3. Matrix Analysis of Graphs
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Eigenvalues of Euclidean Distance Matrices and Rs-Majorization on R2
Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University 2 Talk Eigenvalues of Euclidean distance matrices and rs-majorization on R pp.: 1{4 Eigenvalues of Euclidean Distance Matrices and rs-majorization on R2 Asma Ilkhanizadeh Manesh∗ Department of Pure Mathematics, Vali-e-Asr University of Rafsanjan Alemeh Sheikh Hoseini Department of Pure Mathematics, Shahid Bahonar University of Kerman Abstract Let D1 and D2 be two Euclidean distance matrices (EDMs) with correspond- ing positive semidefinite matrices B1 and B2 respectively. Suppose that λ(A) = ((λ(A)) )n is the vector of eigenvalues of a matrix A such that (λ(A)) ... i i=1 1 ≥ ≥ (λ(A))n. In this paper, the relation between the eigenvalues of EDMs and those of the 2 corresponding positive semidefinite matrices respect to rs, on R will be investigated. ≺ Keywords: Euclidean distance matrices, Rs-majorization. Mathematics Subject Classification [2010]: 34B15, 76A10 1 Introduction An n n nonnegative and symmetric matrix D = (d2 ) with zero diagonal elements is × ij called a predistance matrix. A predistance matrix D is called Euclidean or a Euclidean distance matrix (EDM) if there exist a positive integer r and a set of n points p1, . , pn r 2 2 { } such that p1, . , pn R and d = pi pj (i, j = 1, . , n), where . denotes the ∈ ij k − k k k usual Euclidean norm. The smallest value of r that satisfies the above condition is called the embedding dimension. As is well known, a predistance matrix D is Euclidean if and 1 1 t only if the matrix B = − P DP with P = I ee , where I is the n n identity matrix, 2 n − n n × and e is the vector of all ones, is positive semidefinite matrix. -
Clustering by Left-Stochastic Matrix Factorization
Clustering by Left-Stochastic Matrix Factorization Raman Arora [email protected] Maya R. Gupta [email protected] Amol Kapila [email protected] Maryam Fazel [email protected] University of Washington, Seattle, WA 98103, USA Abstract 1.1. Related Work in Matrix Factorization Some clustering objective functions can be written as We propose clustering samples given their matrix factorization objectives. Let n feature vectors d×n pairwise similarities by factorizing the sim- be gathered into a feature-vector matrix X 2 R . T d×k ilarity matrix into the product of a clus- Consider the model X ≈ FG , where F 2 R can ter probability matrix and its transpose. be interpreted as a matrix with k cluster prototypes n×k We propose a rotation-based algorithm to as its columns, and G 2 R is all zeros except for compute this left-stochastic decomposition one (appropriately scaled) positive entry per row that (LSD). Theoretical results link the LSD clus- indicates the nearest cluster prototype. The k-means tering method to a soft kernel k-means clus- clustering objective follows this model with squared tering, give conditions for when the factor- error, and can be expressed as (Ding et al., 2005): ization and clustering are unique, and pro- T 2 arg min kX − FG kF ; (1) vide error bounds. Experimental results on F;GT G=I simulated and real similarity datasets show G≥0 that the proposed method reliably provides accurate clusterings. where k · kF is the Frobenius norm, and inequality G ≥ 0 is component-wise. This follows because the combined constraints G ≥ 0 and GT G = I force each row of G to have only one positive element. -
Boundary Value Problems on Weighted Paths
Introduction and basic concepts BVP on weighted paths Bibliography Boundary value problems on a weighted path Angeles Carmona, Andr´esM. Encinas and Silvia Gago Depart. Matem`aticaAplicada 3, UPC, Barcelona, SPAIN Midsummer Combinatorial Workshop XIX Prague, July 29th - August 3rd, 2013 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts BVP on weighted paths Bibliography Outline of the talk Notations and definitions Weighted graphs and matrices Schr¨odingerequations Boundary value problems on weighted graphs Green matrix of the BVP Boundary Value Problems on paths Paths with constant potential Orthogonal polynomials Schr¨odingermatrix of the weighted path associated to orthogonal polynomials Two-side Boundary Value Problems in weighted paths MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Weighted graphs A weighted graphΓ=( V ; E; c) is composed by: V is a set of elements called vertices. E is a set of elements called edges. c : V × V −! [0; 1) is an application named conductance associated to the edges. u, v are adjacent, u ∼ v iff c(u; v) = cuv 6= 0. X The degree of a vertex u is du = cuv . v2V c34 u4 u1 c12 u2 c23 u3 c45 c35 c27 u5 c56 u7 c67 u6 MCW 2013, A. Carmona, A.M. Encinas and S.Gago Boundary value problems on a weighted path Introduction and basic concepts Schr¨odingerequations BVP on weighted paths Definition of BVP Bibliography Matrices associated with graphs Definition The weighted Laplacian matrix of a weighted graph Γ is defined as di if i = j; (L)ij = −cij if i 6= j: c34 u4 u c u c u 1 12 2 23 3 0 d1 −c12 0 0 0 0 0 1 c B −c12 d2 −c23 0 0 0 −c27 C 45 B C c B 0 −c23 d3 −c34 −c35 0 0 C 35 B C c27 L = B 0 0 −c34 d4 −c45 0 0 C B C u5 B 0 0 −c35 −c45 d5 −c56 0 C c56 @ 0 0 0 0 −c56 d6 −c67 A 0 −c27 0 0 0 −c67 d7 u7 c67 u6 MCW 2013, A. -
Adjacency and Incidence Matrices
Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. -
Package 'Popbio'
Package ‘popbio’ February 1, 2020 Version 2.7 Author Chris Stubben, Brook Milligan, Patrick Nantel Maintainer Chris Stubben <[email protected]> Title Construction and Analysis of Matrix Population Models License GPL-3 Encoding UTF-8 LazyData true Suggests quadprog Description Construct and analyze projection matrix models from a demography study of marked individu- als classified by age or stage. The package covers methods described in Matrix Population Mod- els by Caswell (2001) and Quantitative Conservation Biology by Morris and Doak (2002). RoxygenNote 6.1.1 NeedsCompilation no Repository CRAN Date/Publication 2020-02-01 20:10:02 UTC R topics documented: 01.Introduction . .3 02.Caswell . .3 03.Morris . .5 aq.census . .6 aq.matrix . .7 aq.trans . .8 betaval............................................9 boot.transitions . 11 calathea . 13 countCDFxt . 14 damping.ratio . 15 eigen.analysis . 16 elasticity . 18 1 2 R topics documented: extCDF . 19 fundamental.matrix . 20 generation.time . 21 grizzly . 22 head2 . 24 hudcorrs . 25 hudmxdef . 25 hudsonia . 26 hudvrs . 27 image2 . 28 Kendall . 29 lambda . 31 lnorms . 32 logi.hist.plot . 33 LTRE ............................................ 34 matplot2 . 37 matrix2 . 38 mean.list . 39 monkeyflower . 40 multiresultm . 42 nematode . 43 net.reproductive.rate . 44 pfister.plot . 45 pop.projection . 46 projection.matrix . 48 QPmat . 50 reproductive.value . 51 resample . 52 secder . 53 sensitivity . 55 splitA . 56 stable.stage . 57 stage.vector.plot . 58 stoch.growth.rate . 59 stoch.projection . 60 stoch.quasi.ext . 62 stoch.sens . 63 stretchbetaval . 64 teasel . 65 test.census . 66 tortoise . 67 var2 ............................................. 68 varEst . 69 vitalsens . 70 vitalsim . 71 whale . 74 woodpecker . 74 Index 76 01.Introduction 3 01.Introduction Introduction to the popbio Package Description Popbio is a package for the construction and analysis of matrix population models. -
Workshop on Modern Applied Mathematics PK 2012 Kraków 23 — 24 November 2012
Workshop on Modern Applied Mathematics PK 2012 Kraków 23 — 24 November 2012 Warsztaty z Nowoczesnej Matematyki i jej Zastosowań Kraków 23 — 24 listopada 2012 Kraków 23 — 24 November 2012 3 Contents 1 Introduction 5 1.1 Organizing Committee . 6 1.2 Scientific Committee . 6 2 List of Participants 7 3 Abstracts 11 3.1 On residually finite groups Orest D. Artemovych . 11 3.2 Some properties of Bernstein-Durrmeyer operators Magdalena Baczyńska . 12 3.3 Some Applications of Gröbner Bases Katarzyna Bolek . 14 3.4 Criteria for normality via C0-semigroups and moment sequences Dariusz Cichoń . 15 3.5 Picard–Fuchs operator for certain one parameter families of Calabi–Yau threefolds Sławomir Cynk . 16 3.6 Inference with Missing data Christiana Drake . 17 3.7 An outer measure on a commutative ring Dariusz Dudzik . 18 3.8 Resampling methods for weakly dependent sequences Elżbieta Gajecka-Mirek . 18 3.9 Graphs with path systems of length k in a hamiltonian cycle Grzegorz Gancarzewicz . 19 3.10 Hausdorff gaps and automorphisms of P(N)=fin Magdalena Grzech . 20 3.11 Functional Regression Models with Structured Penalties Jarosław Harezlak, Madan Kundu . 21 3.12 Approximation of function of two variables from exponential weight spaces Monika Herzog . 22 3.13 The existence of a weak solution of the semilinear first-order differential equation in a Banach space Mariusz Jużyniec . 25 3.14 Bayesian inference for cyclostationary time series Oskar Knapik . 26 4 Workshop on Modern Applied Mathematics PK 2012 3.15 Hausdorff limit in o-minimal structures Beata Kocel–Cynk . 27 3.16 Resampling methods for nonstationary time series Jacek Leśkow . -
Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A
1 Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE Abstract—We define and discuss the utility of two equiv- Consider a graph G = G(A) with adjacency matrix alence graph classes over which a spectral projector-based A 2 CN×N with k ≤ N distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition A = VJV −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of A are Jij, i = 1; : : : k, j = 1; : : : ; gi, up to a permutation on the node labels. Jordan equivalence where gi is the geometric multiplicity of eigenvalue 휆i, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of A − 휆iI. The signal tical topologies and allow a basis-invariant characterization space S can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal s 2 S, the graph Fourier transform (GFT) of [12] is defined as Index Terms—Jordan decomposition, generalized k gi eigenspaces, directed graphs, graph equivalence classes, M M graph isomorphism, signal processing on graphs, networks F : S! Jij i=1 j=1 s ! (s ;:::; s ;:::; s ;:::; s ) ; (1) b11 b1g1 bk1 bkgk I. INTRODUCTION where sij is the (oblique) projection of s onto the Jordan subspace Jij parallel to SnJij. -
Similarity-Based Clustering by Left-Stochastic Matrix Factorization
JournalofMachineLearningResearch14(2013)1715-1746 Submitted 1/12; Revised 11/12; Published 7/13 Similarity-based Clustering by Left-Stochastic Matrix Factorization Raman Arora [email protected] Toyota Technological Institute 6045 S. Kenwood Ave Chicago, IL 60637, USA Maya R. Gupta [email protected] Google 1225 Charleston Rd Mountain View, CA 94301, USA Amol Kapila [email protected] Maryam Fazel [email protected] Department of Electrical Engineering University of Washington Seattle, WA 98195, USA Editor: Inderjit Dhillon Abstract For similarity-based clustering, we propose modeling the entries of a given similarity matrix as the inner products of the unknown cluster probabilities. To estimate the cluster probabilities from the given similarity matrix, we introduce a left-stochastic non-negative matrix factorization problem. A rotation-based algorithm is proposed for the matrix factorization. Conditions for unique matrix factorizations and clusterings are given, and an error bound is provided. The algorithm is partic- ularly efficient for the case of two clusters, which motivates a hierarchical variant for cases where the number of desired clusters is large. Experiments show that the proposed left-stochastic decom- position clustering model produces relatively high within-cluster similarity on most data sets and can match given class labels, and that the efficient hierarchical variant performs surprisingly well. Keywords: clustering, non-negative matrix factorization, rotation, indefinite kernel, similarity, completely positive 1. Introduction Clustering is important in a broad range of applications, from segmenting customers for more ef- fective advertising, to building codebooks for data compression. Many clustering methods can be interpreted in terms of a matrix factorization problem. -
Network Properties Revealed Through Matrix Functions 697
SIAM REVIEW c 2010 Society for Industrial and Applied Mathematics Vol. 52, No. 4, pp. 696–714 Network Properties Revealed ∗ through Matrix Functions † Ernesto Estrada Desmond J. Higham‡ Abstract. The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, com- municability,andbetweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network’s adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks. Key words. centrality measures, clustering methods, communicability, Estrada index, Fiedler vector, graph Laplacian, graph spectrum, power series, resolvent AMS subject classifications. 05C50, 05C82, 91D30 DOI. 10.1137/090761070 1. Motivation. 1.1. Introduction. Connections are important. Across the natural, technologi- cal, and social sciences it often makes sense to focus on the pattern of interactions be- tween individual components in a system [1, 8, 61]. -
Projects and Publications of the Applied Mathematics Division
NATIONAL BUREAU OF STANDARDS REPORT 4546 PROJECTS and PUBLICATIONS of the APPLIED MATHEMATICS DIVISION A Quarterly Report October through December 1 955 U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS FOR OFFICIAL DISTRIBUTION U. S. DEPARTMENT OF COMMERCE Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS The 6cope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant reports and publications, appears on the inside of the back cover of this report. Electricity and Electronics. Resistance and Reactance. Electron Tubes. Electrical Instru- ments. Magnetic Measurements. Process Technology. Engineering Electronics. Electronic Instrumentation. Electrochemistry. Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic Technology. Length. Engineering Metrology. Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines and Lubrication. Engine Fuels. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Nuclear Physics. Radioactivity. X-rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. AEC Radiation Instruments. Chemistry. Organic Coatings. Surface Chemistry. Organic -
Variants of the Graph Laplacian with Applications in Machine Learning
Variants of the Graph Laplacian with Applications in Machine Learning Sven Kurras Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften (Dr. rer. nat.) im Fachbereich Informatik der Fakult¨at f¨urMathematik, Informatik und Naturwissenschaften der Universit¨atHamburg Hamburg, Oktober 2016 Diese Promotion wurde gef¨ordertdurch die Deutsche Forschungsgemeinschaft, Forschergruppe 1735 \Structural Inference in Statistics: Adaptation and Efficiency”. Betreuung der Promotion durch: Prof. Dr. Ulrike von Luxburg Tag der Disputation: 22. M¨arz2017 Vorsitzender des Pr¨ufungsausschusses: Prof. Dr. Matthias Rarey 1. Gutachterin: Prof. Dr. Ulrike von Luxburg 2. Gutachter: Prof. Dr. Wolfgang Menzel Zusammenfassung In s¨amtlichen Lebensbereichen finden sich Graphen. Zum Beispiel verbringen Menschen viel Zeit mit der Kantentraversierung des Internet-Graphen. Weitere Beispiele f¨urGraphen sind soziale Netzwerke, ¨offentlicher Nahverkehr, Molek¨ule, Finanztransaktionen, Fischernetze, Familienstammb¨aume,sowie der Graph, in dem alle Paare nat¨urlicher Zahlen gleicher Quersumme durch eine Kante verbunden sind. Graphen k¨onnendurch ihre Adjazenzmatrix W repr¨asentiert werden. Dar¨uber hinaus existiert eine Vielzahl alternativer Graphmatrizen. Viele strukturelle Eigenschaften von Graphen, beispielsweise ihre Kreisfreiheit, Anzahl Spannb¨aume,oder Random Walk Hitting Times, spiegeln sich auf die ein oder andere Weise in algebraischen Eigenschaften ihrer Graphmatrizen wider. Diese grundlegende Verflechtung erlaubt das Studium von Graphen unter Verwendung s¨amtlicher Resultate der Linearen Algebra, angewandt auf Graphmatrizen. Spektrale Graphentheorie studiert Graphen insbesondere anhand der Eigenwerte und Eigenvektoren ihrer Graphmatrizen. Dabei ist vor allem die Laplace-Matrix L = D − W von Bedeutung, aber es gibt derer viele Varianten, zum Beispiel die normalisierte Laplacian, die vorzeichenlose Laplacian und die Diplacian. Die meisten Varianten basieren auf einer \syntaktisch kleinen" Anderung¨ von L, etwa D +W anstelle von D −W . -
Stabilization, Estimation and Control of Linear Dynamical Systems with Positivity and Symmetry Constraints
Stabilization, Estimation and Control of Linear Dynamical Systems with Positivity and Symmetry Constraints A Dissertation Presented by Amirreza Oghbaee to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Northeastern University Boston, Massachusetts April 2018 To my parents for their endless love and support i Contents List of Figures vi Acknowledgments vii Abstract of the Dissertation viii 1 Introduction 1 2 Matrices with Special Structures 4 2.1 Nonnegative (Positive) and Metzler Matrices . 4 2.1.1 Nonnegative Matrices and Eigenvalue Characterization . 6 2.1.2 Metzler Matrices . 8 2.1.3 Z-Matrices . 10 2.1.4 M-Matrices . 10 2.1.5 Totally Nonnegative (Positive) Matrices and Strictly Metzler Matrices . 12 2.2 Symmetric Matrices . 14 2.2.1 Properties of Symmetric Matrices . 14 2.2.2 Symmetrizer and Symmetrization . 15 2.2.3 Quadratic Form and Eigenvalues Characterization of Symmetric Matrices . 19 2.3 Nonnegative and Metzler Symmetric Matrices . 22 3 Positive and Symmetric Systems 27 3.1 Positive Systems . 27 3.1.1 Externally Positive Systems . 27 3.1.2 Internally Positive Systems . 29 3.1.3 Asymptotic Stability . 33 3.1.4 Bounded-Input Bounded-Output (BIBO) Stability . 34 3.1.5 Asymptotic Stability using Lyapunov Equation . 37 3.1.6 Robust Stability of Perturbed Systems . 38 3.1.7 Stability Radius . 40 3.2 Symmetric Systems . 43 3.3 Positive Symmetric Systems . 47 ii 4 Positive Stabilization of Dynamic Systems 50 4.1 Metzlerian Stabilization . 50 4.2 Maximizing the stability radius by state feedback .