Eigenvector Localization Research Articles

Spectral analysis of gene co-expression network of Zebrafish

We analyze the gene expression data of Zebrafish under the combined framework of complex networks and random matrix theory. The nearest-neighbor spacing distribution of the corresponding matrix spectra follows random matrix predictions of Gaussian orthogonal statistics. Based on the eigenvector analysis we can divide the spectra into two parts, the first part for which the eigenvector localization properties match with the random matrix theory predictions, and the second part for which they show deviation from the theory and hence are useful to understand the system-dependent properties. Spectra with the localized eigenvectors can be characterized into three groups based on the eigenvalues. We explore the position of localized nodes from these different categories. Using an overlap measure, we find that the top contributing nodes in the different groups carry distinguished structural features. Furthermore, the top contributing nodes of the different localized eigenvectors corresponding to the lower eigenvalue regime form different densely connected structure well separated from each other. Preliminary biological interpretation of the genes, associated with the top contributing nodes in the localized eigenvectors, suggests that the genes corresponding to same vector share common features.

Spectral properties of directed random networks with modular structure

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries. Spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, whereas networks with correlated entries have localized eigenvectors. In order to understand the origin of localization we track the spectra as a function of connection probability and directionality. As connections are made directed, eigenstates start occurring in complex-conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern. Moreover, for a very well distinguished community structure, the whole spectrum is localized except few eigenstates at the boundary of the circular distribution. As the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure. We search for this effect for the whole range of correlation strengths and for different community configurations. Furthermore, we investigate spectral properties of a metabolic network of zebrafish and compare them with those of the model networks.

The connectome of the rat central nervous system in neuroVIISAS

Event Abstract Back to Event The connectome of the rat central nervous system in neuroVIISAS Oliver Schmitt1*, Peter Eipert1, Andreas Wree1 and Klaus-Peter Schmitz2 1 University of Rostock, Dep. of Anatomy, Germany 2 University of Rostock, Dep. of Biomedical Engeneering, Germany Background. Most tracing studies have been performed in the rat central nervous system. However, an integrative approach to analyze and compare these tracing data on all levels of the rat central nervous system is not available. The generic platform neuroVIISAS allows to integrate neuroontological, atlas, visualization, connectivity analysis and population based simulations techniques for all types of nervous systems. Material. In an ongoing metastudy 830 tracing publications have been evaluated with regard to ipsi-, contralateral connections, projection densities, projection directions and neurochemical features. 56000 conenctions of the rat central nervous system are available in neuroVIISAS. The connectivity data are related to 8400 neuroanatomic regions that are organized hierarchically. The topographic relations between regions are defined by ontological relations like e.g. „part_of“. The regions are mapped to a high resolution affine linear and elastic registered histological dataset with an isotropic resolution of 5 x 5 x 5 µm³ and to the atlas dataset of the Paxinos and Watson stereotactic rat brain atlas (2007). Methods. neuroVIISAS has been developed in JAVA (Java Development Kit (JDK)) and is running on 32 and 64 Bit Windows and Linux systems. The Visualization Toolkit (VTK) is used for 3D-visualization of brain region surfaces within neuroVIISAS. VTK consists of a C++ class library that is accessed via a JAVAwrapper. Further packages are InfoNode Docking Windows (v1.5), which is a JAVA Swing based docking windows framework and the icon library Fugue Icons (v2.0). Furthermore, Python, PyNEST and NEST are necessary to generate simulation scripts under neuroVIISAS and for performing the simulation in a Linux environment. Result. The connectome of the rat nervous system has features of a scale-free Watts-Strogatz network. The average cluster coefficient is 0.112 and the avergae path length is 4.844. The average valency is 8.975 and the centrality is 0.081. Motif-analysis of 3 node directed motifs showed significant frequencies for divergent, convergent and reciprocal motifs. The lateral hypothalamic area possesses largest in- and outdegrees. We will present results of local connectivity analysis, e.g., cluster coefficient, eigenvector centrality, Shapley values, modularity analysis and communicability features of the connectome components. Based on these parameters the extrapyramidal system has been selected and visualized for an examplary simulation based on real neuroanatomic connectivity (Fig. 1). Outlook. neuroVIISAS integrates basic components of neuroinformatic techniques like neuroontological data structures, attributes, hierarchies, mapping/atlasing, multimodal data integration (histology, atlas, MRI datasets), connectivity analysis, multidimensional visualization, simulation definition/control/analysis/visualization in a generic platform in combination with the largest connectivity database. Currently tools are under development that allows to match tractographic data with connectivity data to compute precise spatial pathways. Multiscale techniques for visualization, connectivity filtering and simulation analysis are of particular interest to understand effects of scaling within simulations. Finally, multicompartment models of neurons for more specific local circuit definitions are under development in order to simulate complex functions like motor control by extrapyramidal and cerebellar subsystems. Figure 1 Keywords: computational neuroscience Conference: 4th INCF Congress of Neuroinformatics, Boston, United States, 4 Sep - 6 Sep, 2011. Presentation Type: Poster Presentation Topic: Computational neuroscience Citation: Schmitt O, Eipert P, Wree A and Schmitz K (2011). The connectome of the rat central nervous system in neuroVIISAS. Front. Neuroinform. Conference Abstract: 4th INCF Congress of Neuroinformatics. doi: 10.3389/conf.fninf.2011.08.00025 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 17 Oct 2011; Published Online: 19 Oct 2011. * Correspondence: Dr. Oliver Schmitt, University of Rostock, Dep. of Anatomy, Rostock, Germany, schmitt@med.uni-rostock.de Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Oliver Schmitt Peter Eipert Andreas Wree Klaus-Peter Schmitz Google Oliver Schmitt Peter Eipert Andreas Wree Klaus-Peter Schmitz Google Scholar Oliver Schmitt Peter Eipert Andreas Wree Klaus-Peter Schmitz PubMed Oliver Schmitt Peter Eipert Andreas Wree Klaus-Peter Schmitz Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

EIGENVECTOR LOCALIZATION AS A TOOL TO STUDY SMALL COMMUNITIES IN ONLINE SOCIAL NETWORKS

We present and discuss a mathematical procedure for identification of small "communities" or segments within large bipartite networks. The procedure is based on spectral analysis of the matrix encoding network structure. The principal tool here is localization of eigenvectors of the matrix, by means of which the relevant network segments become visible. We exemplified our approach by analyzing the data related to product reviewing on Amazon.com. We found several segments, a kind of hybrid communities of densely interlinked reviewers and products, which we were able to meaningfully interpret in terms of the type and thematic categorization of reviewed items. The method provides a complementary approach to other ways of community detection, typically aiming at identification of large network modules.

Disk-annulus transition and localization in random non-Hermitian tridiagonal matrices

Eigenvalues and localization of eigenvectors of non-Hermitian tridiagonal periodic random matrices are studied by means of the Hatano–Nelson deformation. The support of the spectrum undergoes a disk to annulus transition, with inner radius measured by the complex Thouless formula. The inner bounding circle and the annular halo are structures that correspond to the two arcs and wings observed by Hatano and Nelson in deformed Hermitian models, and are explained in terms of localization of eigenstates via a spectral duality and the argument principle. This disk-annulus transition is reminiscent of Feinberg and Zee's transition observed in full complex random matrices.

Eigenvector Localization for Random Band Matrices with Power Law Band Width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width W raised to a power μ remains smaller than the matrix size N. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate μ ≤ 8 holds.

Turbulence-chemistry interactions in CFD modelling of diesel engines

The use of detailed mechanisms is indispensable in predicting intermediate species in practical combustion devices. Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions. Their size makes it prohibitive to use them in three-dimensional combustion simulations of real-life equipment. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. In such situations, the ILDM (Intrinsic Low Dimensional Manifold) method gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence. It is a method of automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time-scales identified by a local eigenvector analysis. The KIVA-III code is used to simulate the flow in a single cylinder of a DI diesel engine. A RNG (Renormalized Group Theory) k-ϵ model is added to the code to model the turbulent flow field. Turbulence–chemistry interactions are taken into account by integrating the chemical source terms over a presumed probability density function (PDF). A standard approach can lead to unphysical states and rather poor heat-release and temperature profiles. An improvised approach to eliminate this problem is outlined in this work. Since no equilibrium or partial-equilibrium approach for mechanism reduction is involved, accurate concentrations of intermediate species such as O radicals and soot-precursors (C2H2 and C3H3) can be obtained from the ILDM. This improves the accuracy of pollutant predictions. The goal of this work is to establish the methods used as industrially acceptable ways of coupling the chemistry with turbulence.

Properties of near-zero modes and chiral symmetry breaking

We study localization and chirality properties of eigenvectors of the lattice Dirac operator. In particular, we focus on the dependence of our observables on the size of the corresponding eigenvalue, which allows us to study the transition of a near-zero mode into a bulk mode. We analyze ensembles of quenched SU(3) configurations using a Dirac operator which is a systematic expansion in path length of a solution of the Ginsparg–Wilson equation. Our results support the interpretation of the excitations relevant for chiral symmetry breaking as interacting instantons and anti-instantons.

On kink states of ferromagnetic chains

We study kink states of quantum, ferromagnetic, easy axis spin 1 2 chains at zero temperature. These are produced by applying opposite magnetic fields on the two end sites of the chain. For sufficiently strong anisotropy and boundary field, we obtain estimates on the wave function of the lowest energy states in sectors with fixed third component of the total spin. These estimates imply that the magnetization profile has a kink structure with a well-defined location and a finite width. The energies of kink states in different sectors are exponentially close as long as they are not located near the boundaries. The basic tool that we use here is the principle of exponential localization of eigenvectors. We illustrate the method in the simplest case of the Heisenberg XXZ model and then show how it can be generalized to more complicated models. In the particular case of the Heisenberg XXZ model our results are consistent with the exact kink wave functions known for a special value of the boundary magnetic field.

Monte Carlo methodologies for enhanced configurational sampling of dense systems: motion of a spherical solute in a polymer melt as a model problem

A number of traditional and novel Monte Carlo (MC) methodologies for configurational sampling in condensed phases are studied. The stochastic motion of a spherical solute molecule in a melt of short polyethylene chains is used as a model problem to assess the efficiency of the MC algorithms. Traditional MC methods, such as Metropolis MC and force-bias MC with or without preferential sampling, are inefficient in imparting significant mobility to the guest in the dense many-chain system. Two novel MC algorithms, based on local-Hessian information, are introduced here for the first time. Multidimensional force- or anti-force-bias along local eigenvector directions, and Metropolis MC with eigenvalue-scaling are found surprisingly inefficient for the problem at hand. Significant mobilities are achieved only with a new energy-biased MC method, which ignores the existing barriers and performs a coarse-grained random walk over local energy minima. As well as evaluating the various MC algorithms, this work also ad...

Assessing quantification for the EMS algorithm

A framework is developed for assessing the potential of the EMS (estimate, maximize, and smooth) algorithm to correctly quantify the emission it aims to reconstruct. Maximal eigenvector localization bounds of Perron-Frobenius theory are used to show that the ability of the EMS algorithm to quantify differences in emission is fundamentally limited by the smoothing imposed and the properties of the underlying model matrix in the algorithm. In particular, it is established that, for nonnegative irreducible smoothing, the EMS algorithm will always fail to correctly reconstruct zero emission intensity.

Absence of Cantor spectrum for a class of Schrödinger operators

It is shown that the complete localization of eigenvectors for the almost Mathieu operator entails the absence of Cantor spectrum for this operator.

The spectral moments method

A systematic analysis of the spectral moments method is presented and developed to compute the response functions of very large harmonic systems. Convergence of the algorithms is discussed and solutions are proposed to improve the results obtained. New developments are proposed. They concern, on the one hand, the determination of the Green functions or correlation functions of the system, and the localization of eigenvectors and, on the other hand, the determination of the spectral density of very large homogeneous matrices by a very simple and powerful technique. These results are illustrated by several examples taken from the main subjects studied by the authors: conducting polymers, fractals and quasi-crystals. Then comparison with other methods is discussed.

Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds

A general procedure for simplifying chemical kinetics and its use in reacting flow models is developed, which is based on the dynamical systems approach. In contrast to conventional reduced mechanisms no information is required concerning which reactions are to be assumed to be in partial equilibrium nor which species are assumed to be in steady state. Based on a local eigenvector analysis, the method identifies the fast time scales of the chemical reaction systems, which differ typically by orders of magnitude. Assuming that the fastest relaxation processes in chemical reactions proceed infinitely fast (i.e., are in local equilibrium), it is then possible to reduce the state space globally, such that it can be described by means of only a small number of reaction progress variables. The only “inputs” to the procedure are the detailed kinetics mechanism and the number of degrees of freedom required in the simplified scheme. Then the state properties given by the simplified scheme are automatically determined as functions of the coordinates associated with the degrees of freedom. A tabulation procedure allows an efficient use of the results in CFD codes. Furthermore a general procedure for coupling the reduced mechanism with other than chemical processes like flow and molecular transport is discussed. Results are presented for the CO/H 2 /air system both for a simple homogeneous closed system and a flow reactor.

Heterogeneous and homogeneous critical points of polymer distributions

General equations for the convergence of more than two phases (heterogeneous multiple critical points) and the merging of critical points associated with phase regions (homogeneous multiple critical points) are developed in the first half of this paper and are applied to three polymer distributions in the second half. At a heterogeneous point the spinodal surface is locally translation invariant along a local zero eigenvector field of the stability matrix and at a homogeneous point the gradient of the spinodal surface is null. Local invariance is extended to (1) field parameters (as temperature or pressure) along the streamlines of the zero eigenvector field on the spinodal surface and to (2) the classical approach with Legendre transform free energy of one composition variable. The heterogeneous critical conditions of Solc for a mixture of two homopolymers with molecular weight distribution are confirmed. For a copolymer with mer distribution, or equivalently a mixture of regularly interacting components, these equations are just successive moment expressions of a Gaussian distribution. The last example concerns the local destabilization of a homopolymer/copolymer mixture by copolymer mer distribution. Solc’s formalism can be extended to match these results also. Homogeneous multiple critical points are forbidden by molecular weight distribution in the first example and by a mer distribution that represents more than two components in the last two examples. A simple Landau expansion describing the local convergence of phases and critical points about a ternary tricritical point emphasizes the invariance of the critical point hierarchy with eigenvector scaling that was employed in developing all examples.

Phase transitions in the Ising model with transverse field

We show the existence of a phase transition in the Ising model with transverse field for dimensionsv ≥ 2 provided the transverse term is sufficiently small. This is done by proving long-range order occurs using the reflection positivity of the Hamiltonian and localization of eigenvectors.

Krein's formula and one-dimensional multiple-well

It is shown that all the discrete eigenvalues of a one-dimensional Schrödinger operator with a multiple well potential possess an asymptotic expansions in power of h ̵ 1 2 when h ̵ → 0 . A formula for all the coefficients of these expansions is given. The method uses two main tools: Perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article. As a by-product of this work, the exponential localization of eigenvectors when h ̵ goes to zero can be proved.

Comments on “absence of decay and eigenvector localization in a soluble one-dimensional system”

The localized states near E = 0 found by Stey and Gusman for the one-dimensional Weaire-Thorpe tight-binding model are found to be surface states, which disappear if the system has translational invariance or satisfies periodic boundary condition.

Absence of decay and eigenvector localization in a soluble one-dimensional system

Abstract The relation between the presence of spatially localized eigenvectors and the absence of decay of a state is illustrated in terms of the spectrum of a soluble one-dimensional model.