of the model.

Approximation of optimal surface parameterizations and the application in cerebral cortex mapping

Optimal parameterizations of surface meshes are useful in the mapping and visualization of the cerebral cortex, the outer layer of the human brain. We propose two new methods to compute approximations of the optimal parameterizations, and apply these methods to human cortical surface meshes extracted from magnetic resonance images. Our methods approximate the parameterizations in a low-dimensional subspace spanned by the coordinate vectors of an initial parameterization and the low-frequency eigenvectors of a mesh Laplacian. This low-dimensional approximation reduces the computational complexity while minimizing the error.

Laplacian spectra as a diagnostic tool for network structure and dynamics

We examine numerically the three-way relationships among structure, Laplacian spectra, and frequency synchronization dynamics on complex networks. We study the effects of clustering, degree distribution, and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks. We find that these topological factors produce marked signatures in the Laplacian eigenvalue distribution and in the localization properties of individual eigenvectors. Using a set of coordinates based on the Laplacian eigenvectors as a diagnostic tool for synchronization dynamics, we find that the process of frequency synchronization can be visualized as a series of quasi-independent transitions involving different normal modes. Particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes. For example, there are important partially synchronized states in which a set of low-lying modes remain unlocked while those in the main spectral peak are locked. We find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue.

Image Processing with Nonlocal Spectral Bases

This article studies regularization schemes that are defined using a lifting of the image pixels in a high-dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low-dimensional smooth manifold. The construction of differential operators on this lifted space allows one to compute PDE flows and perform variational optimizations. All of these schemes lead to regularizations that exploit the manifold structure of the lifted image. Depending on the specific definition of the lifting, one recovers several well-known semilocal and nonlocal denoising algorithms that can be interpreted as local estimators over a semilocal or a nonlocal manifold. This framework also allows one to define thresholding operators in adapted orthogonal bases. These bases are eigenvectors of the discrete Laplacian on a manifold adapted to the geometry of the image. Numerical results compare the efficiency of PDE flows, energy minimizations, and thresholdings in the semilocal and nonlocal settings. The superiority of the nonlocal computations is studied through the performance of nonlinear approximation in orthogonal bases.