Francesco Sorrentino Department of Mechanical Engineering

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Francesco Sorrentino Department of Mechanical Engineering"

Transcription

1 Master stability function approaches to analyze stability of the synchronous evolution for hypernetworks and of synchronized clusters for networks with symmetries Francesco Sorrentino Department of Mechanical Engineering 203 IEEE International Workshop on Complex Systems and Networks, Vancouver, December the 2 th 203

2 Synchronization of Networks x ( t) F( x ( t)) x ( t) F( x2( 2 t )) x i ( t) F( x ( t)) i x N ( t) F( x ( t)) N The systems are characterized by the same What is the effect of the network structure on the dynamics (possibly chaotic), when uncoupled. dynamics? How do faults affect the network dynamics?

3 Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation:

4 Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: x i ( t) F( x ( t)) i Individual dynamics (chaotic)

5 Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: x i ( t) F( x ( t)) i N j A ij t Hx t H x j i Coupling term The matrix A{ A ij } is the network (weighted) adjacency matrix. The parameter measures the strengths of the network connections. The function H is an output function at each node. In principle H can be any function.

6 Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: is the Laplacian matrix N j j ij i i t x H L t x F t x )) ( ( ) ( { ij } L L N N N N N d A A A d A A A d L N j d i A ij

7 Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: is the Laplacian matrix N j j ij i i t x H L t x F t x )) ( ( ) ( { ij } L L L L0 N N 2 0 The network equations admit a synchronous solution: t x t x t x t x s N 2. t F x t x s s where

8 Master Stability Function Reduced Linearized equation: [L. M. Pecora, T. L. Carroll, 98] s i s The Master Stability Function (MSF) approach studies transversal stability of the synchronous trajectory The sync solution is linearly stable iff the MSF is negative. Depending on the individual nodes dynamics, we can have three different kinds of MSF. Maximum Transverse Lyapunov exponent Λ(ν) DF( x ( t)) DH( x ( t)) x, x σλ 2,...,σλ N i

9 Parallel and transversal perturbations By construction there is always one eigenvalue 0 This eigenvalue is associated with the eigenvector [,,,], that is parallel to the synchronization manifold In order to evaluate stability, we are only concerned with transversal perturbations

10 Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES )Pinning control of networks. F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, Phys. Rev.E, 75, 4, (2007).

11 Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 2) Network Synchronization of Groups F. Sorrentino and Edward Ott, Network Synchronization of Groups, Phys. Rev. E, 76, 0564 (2007).

12 Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 3) Adaptive synchronization of a time-evolving dynamical networks/ sensor network application F. Sorrentino, G. Barlev, A. B. Cohen, and E. Ott, Chaos, 20, 0303 (200) A. Cohen, B. Ravoori, F. Sorrentino, T. Murphy, E. Ott, and R. Roy, Chaos, 20, (200).

13 Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 4) Networks with couplings of different types F. Sorrentino, Synchronization of hypernetworks of coupled dynamical systems, New Journal of Physics, 4, (202) D. Irving and F. Sorrentino, Phys. Rev. E, Phys. Rev. E, 86, (202).

14 Motivation: chemical synapses and electrical gap junctions How does synchronization arises in this networks? Is a low dimensional approach still possible?

15 Mathematical formulation Two different coupling mechanisms: Stability?

16 Stability analysis Problem: In general, it is not possible to simultaneously diagonalize two matrices Question: Are there conditions under which these equations can be reduced in this form? The answer is yes in three cases.

17 Case I The matrices L A and L B commute Question: What are the graph properties for the Laplacians to commute?

18 Case II One of the two networks is fully connected and unweighted:

19 Case III One of the two networks is such that all the links originating from the same node have equal weights:

20 SBD approach What if none of these conditions (I-III) is satisfied? Danny Irving and I proposed an alternative approach based on simultaneous block-diagonalization (SBD) of matrices PROBLEM: Given the set of N-square real matrices = {L (),L (2),...,L (M) }, find the finest simultaneous block diagonalization (SBD) of. APPROACH: Find an invertible matrix P, such that P - L (i) P= j=,..n B j i and the dimensions of the blocks are minimal.

21 Method by Maehara & Murota (i) Let O (i) be the N 2 -matrix O (i) = I N L(i) L(i) I N. (ii) Construct the matrix S = Σ i O (i)t O (i) (iii) Let y be any N 2 -vector in the null subspace of the matrix S. Let y = [u T,u 2T,...,u NT ] T. (iv) Construct the matrix U=[u,u 2,,u N ]. (v) Output the matrix P whose columns are the eigenvectors of U. This procedure provides the finest simultaneous block diagonalization it provides the maximum reduction of the sync stability problem

22 Example I

23 Example II

24 Example III: undirected unweighted 3-motifs

25 Cluster synchronization in networks with symmetries L. Pecora, F. Sorrentino, A. Hagerstrom, T. Murphy, R. Roy Consider the following general equations: Where now the matrix A is completely arbitrary, e.g., non constant-row-sum QUESTION: Can synchronization be achieved? ANSWER: These equations are compatible with cluster synchronization: M synchronized motions {s,, s M }, where the clusters are determined by the network structure

26 Symmetries and Clusters Three -node random networks with the same number of edges These three networks are characterized by different numbers of symmetries (automorphisms) Equivalence relation partition into maximal disjoint sets of nodes

27 Sync solutions Graph symmetries Dynamical solutions: nodes that belong to the same cluster can synchronize, nodes that belong to different clusters cannot. The symmetries of the network form a group G. Each symmetry of the group can be described by a permutation matrix R g that re-orders the nodes in a way that leaves the dynamical equations unchanged (i.e., each R g commutes with A). Computation of the graph automorphisms enables us to find the possible cluster sync solutions corresponding to a given network structure - Are these solutions stable?

28 Sync solutions - stability Group theory provides a powerful way to transform the variational equations to a new coordinate system (the irreducible representation) in which the transformed coupling matrix B = TAT - has a block diagonal form that matches the cluster structure.

29 Experimental setup The dynamical oscillators that form the network are realized as square patches of pixels selected from a 32 x 32 tiling of the SLM array.

30 Stability results and Isolated Desynchronization

31 Application to power networks Geographical diagram of the Nepal Power Grid Network

32 Conclusions We are pushing the master stability function approach [Pecora and Carroll] to its limits In the case of networks with multiple type of connections, the stability problem can be reduced in a low-dimensional form by using SBD This guarantees that the problem is reduced in the lowest possible dimensional form In the case of arbitrary adjacency matrices, clusters may arise that correspond to a partition of the nodes based on the network symmetries Stability of each cluster can be studied independently from other clusters unless clusters are intertwined As a consequence we can experience isolated desynchronization of the cluster solutions Possible applications to networks of neurons and power grids

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II. Matrix Algorithms DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

More information

NETZCOPE - a tool to analyze and display complex R&D collaboration networks

NETZCOPE - a tool to analyze and display complex R&D collaboration networks The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots NETZCOPE - a tool to analyze and display complex R&D collaboration networks L. Streit & O. Strogan BiBoS, Univ.

More information

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated

SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET. Action Taken (Please Check One) New Course Initiated SALEM COMMUNITY COLLEGE Carneys Point, New Jersey 08069 COURSE SYLLABUS COVER SHEET Course Title Course Number Department Linear Algebra Mathematics MAT-240 Action Taken (Please Check One) New Course Initiated

More information

Rank one SVD: un algorithm pour la visualisation d une matrice non négative

Rank one SVD: un algorithm pour la visualisation d une matrice non négative Rank one SVD: un algorithm pour la visualisation d une matrice non négative L. Labiod and M. Nadif LIPADE - Universite ParisDescartes, France ECAIS 2013 November 7, 2013 Outline Outline 1 Data visualization

More information

Synchronization in Electrical Power Network. Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi

Synchronization in Electrical Power Network. Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi Synchronization in Electrical Power Network Oren Lee, Thomas Taylor, Tianye Chi, Austin Gubler, Maha Alsairafi Contents Introduction 1 Synchronization 1 Enhancement of synchronization stability 3 Future

More information

Chapter 7. Lyapunov Exponents. 7.1 Maps

Chapter 7. Lyapunov Exponents. 7.1 Maps Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

More information

Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12

Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12 Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer

More information

Part 2: Community Detection

Part 2: Community Detection Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection - Social networks -

More information

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1 (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

some algebra prelim solutions

some algebra prelim solutions some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

More information

Social Media Mining. Network Measures

Social Media Mining. Network Measures Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

More information

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This

More information

Classification of Cartan matrices

Classification of Cartan matrices Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

More information

Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010 Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

More information

STUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE

STUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson Addison-Wesley. All rights reserved. Reproduced by Pearson Addison-Wesley

More information

Component Ordering in Independent Component Analysis Based on Data Power

Component Ordering in Independent Component Analysis Based on Data Power Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Chapter 6. Orthogonality

Chapter 6. Orthogonality 6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

More information

University of Ottawa

University of Ottawa University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)

More information

MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

More information

CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

More information

1 Eigenvalues and Eigenvectors

1 Eigenvalues and Eigenvectors Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

More information

Subspace Analysis and Optimization for AAM Based Face Alignment

Subspace Analysis and Optimization for AAM Based Face Alignment Subspace Analysis and Optimization for AAM Based Face Alignment Ming Zhao Chun Chen College of Computer Science Zhejiang University Hangzhou, 310027, P.R.China zhaoming1999@zju.edu.cn Stan Z. Li Microsoft

More information

Linear Algebra and TI 89

Linear Algebra and TI 89 Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing

More information

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010 Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

More information

C 1 x(t) = e ta C = e C n. 2! A2 + t3

C 1 x(t) = e ta C = e C n. 2! A2 + t3 Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

More information

Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012 CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

More information

Walk-Based Centrality and Communicability Measures for Network Analysis

Walk-Based Centrality and Communicability Measures for Network Analysis Walk-Based Centrality and Communicability Measures for Network Analysis Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, Georgia, USA Workshop on Innovative Clustering

More information

Similar matrices and Jordan form

Similar matrices and Jordan form Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

More information

Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

Comparison of Non-linear Dimensionality Reduction Techniques for Classification with Gene Expression Microarray Data

Comparison of Non-linear Dimensionality Reduction Techniques for Classification with Gene Expression Microarray Data CMPE 59H Comparison of Non-linear Dimensionality Reduction Techniques for Classification with Gene Expression Microarray Data Term Project Report Fatma Güney, Kübra Kalkan 1/15/2013 Keywords: Non-linear

More information

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

More information

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V. .4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Nonlinear Iterative Partial Least Squares Method

Nonlinear Iterative Partial Least Squares Method Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

More information

4.1 VECTOR SPACES AND SUBSPACES

4.1 VECTOR SPACES AND SUBSPACES 4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Presentation 3: Eigenvalues and Eigenvectors of a Matrix Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

More information

SECTIONS 1.5-1.6 NOTES ON GRAPH THEORY NOTATION AND ITS USE IN THE STUDY OF SPARSE SYMMETRIC MATRICES

SECTIONS 1.5-1.6 NOTES ON GRAPH THEORY NOTATION AND ITS USE IN THE STUDY OF SPARSE SYMMETRIC MATRICES SECIONS.5-.6 NOES ON GRPH HEORY NOION ND IS USE IN HE SUDY OF SPRSE SYMMERIC MRICES graph G ( X, E) consists of a finite set of nodes or vertices X and edges E. EXMPLE : road map of part of British Columbia

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

5.04 Principles of Inorganic Chemistry II

5.04 Principles of Inorganic Chemistry II MIT OpenourseWare http://ocw.mit.edu 5.4 Principles of Inorganic hemistry II Fall 8 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.4, Principles of

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

Principal Component Analysis Application to images

Principal Component Analysis Application to images Principal Component Analysis Application to images Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/

More information

1 Spherical Kinematics

1 Spherical Kinematics ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

More information

Analysis of Internet Topologies: A Historical View

Analysis of Internet Topologies: A Historical View Analysis of Internet Topologies: A Historical View Mohamadreza Najiminaini, Laxmi Subedi, and Ljiljana Trajković Communication Networks Laboratory http://www.ensc.sfu.ca/cnl Simon Fraser University Vancouver,

More information

Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

More information

COMMUNICATION AND SYNCHRONIZATION IN DISCONNECTED NETWORKS WITH DYNAMIC TOPOLOGY: MOVING NEIGHBORHOOD NETWORKS. Joseph D. Skufca. Erik M.

COMMUNICATION AND SYNCHRONIZATION IN DISCONNECTED NETWORKS WITH DYNAMIC TOPOLOGY: MOVING NEIGHBORHOOD NETWORKS. Joseph D. Skufca. Erik M. MATHEMATICAL BIOSCIENCES AND ENGINEERING Volume 1, Number 2, July 2004 http://math.asu.edu/ mbe/ pp. COMMUNICATION AND SYNCHRONIZATION IN DISCONNECTED NETWORKS WITH DYNAMIC TOPOLOGY: MOVING NEIGHBORHOOD

More information

NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES

NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES NEW VERSION OF DECISION SUPPORT SYSTEM FOR EVALUATING TAKEOVER BIDS IN PRIVATIZATION OF THE PUBLIC ENTERPRISES AND SERVICES Silvija Vlah Kristina Soric Visnja Vojvodic Rosenzweig Department of Mathematics

More information

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS. Mikhail Tsitsvero and Sergio Barbarossa

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS. Mikhail Tsitsvero and Sergio Barbarossa ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS Mikhail Tsitsvero and Sergio Barbarossa Sapienza Univ. of Rome, DIET Dept., Via Eudossiana 18, 00184 Rome, Italy E-mail: tsitsvero@gmail.com, sergio.barbarossa@uniroma1.it

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc. 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

More information

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, Chapter 1: Linear Equations and Matrices MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

More information

7 Communication Classes

7 Communication Classes this version: 26 February 2009 7 Communication Classes Perhaps surprisingly, we can learn much about the long-run behavior of a Markov chain merely from the zero pattern of its transition matrix. In the

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Relations Graphical View

Relations Graphical View Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

More information

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional

More information

Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues

Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues By Peter Woolf pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS

USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

Why graph clustering is useful?

Why graph clustering is useful? Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management

More information

Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation.

Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. Hello, my name is Olga Michasova and I present the work The generalized model of economic growth with human capital accumulation. 1 Without any doubts human capital is a key factor of economic growth because

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

1.5 Elementary Matrices and a Method for Finding the Inverse

1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

More information

A Simple Feature Extraction Technique of a Pattern By Hopfield Network

A Simple Feature Extraction Technique of a Pattern By Hopfield Network A Simple Feature Extraction Technique of a Pattern By Hopfield Network A.Nag!, S. Biswas *, D. Sarkar *, P.P. Sarkar *, B. Gupta **! Academy of Technology, Hoogly - 722 *USIC, University of Kalyani, Kalyani

More information

System Identification for Acoustic Comms.:

System Identification for Acoustic Comms.: System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation

More information

Analysis of Internet Topologies

Analysis of Internet Topologies Analysis of Internet Topologies Ljiljana Trajković ljilja@cs.sfu.ca Communication Networks Laboratory http://www.ensc.sfu.ca/cnl School of Engineering Science Simon Fraser University, Vancouver, British

More information

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

More information

Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

More information

Lecture 5: Singular Value Decomposition SVD (1)

Lecture 5: Singular Value Decomposition SVD (1) EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

More information

Math Practice Problems for Test 1

Math Practice Problems for Test 1 Math 290 - Practice Problems for Test 1 UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT. 3 4 5 1. Let c 1 and c 2 be the columns of A 5 2 and b 1. Show that b Span{c 1, c 2 } by 6 6 6 writing b as a linear

More information

Problems for Advanced Linear Algebra Fall 2012

Problems for Advanced Linear Algebra Fall 2012 Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are

More information

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S. ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

More information

Link-based Analysis on Large Graphs. Presented by Weiren Yu Mar 01, 2011

Link-based Analysis on Large Graphs. Presented by Weiren Yu Mar 01, 2011 Link-based Analysis on Large Graphs Presented by Weiren Yu Mar 01, 2011 Overview 1 Introduction 2 Problem Definition 3 Optimization Techniques 4 Experimental Results 2 1. Introduction Many applications

More information

In the following we will only consider undirected networks.

In the following we will only consider undirected networks. Roles in Networks Roles in Networks Motivation for work: Let topology define network roles. Work by Kleinberg on directed graphs, used topology to define two types of roles: authorities and hubs. (Each

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

Using the Theory of Reals in. Analyzing Continuous and Hybrid Systems

Using the Theory of Reals in. Analyzing Continuous and Hybrid Systems Using the Theory of Reals in Analyzing Continuous and Hybrid Systems Ashish Tiwari Computer Science Laboratory (CSL) SRI International (SRI) Menlo Park, CA 94025 Email: ashish.tiwari@sri.com Ashish Tiwari

More information

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks H. T. Kung Dario Vlah {htk, dario}@eecs.harvard.edu Harvard School of Engineering and Applied Sciences

More information

Computation of crystal growth. using sharp interface methods

Computation of crystal growth. using sharp interface methods Efficient computation of crystal growth using sharp interface methods University of Regensburg joint with John Barrett (London) Robert Nürnberg (London) July 2010 Outline 1 Curvature driven interface motion

More information

Mining Social-Network Graphs

Mining Social-Network Graphs 342 Chapter 10 Mining Social-Network Graphs There is much information to be gained by analyzing the large-scale data that is derived from social networks. The best-known example of a social network is

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear

More information

Applied Linear Algebra I Review page 1

Applied Linear Algebra I Review page 1 Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

APPM4720/5720: Fast algorithms for big data. Gunnar Martinsson The University of Colorado at Boulder

APPM4720/5720: Fast algorithms for big data. Gunnar Martinsson The University of Colorado at Boulder APPM4720/5720: Fast algorithms for big data Gunnar Martinsson The University of Colorado at Boulder Course objectives: The purpose of this course is to teach efficient algorithms for processing very large

More information

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form

A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

More information

Intrinsic Low-Dimensional Manifold Method for Rational Simplification of Chemical Kinetics

Intrinsic Low-Dimensional Manifold Method for Rational Simplification of Chemical Kinetics Intrinsic Low-Dimensional Manifold Method for Rational Simplification of Chemical Kinetics University of Notre Dame Department of Aerospace and Mechanical Engineering prepared by: Nicholas J. Glassmaker

More information

H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the

H = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the 1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information

More information

Facts About Eigenvalues

Facts About Eigenvalues Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

(January 14, 2009) End k (V ) End k (V/W )

(January 14, 2009) End k (V ) End k (V/W ) (January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

More information

Group Theory. 1 Cartan Subalgebra and the Roots. November 23, 2011. 1.1 Cartan Subalgebra. 1.2 Root system

Group Theory. 1 Cartan Subalgebra and the Roots. November 23, 2011. 1.1 Cartan Subalgebra. 1.2 Root system Group Theory November 23, 2011 1 Cartan Subalgebra and the Roots 1.1 Cartan Subalgebra Let G be the Lie algebra, if h G it is called a subalgebra of G. Now we seek a basis in which [x, T a ] = ζ a T a

More information

Applied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama

Applied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama Applied Finite Element Analysis M. E. Barkey Aerospace Engineering and Mechanics The University of Alabama M. E. Barkey Applied Finite Element Analysis 1 Course Objectives To introduce the graduate students

More information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information