Lecture 1 and 2: Introduction and Graph theory basics. Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012

Size: px
Start display at page:

Download "Lecture 1 and 2: Introduction and Graph theory basics. Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012"

Transcription

1 Lecture 1 and 2: Introduction and Graph theory basics Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012

2 Spring 2012: EE Networked estimation and control Schedule Monday and Wednesday, 4:30pm - 5:45pm, Halligan 108 Instructor Prof. Usman Khan 135 Halligan khan@ece.tufts.edu Office hours: M-W, 6:00pm 7:00pm (or by ) Web: Helpful references Modern Graph Theory, B. Bollobás, Springer, New York, NY, Algebraic Graph Theory, C. Godsil and G. F. Royle, Springer, New York, NY, Matrix Computations, G. H. Golub and C. F. Van Loan, Johns Hopkins Press, Mathematical Statistics, P. J. Bickel and K. A. Docksum, Prentice Hall, Statistical Signal Processing, L. Scahrf, Prentice Hall, A Course in Robust Control Theory: A Convex Approach, G. E. Dullerud and F. Paganini, Springer, Feedback Control of Dynamic Systems, G. F. Franklin, J. D. Powell and A. Emami-Naieni, Prentice Hall, Assignments Class presentations Based on a series of readings comprising of recent research papers related to the course Homework assignments Up to five problem sets distributed throughout the course Final project There will be a course project in place of the final exam. The project goals and nature of study has to be approved by the instructor. A team of up to 2 members is required. Deliverable: A 4-page research paper submitted by Apr. 30, Course Structure Class participation and presentations: 30% HW assignments: 30% Final project: 40% Active class participation and engagement in creative discussions is absolutely required.

3 EE Networked Estimation and Control Course Schedule Assigned Due Misc. Jan-23 (Mon) Lecture 01 No office Hours Jan-25 (Wed) Lecture 02 Jan-30 (Mon) Lecture 03 Feb-01 (Wed) Lecture 04 Feb-06 (Mon) Lecture 05 Feb-08 (Wed) Lecture 06 Feb-13 (Mon) Lecture 07 Feb-15 (Wed) Lecture 08 Feb-20 (Mon) Feb-22 (Wed) No classes (President's Day) Lecture 09 Feb-23 (Thu) Lecture 10 Substitute Lec Feb-27 (Mon) Lecture 11 Feb-29 (Wed) Lecture 12 Mar-05 (Mon) Lecture 13 Mar-07 (Wed) Lecture 14 Mar-12 (Mon) Lecture 15 Mar-14 (Wed) Lecture 16 Mar-19 (Mon) Mar-21 (Wed) Mar-26 (Mon) No classes (Spring Recess) No classes (Spring Recess) Lecture 17 Mar-28 (Wed) Lecture 18 Apr-02 (Mon) Lecture 19 Apr-04 (Wed) Lecture 20 Apr-09 (Mon) Lecture 21 Apr-11 (Wed) Lecture 22 Apr-16 (Mon) Apr-18 (Wed) No classes (Patriot's Day) Lecture 23 Apr-23 (Mon) Lecture 24 Apr-25 (Wed) Lecture 25 Apr-30 (Mon) Lecture 26 Prof. Usman Khan Spring 2012 ECE Dept., Tufts University

4 4 I. INTRODUCTION Estimation theory Control theory Distributed algorithms Dynamical system DT-LTI or DT-LTV x k+1 = A k x k + v k y k = C k x k + r k. Given the statistics of v k and r k, what is the optimal estimate, x k k, of x k, when we only observe y k? Dynamical system DT-LTI or DT-LTV x k+1 = A k x k + Bu k y k = C k x k + r k. Given the optimal estimate, x k k, of x k, what is the optimal control input (or actuation), u k = K k x k k, when we only observe y k? Suppose the observations (system measurements) are given as y i k = C i kx k + r i k, where i is the ith sensor. How do we modify optimal estimation (control) when yk i s are not available at one location but the sensors interact over a network? Agents (sensors) interacting over a network can be modeled as nodes interacting over a graph. Graph theory deals with: Formal models of such interactions. Information flow within such interactions. Example 1: Suppose we have N agents, i.e., i = 1,..., N. Each agent possesses a quantity denoted by y i. We are interested in computing the following functions of y i s, y = max i y i, y = min i y i, ỹ = 1 N N y i, when there is no central location where all y i s are available. Each agent i interacts with a subset of other agents over a network. A typical distributed algorithm is i=1 z i k+1 = j N i w ij z j k, zi 0 = y i, i, where N i is the set of neighbors of agent i. Where does the above converge (lim k z i k ) and under what conditions on w ij and agent connectivity? How fast is the convergence? How does the convergence speed change as a function of the agent connectivity? Graph theory provides a means to design and analyze such computations.

5 5 II. GRAPH THEORY A graph, G, is defined to be a collection of two sets: (i) a vertex-set, V = {1,..., N}, that is a collection of nodes (vertices); and an edge-set, E V V, that is a collection of edges. The edge-set, E, is defined as a set of ordered pairs (i, j) with i, j V such that j is connected to i to be interpreted as j can send information to i. Formally, and a graph is denoted ny G = (V, E). E = {(i, j) j i}, (1) A graph is said to be undirected if (i, j) E (j, i) E for all i and j. A graph that does not satisfy this property is called a directed graph or a digraph. Unless otherwise stated, we deal explicitly with undirected graphs in the following. The neighborhood of a node i is defined as N i = {j (i, j) E}. (2) Sometimes, the above is called the open neighborhood of a node, whereas the closed neighborhood is defined as N i {i}. The degree of a node i is defined as the number of nodes that can send information to node i, i.e., N i. A. Graph theory and Linear algebra Analysis of graphs is typically carried out via matrix theory. For this purpose, we define matrices that can define a graph (as opposed to the set notation earlier). The adjacency matrix, A = {a ij }, of a graph is defined as a ij = { 1, j i, 0, otw. Sometimes it is assumed that (i, i) E. With this assumption, the adjacency matrix has all 1 s on the main diagonal and is called extended adjacency. Remark 1: The adjacency matrix of an undirected graph is symmetric. The incidence matrix, C = c ij, of a graph is defined as an N M matrix (where M is the total number of edges) such that for the mth edge (i, j) E, the mth column of C has a 1 at the ith location, a 1 at the jth location, and zeros everywhere else. Note that if (i, j) is counted then (j, i) is not counted. We fix any one arbitrary orientation, i.e., either incoming or outgoing. (3)

6 6 The degree matrix, D, is defined as a diagonal matrix that has N i as the ith element on the main diagonal. The following definitions of a graph Laplacian, L = {l ij }, are equivalent: (i) L = D A. (4) N i, j = i, (ii) l ij = 1, i j, j i, (5) 0, otw. (iii) L = CC T. (6) Remark 2: The Laplacian, L, is symmetric and positive-semidefinite. Proof: Obvious from definition (iii). The eigenvalues of L are denoted by λ 1, λ 2,..., λ N ; the following conventional is typically employed, 0 = λ 1 λ 2... λ N. Remark 3: The Laplacian, L, is singular (rank-deficient), i.e., it has at least one 0 eigenvalue. Proof: Row-sum is 0. A path between node i 1 V and node i K+1 V of length K is defined as a sequence of edges (i 1, i 2 ), (i 2, i 3 ),... (i K, i K+1 ) in E for any distinct i 2,..., i K. An undirected graph is said to be connected if there exists a path from each i V to each j V. A graph is said to be fully-connected or an all-to-all networkif (i, j) E, for all i and j. If a graph is not connected then it can be partitioned into connected components.

7 7 Lec 2: Wednesday, Jan. 25, 2012 A diagonally-dominant matrix, A, is such that a ii j i a ij, i. A strictly diagonallydominant, A, is such that a ii > j i a ij, i. Lemma 1 (Gershgorin circle theorem): Let A = {a ij } C N N. Let D i be the closed disc centered at a ii with radius j i a ij. Then every eigenvalue of A lies in i D i. Corollary 1: A symmetric diagonally-dominant matrix with non-negative diagonals is PSD. Proof: Follows from Gershgorin circle theorem. Corollary 2: A Laplacian matrix is PSD. Proof: Laplacian matrices are symmetric diagonally-dominant with non-negative elements on the main diagonal. Lemma 2: Let G be connected and let λ 1 λ 2... λ N be the Laplacian eigenvalues. Then λ 2 > 0. Proof: Let u = [u 1, u 2,..., u N ] T be an eigenvector of L with eigenvalue 0. Since Lu = 0 and u T Lu = u T CC T u, we have C T u = 0. Now C T u u i u j = 0, (i, j) E. (7) This implies that u i = u j for all (i, j) E. As the graph is connected, we have u i = u j for all i, j V and the only normalized eigenvector that satisfies Lu = 0 is u = 1 [1, 1,..., 1] T. (8) N }{{} N elements Hence, the there is only one 0 eigenvalue and λ 2 > 0 since L is PSD. Lemma 3: The number of connected components equals to the multiplicity of 0 eigenvalues in its Laplacian. Proof: A disconnected graph is a union of some number of connected components. Each of such component is a connected graph on its own and has exactly one 0 eigenvalue. Example 2: Consider a network with N nodes and no edges. There are N connected components (each node). From the above lemma, the Laplacian should have N 0 eigenvalues. Can be verified as the Laplacian in this case is a 0 matrix. An irreducible matrix is such that it cannot be transformed into a block-diagonal matrix with any row-column permutation. Remark 4: A matrix is irreducible if and only if its associated graph is strongly-connected. A symmetric matrix is irreducible if and only if its associated graph is connected.

8 8 A primitive matrix is such that it is non-negative, square, and its pth power (p > 0) has all positive elements. Remark 5: A primitive matrix is irreducible. Proof: Can be proved by contradiction. The following statements can be proved. (i) A graph is connected if and only if its Laplacian is irreducible. (ii) For a fully-connected graph, λ 2 =,..., λ N = N. The algebraic connectivity of the graph is defined as the second-smallest eigenvalue of its Laplacian, i.e., λ 2. For connected graphs, this measures the strength of connectivity. Remark 6: In a connected graph, adding an edge does not decrease λ 2. B. Types of graphs A k-regular graph is such that each node is connected to exactly k other nodes. A nearest neighbor graph is such that each node is connected to all the nodes within a certain communication radius. An m-circulant graph is such that each node is connected to m forward and m backward neighbors. Remark 7: The adjacency and Laplacian matrices of a circulant graph are circulant matrices. The eigenvalues and eigenvectors of a circulant matrix are known in closed-form. The above graphs are referred to as structured graphs. Typically such graphs are highly clustered (how many of my neighbors are neighbors of each other) but have a large average shortest path. Can also be related to graph diameter (largest shortest path). A random graph with 1 p 0 is such that every two nodes, i and j, are connected with a probability p. Random graphs have smaller average shortest path but suffer from weak clustering. Example 3: The above is one of the Erdös-Renýi graph generating model. The other is to randomly pick (with uniform probability) one graph out of all possible graphs with N nodes and K edges. Consider the following graph generation: Take a structured graph and a positive number 0 p 1. For each edge in the graph, rewire it to a randomly chosen (uniform probability) node with the probability p. Watts-Strogatz model: When p is small and the starting graph is circulant, the resulting graph is shown to exhibit the small-world principle, i.e., small average shortest path and large clustering. Example 4: Transportation network, electric power grids, network of brain neurons, social networks, six degrees of separation. The sending experiment. The author collaboration

9 9 network, the famous Erdös number (mine is is 5 from three paths, may be 4-cannot prove), mean is 4.65: Khan, U. Moura, J. Püschel, M. Beth, T. Mullin, R. Erdös, P.)

Lecture 18 and 19. and a graph is denoted by G = (V, E).

Lecture 18 and 19. and a graph is denoted by G = (V, E). 1 Lecture 18 and 19 Spring 2013 - EE 194, Advanced Control (Prof. Khan) Mar. 27 (Wed.) and Apr. 01 (Mon.), 2013 I. GRAPH THEORY A graph, G, is defined to be a collection of two sets: (i) a vertex-set,

More information

Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian

Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =

More information

Data Mining and Analysis: Fundamental Concepts and Algorithms

Data Mining and Analysis: Fundamental Concepts and Algorithms : Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA 2 Department of Computer

More information

arxiv:quant-ph/ v1 22 Aug 2005

arxiv:quant-ph/ v1 22 Aug 2005 Conditions for separability in generalized Laplacian matrices and nonnegative matrices as density matrices arxiv:quant-ph/58163v1 22 Aug 25 Abstract Chai Wah Wu IBM Research Division, Thomas J. Watson

More information

Lecture 4: Introduction to Graph Theory and Consensus. Cooperative Control Applications

Lecture 4: Introduction to Graph Theory and Consensus. Cooperative Control Applications Lecture 4: Introduction to Graph Theory and Consensus Richard M. Murray Caltech Control and Dynamical Systems 16 March 2009 Goals Introduce some motivating cooperative control problems Describe basic concepts

More information

NCS Lecture 8 A Primer on Graph Theory. Cooperative Control Applications

NCS Lecture 8 A Primer on Graph Theory. Cooperative Control Applications NCS Lecture 8 A Primer on Graph Theory Richard M. Murray Control and Dynamical Systems California Institute of Technology Goals Introduce some motivating cooperative control problems Describe basic concepts

More information

Lecture 3: graph theory

Lecture 3: graph theory CONTENTS 1 BASIC NOTIONS Lecture 3: graph theory Sonia Martínez October 15, 2014 Abstract The notion of graph is at the core of cooperative control. Essentially, it allows us to model the interaction topology

More information

Multi-Robotic Systems

Multi-Robotic Systems CHAPTER 9 Multi-Robotic Systems The topic of multi-robotic systems is quite popular now. It is believed that such systems can have the following benefits: Improved performance ( winning by numbers ) Distributed

More information

Note on deleting a vertex and weak interlacing of the Laplacian spectrum

Note on deleting a vertex and weak interlacing of the Laplacian spectrum Electronic Journal of Linear Algebra Volume 16 Article 6 2007 Note on deleting a vertex and weak interlacing of the Laplacian spectrum Zvi Lotker zvilo@cse.bgu.ac.il Follow this and additional works at:

More information

Lecture 4 and 5 Controllability and Observability: Kalman decompositions

Lecture 4 and 5 Controllability and Observability: Kalman decompositions 1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS

More information

Graph fundamentals. Matrices associated with a graph

Graph fundamentals. Matrices associated with a graph Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n

More information

6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search

6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search 6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search Daron Acemoglu and Asu Ozdaglar MIT September 30, 2009 1 Networks: Lecture 7 Outline Navigation (or decentralized search)

More information

Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control

Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control Outline Background Preliminaries Consensus Numerical simulations Conclusions Average-Consensus of Multi-Agent Systems with Direct Topology Based on Event-Triggered Control Email: lzhx@nankai.edu.cn, chenzq@nankai.edu.cn

More information

A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo

A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A. E. Brouwer & W. H. Haemers 2008-02-28 Abstract We show that if µ j is the j-th largest Laplacian eigenvalue, and d

More information

AS 101: The Solar System (Spring 2017) Course Syllabus

AS 101: The Solar System (Spring 2017) Course Syllabus AS 101: The Solar System (Spring 2017) Course Syllabus Instructor: Professor Wen Li Office: CAS 501 Phone: 617-353-7439 Email: wenli77@bu.edu Office hours: Mondays 3:30 5:00 pm, Wednesdays 3:30 5:00 pm,

More information

COMPSCI 514: Algorithms for Data Science

COMPSCI 514: Algorithms for Data Science COMPSCI 514: Algorithms for Data Science Arya Mazumdar University of Massachusetts at Amherst Fall 2018 Lecture 8 Spectral Clustering Spectral clustering Curse of dimensionality Dimensionality Reduction

More information

Consensus, Flocking and Opinion Dynamics

Consensus, Flocking and Opinion Dynamics Consensus, Flocking and Opinion Dynamics Antoine Girard Laboratoire Jean Kuntzmann, Université de Grenoble antoine.girard@imag.fr International Summer School of Automatic Control GIPSA Lab, Grenoble, France,

More information

No class on Thursday, October 1. No office hours on Tuesday, September 29 and Thursday, October 1.

No class on Thursday, October 1. No office hours on Tuesday, September 29 and Thursday, October 1. Stationary Distributions Monday, September 28, 2015 2:02 PM No class on Thursday, October 1. No office hours on Tuesday, September 29 and Thursday, October 1. Homework 1 due Friday, October 2 at 5 PM strongly

More information

Aditya Bhaskara CS 5968/6968, Lecture 1: Introduction and Review 12 January 2016

Aditya Bhaskara CS 5968/6968, Lecture 1: Introduction and Review 12 January 2016 Lecture 1: Introduction and Review We begin with a short introduction to the course, and logistics. We then survey some basics about approximation algorithms and probability. We also introduce some of

More information

18.312: Algebraic Combinatorics Lionel Levine. Lecture 19

18.312: Algebraic Combinatorics Lionel Levine. Lecture 19 832: Algebraic Combinatorics Lionel Levine Lecture date: April 2, 20 Lecture 9 Notes by: David Witmer Matrix-Tree Theorem Undirected Graphs Let G = (V, E) be a connected, undirected graph with n vertices,

More information

Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies

Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED FOR PUBLICATION AS A TECHNICAL NOTE. 1 Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies Wei Ren, Student Member,

More information

High-dimensional probability and statistics for the data sciences

High-dimensional probability and statistics for the data sciences High-dimensional probability and statistics for the data sciences Larry Goldstein and Mahdi Soltanolkotabi Motivation August 21, 2017 Motivation August 21, 2017 Please ask questions! Traditionally mathematics

More information

Machine Learning for Data Science (CS4786) Lecture 11

Machine Learning for Data Science (CS4786) Lecture 11 Machine Learning for Data Science (CS4786) Lecture 11 Spectral clustering Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016sp/ ANNOUNCEMENT 1 Assignment P1 the Diagnostic assignment 1 will

More information

Consensus of Information Under Dynamically Changing Interaction Topologies

Consensus of Information Under Dynamically Changing Interaction Topologies Consensus of Information Under Dynamically Changing Interaction Topologies Wei Ren and Randal W. Beard Abstract This paper considers the problem of information consensus among multiple agents in the presence

More information

CSEP 521 Applied Algorithms. Richard Anderson Winter 2013 Lecture 1

CSEP 521 Applied Algorithms. Richard Anderson Winter 2013 Lecture 1 CSEP 521 Applied Algorithms Richard Anderson Winter 2013 Lecture 1 CSEP 521 Course Introduction CSEP 521, Applied Algorithms Monday s, 6:30-9:20 pm CSE 305 and Microsoft Building 99 Instructor Richard

More information

Lecture 2 and 3: Controllability of DT-LTI systems

Lecture 2 and 3: Controllability of DT-LTI systems 1 Lecture 2 and 3: Controllability of DT-LTI systems Spring 2013 - EE 194, Advanced Control (Prof Khan) January 23 (Wed) and 28 (Mon), 2013 I LTI SYSTEMS Recall that continuous-time LTI systems can be

More information

Clustering compiled by Alvin Wan from Professor Benjamin Recht s lecture, Samaneh s discussion

Clustering compiled by Alvin Wan from Professor Benjamin Recht s lecture, Samaneh s discussion Clustering compiled by Alvin Wan from Professor Benjamin Recht s lecture, Samaneh s discussion 1 Overview With clustering, we have several key motivations: archetypes (factor analysis) segmentation hierarchy

More information

Basic graph theory 18.S995 - L26.

Basic graph theory 18.S995 - L26. Basic graph theory 18.S995 - L26 dunkel@math.mit.edu http://java.dzone.com/articles/algorithm-week-graphs-and no cycles Isomorphic graphs f(a) = 1 f(b) = 6 f(c) = 8 f(d) = 3 f(g) = 5 f(h) = 2 f(i) = 4

More information

Algebraic Representation of Networks

Algebraic Representation of Networks Algebraic Representation of Networks 0 1 2 1 1 0 0 1 2 0 0 1 1 1 1 1 Hiroki Sayama sayama@binghamton.edu Describing networks with matrices (1) Adjacency matrix A matrix with rows and columns labeled by

More information

Decentralized Stabilization of Heterogeneous Linear Multi-Agent Systems

Decentralized Stabilization of Heterogeneous Linear Multi-Agent Systems 1 Decentralized Stabilization of Heterogeneous Linear Multi-Agent Systems Mauro Franceschelli, Andrea Gasparri, Alessandro Giua, and Giovanni Ulivi Abstract In this paper the formation stabilization problem

More information

Measurement partitioning and observational. equivalence in state estimation

Measurement partitioning and observational. equivalence in state estimation Measurement partitioning and observational 1 equivalence in state estimation Mohammadreza Doostmohammadian, Student Member, IEEE, and Usman A. Khan, Senior Member, IEEE arxiv:1412.5111v1 [cs.it] 16 Dec

More information

Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5

Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5 Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 5 Instructor: Farid Alizadeh Scribe: Anton Riabov 10/08/2001 1 Overview We continue studying the maximum eigenvalue SDP, and generalize

More information

IBM Research Report. Conditions for Separability in Generalized Laplacian Matrices and Diagonally Dominant Matrices as Density Matrices

IBM Research Report. Conditions for Separability in Generalized Laplacian Matrices and Diagonally Dominant Matrices as Density Matrices RC23758 (W58-118) October 18, 25 Physics IBM Research Report Conditions for Separability in Generalized Laplacian Matrices and Diagonally Dominant Matrices as Density Matrices Chai Wah Wu IBM Research

More information

Data Mining and Analysis: Fundamental Concepts and Algorithms

Data Mining and Analysis: Fundamental Concepts and Algorithms Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA

More information

ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact.

ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact. ORIE 6334 Spectral Graph Theory September 8, 2016 Lecture 6 Lecturer: David P. Williamson Scribe: Faisal Alkaabneh 1 The Matrix-Tree Theorem In this lecture, we continue to see the usefulness of the graph

More information

Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank

Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank David Glickenstein November 3, 4 Representing graphs as matrices It will sometimes be useful to represent graphs

More information

A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching 1

A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching 1 CHAPTER-3 A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching Graph matching problem has found many applications in areas as diverse as chemical structure analysis, pattern

More information

A Questionable Distance-Regular Graph

A Questionable Distance-Regular Graph A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection

More information

Math.3336: Discrete Mathematics. Chapter 9 Relations

Math.3336: Discrete Mathematics. Chapter 9 Relations Math.3336: Discrete Mathematics Chapter 9 Relations Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018

More information

Chapter 7 Network Flow Problems, I

Chapter 7 Network Flow Problems, I Chapter 7 Network Flow Problems, I Network flow problems are the most frequently solved linear programming problems. They include as special cases, the assignment, transportation, maximum flow, and shortest

More information

Fiedler s Theorems on Nodal Domains

Fiedler s Theorems on Nodal Domains Spectral Graph Theory Lecture 7 Fiedler s Theorems on Nodal Domains Daniel A. Spielman September 19, 2018 7.1 Overview In today s lecture we will justify some of the behavior we observed when using eigenvectors

More information

Lecture 13: Spectral Graph Theory

Lecture 13: Spectral Graph Theory CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 13: Spectral Graph Theory Lecturer: Shayan Oveis Gharan 11/14/18 Disclaimer: These notes have not been subjected to the usual scrutiny reserved

More information

Consensus Problems on Small World Graphs: A Structural Study

Consensus Problems on Small World Graphs: A Structural Study Consensus Problems on Small World Graphs: A Structural Study Pedram Hovareshti and John S. Baras 1 Department of Electrical and Computer Engineering and the Institute for Systems Research, University of

More information

Lecture 7 and 8. Fall EE 105, Feedback Control Systems (Prof. Khan) September 30 and October 05, 2015

Lecture 7 and 8. Fall EE 105, Feedback Control Systems (Prof. Khan) September 30 and October 05, 2015 1 Lecture 7 and 8 Fall 2015 - EE 105, Feedback Control Systems (Prof Khan) September 30 and October 05, 2015 I CONTROLLABILITY OF AN DT-LTI SYSTEM IN k TIME-STEPS The DT-LTI system is given by the following

More information

Spectral Graph Theory Lecture 2. The Laplacian. Daniel A. Spielman September 4, x T M x. ψ i = arg min

Spectral Graph Theory Lecture 2. The Laplacian. Daniel A. Spielman September 4, x T M x. ψ i = arg min Spectral Graph Theory Lecture 2 The Laplacian Daniel A. Spielman September 4, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class. The notes written before

More information

Lecture 8 : Eigenvalues and Eigenvectors

Lecture 8 : Eigenvalues and Eigenvectors CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with

More information

CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory

CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory Tim Roughgarden & Gregory Valiant May 2, 2016 Spectral graph theory is the powerful and beautiful theory that arises from

More information

Lecture 9: Laplacian Eigenmaps

Lecture 9: Laplacian Eigenmaps Lecture 9: Radu Balan Department of Mathematics, AMSC, CSCAMM and NWC University of Maryland, College Park, MD April 18, 2017 Optimization Criteria Assume G = (V, W ) is a undirected weighted graph with

More information

Fiedler s Theorems on Nodal Domains

Fiedler s Theorems on Nodal Domains Spectral Graph Theory Lecture 7 Fiedler s Theorems on Nodal Domains Daniel A Spielman September 9, 202 7 About these notes These notes are not necessarily an accurate representation of what happened in

More information

An Introduction to Spectral Graph Theory

An Introduction to Spectral Graph Theory An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca Outline Outline 1. How many walks are there from vertices v i to v j

More information

Algebraic Methods in Combinatorics

Algebraic Methods in Combinatorics Algebraic Methods in Combinatorics Po-Shen Loh 27 June 2008 1 Warm-up 1. (A result of Bourbaki on finite geometries, from Răzvan) Let X be a finite set, and let F be a family of distinct proper subsets

More information

Lecture: Modeling graphs with electrical networks

Lecture: Modeling graphs with electrical networks Stat260/CS294: Spectral Graph Methods Lecture 16-03/17/2015 Lecture: Modeling graphs with electrical networks Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these notes are still very rough.

More information

New feasibility conditions for directed strongly regular graphs

New feasibility conditions for directed strongly regular graphs New feasibility conditions for directed strongly regular graphs Sylvia A. Hobart Jason Williford Department of Mathematics University of Wyoming Laramie, Wyoming, U.S.A sahobart@uwyo.edu, jwillif1@uwyo.edu

More information

An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings

An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings Also available at http://amc-journal.eu ISSN 855-3966 (printed edn.), ISSN 855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 2 (207) 205 27 An algebraic proof of the Erdős-Ko-Rado theorem for intersecting

More information

Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies

Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED FOR PUBLICATION AS A TECHNICAL NOTE. Consensus Seeking in Multi-agent Systems Under Dynamically Changing Interaction Topologies Wei Ren, Student Member,

More information

Fast Linear Iterations for Distributed Averaging 1

Fast Linear Iterations for Distributed Averaging 1 Fast Linear Iterations for Distributed Averaging 1 Lin Xiao Stephen Boyd Information Systems Laboratory, Stanford University Stanford, CA 943-91 lxiao@stanford.edu, boyd@stanford.edu Abstract We consider

More information

Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality

Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality CSE 521: Design and Analysis of Algorithms I Spring 2016 Lecture 12: Introduction to Spectral Graph Theory, Cheeger s inequality Lecturer: Shayan Oveis Gharan May 4th Scribe: Gabriel Cadamuro Disclaimer:

More information

On the convergence of weighted-average consensus

On the convergence of weighted-average consensus On the convergence of weighted-average consensus Francisco Pedroche Miguel Rebollo Carlos Carrascosa Alberto Palomares arxiv:307.7562v [math.oc] 29 Jul 203 Abstract In this note we give sufficient conditions

More information

Eigenvectors Via Graph Theory

Eigenvectors Via Graph Theory Eigenvectors Via Graph Theory Jennifer Harris Advisor: Dr. David Garth October 3, 2009 Introduction There is no problem in all mathematics that cannot be solved by direct counting. -Ernst Mach The goal

More information

DS504/CS586: Big Data Analytics Graph Mining II

DS504/CS586: Big Data Analytics Graph Mining II Welcome to DS504/CS586: Big Data Analytics Graph Mining II Prof. Yanhua Li Time: 6:00pm 8:50pm Mon. and Wed. Location: SL105 Spring 2016 Reading assignments We will increase the bar a little bit Please

More information

1 Matrix notation and preliminaries from spectral graph theory

1 Matrix notation and preliminaries from spectral graph theory Graph clustering (or community detection or graph partitioning) is one of the most studied problems in network analysis. One reason for this is that there are a variety of ways to define a cluster or community.

More information

Data Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings

Data Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings Data Analysis and Manifold Learning Lecture 3: Graphs, Graph Matrices, and Graph Embeddings Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline

More information

This section is an introduction to the basic themes of the course.

This section is an introduction to the basic themes of the course. Chapter 1 Matrices and Graphs 1.1 The Adjacency Matrix This section is an introduction to the basic themes of the course. Definition 1.1.1. A simple undirected graph G = (V, E) consists of a non-empty

More information

EE595A Submodular functions, their optimization and applications Spring 2011

EE595A Submodular functions, their optimization and applications Spring 2011 EE595A Submodular functions, their optimization and applications Spring 2011 Prof. Jeff Bilmes University of Washington, Seattle Department of Electrical Engineering Winter Quarter, 2011 http://ee.washington.edu/class/235/2011wtr/index.html

More information

HONORS LINEAR ALGEBRA (MATH V 2020) SPRING 2013

HONORS LINEAR ALGEBRA (MATH V 2020) SPRING 2013 HONORS LINEAR ALGEBRA (MATH V 2020) SPRING 2013 PROFESSOR HENRY C. PINKHAM 1. Prerequisites The only prerequisite is Calculus III (Math 1201) or the equivalent: the first semester of multivariable calculus.

More information

Distributed intelligence in multi agent systems

Distributed intelligence in multi agent systems Distributed intelligence in multi agent systems Usman Khan Department of Electrical and Computer Engineering Tufts University Workshop on Distributed Optimization, Information Processing, and Learning

More information

Lecture 1 and 2: Random Spanning Trees

Lecture 1 and 2: Random Spanning Trees Recent Advances in Approximation Algorithms Spring 2015 Lecture 1 and 2: Random Spanning Trees Lecturer: Shayan Oveis Gharan March 31st Disclaimer: These notes have not been subjected to the usual scrutiny

More information

MATH 829: Introduction to Data Mining and Analysis Clustering II

MATH 829: Introduction to Data Mining and Analysis Clustering II his lecture is based on U. von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing, 17 (4), 2007. MATH 829: Introduction to Data Mining and Analysis Clustering II Dominique Guillot Departments

More information

17.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs

17.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs Chapter 17 Graphs and Graph Laplacians 17.1 Directed Graphs, Undirected Graphs, Incidence Matrices, Adjacency Matrices, Weighted Graphs Definition 17.1. A directed graph isa pairg =(V,E), where V = {v

More information

Topics in Graph Theory

Topics in Graph Theory Topics in Graph Theory September 4, 2018 1 Preliminaries A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End

More information

Diffusion and random walks on graphs

Diffusion and random walks on graphs Diffusion and random walks on graphs Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural

More information

Spectral Graph Theory Tools. Analysis of Complex Networks

Spectral Graph Theory Tools. Analysis of Complex Networks Spectral Graph Theory Tools for the Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory) Ernesto Estrada (Strathclyde, UK) Support:

More information

Applications to network analysis: Eigenvector centrality indices Lecture notes

Applications to network analysis: Eigenvector centrality indices Lecture notes Applications to network analysis: Eigenvector centrality indices Lecture notes Dario Fasino, University of Udine (Italy) Lecture notes for the second part of the course Nonnegative and spectral matrix

More information

Lecture 5: January 30

Lecture 5: January 30 CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 5: January 30 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They

More information

MULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY

MULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY Jrl Syst Sci & Complexity (2009) 22: 722 731 MULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY Yiguang HONG Xiaoli WANG Received: 11 May 2009 / Revised: 16 June 2009 c 2009

More information

Markov Chains, Random Walks on Graphs, and the Laplacian

Markov Chains, Random Walks on Graphs, and the Laplacian Markov Chains, Random Walks on Graphs, and the Laplacian CMPSCI 791BB: Advanced ML Sridhar Mahadevan Random Walks! There is significant interest in the problem of random walks! Markov chain analysis! Computer

More information

Course : Algebraic Combinatorics

Course : Algebraic Combinatorics Course 18.312: Algebraic Combinatorics Lecture Notes #29-31 Addendum by Gregg Musiker April 24th - 29th, 2009 The following material can be found in several sources including Sections 14.9 14.13 of Algebraic

More information

Distributed Coordinated Tracking With Reduced Interaction via a Variable Structure Approach Yongcan Cao, Member, IEEE, and Wei Ren, Member, IEEE

Distributed Coordinated Tracking With Reduced Interaction via a Variable Structure Approach Yongcan Cao, Member, IEEE, and Wei Ren, Member, IEEE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 1, JANUARY 2012 33 Distributed Coordinated Tracking With Reduced Interaction via a Variable Structure Approach Yongcan Cao, Member, IEEE, and Wei Ren,

More information

Applications of the Lopsided Lovász Local Lemma Regarding Hypergraphs

Applications of the Lopsided Lovász Local Lemma Regarding Hypergraphs Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided

More information

Spectral Theory of Unsigned and Signed Graphs Applications to Graph Clustering. Some Slides

Spectral Theory of Unsigned and Signed Graphs Applications to Graph Clustering. Some Slides Spectral Theory of Unsigned and Signed Graphs Applications to Graph Clustering Some Slides Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104,

More information

Gershgorin s Circle Theorem for Estimating the Eigenvalues of a Matrix with Known Error Bounds

Gershgorin s Circle Theorem for Estimating the Eigenvalues of a Matrix with Known Error Bounds Gershgorin s Circle Theorem for Estimating the Eigenvalues of a Matrix with Known Error Bounds Author: David Marquis Advisors: Professor Hans De Moor Dr. Kathryn Porter Reader: Dr. Michael Nathanson May

More information

Lecture 1: From Data to Graphs, Weighted Graphs and Graph Laplacian

Lecture 1: From Data to Graphs, Weighted Graphs and Graph Laplacian Lecture 1: From Data to Graphs, Weighted Graphs and Graph Laplacian Radu Balan February 5, 2018 Datasets diversity: Social Networks: Set of individuals ( agents, actors ) interacting with each other (e.g.,

More information

Lecture 4: An FPTAS for Knapsack, and K-Center

Lecture 4: An FPTAS for Knapsack, and K-Center Comp 260: Advanced Algorithms Tufts University, Spring 2016 Prof. Lenore Cowen Scribe: Eric Bailey Lecture 4: An FPTAS for Knapsack, and K-Center 1 Introduction Definition 1.0.1. The Knapsack problem (restated)

More information

Chap 3. Linear Algebra

Chap 3. Linear Algebra Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions

More information

On the distance signless Laplacian spectral radius of graphs and digraphs

On the distance signless Laplacian spectral radius of graphs and digraphs Electronic Journal of Linear Algebra Volume 3 Volume 3 (017) Article 3 017 On the distance signless Laplacian spectral radius of graphs and digraphs Dan Li Xinjiang University,Urumqi, ldxjedu@163.com Guoping

More information

MAT188H1S LINEAR ALGEBRA: Course Information as of February 2, Calendar Description:

MAT188H1S LINEAR ALGEBRA: Course Information as of February 2, Calendar Description: MAT188H1S LINEAR ALGEBRA: Course Information as of February 2, 2019 2018-2019 Calendar Description: This course covers systems of linear equations and Gaussian elimination, applications; vectors in R n,

More information

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic 1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What

More information

DS504/CS586: Big Data Analytics Graph Mining II

DS504/CS586: Big Data Analytics Graph Mining II Welcome to DS504/CS586: Big Data Analytics Graph Mining II Prof. Yanhua Li Time: 6-8:50PM Thursday Location: AK233 Spring 2018 v Course Project I has been graded. Grading was based on v 1. Project report

More information

Math Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012.

Math Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012. Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems

More information

Numerical Schemes from the Perspective of Consensus

Numerical Schemes from the Perspective of Consensus Numerical Schemes from the Perspective of Consensus Exploring Connections between Agreement Problems and PDEs Department of Electrical Engineering and Computer Sciences University of California, Berkeley

More information

Deterministic Decentralized Search in Random Graphs

Deterministic Decentralized Search in Random Graphs Deterministic Decentralized Search in Random Graphs Esteban Arcaute 1,, Ning Chen 2,, Ravi Kumar 3, David Liben-Nowell 4,, Mohammad Mahdian 3, Hamid Nazerzadeh 1,, and Ying Xu 1, 1 Stanford University.

More information

Learning from Sensor Data: Set II. Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University

Learning from Sensor Data: Set II. Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University Learning from Sensor Data: Set II Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University 1 6. Data Representation The approach for learning from data Probabilistic

More information

Modeling and Stability Analysis of a Communication Network System

Modeling and Stability Analysis of a Communication Network System Modeling and Stability Analysis of a Communication Network System Zvi Retchkiman Königsberg Instituto Politecnico Nacional e-mail: mzvi@cic.ipn.mx Abstract In this work, the modeling and stability problem

More information

Data Analysis and Manifold Learning Lecture 7: Spectral Clustering

Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Data Analysis and Manifold Learning Lecture 7: Spectral Clustering Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture 7 What is spectral

More information

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS Abstract. We present elementary proofs of the Cauchy-Binet Theorem on determinants and of the fact that the eigenvalues of a matrix

More information

Eigenvalues of 2-edge-coverings

Eigenvalues of 2-edge-coverings Eigenvalues of -edge-coverings Steve Butler October 3, 007 Abstract A -edge-covering between G and H is an onto homomorphism from the vertices of G to the vertices of H so that each edge is covered twice

More information

STA141C: Big Data & High Performance Statistical Computing

STA141C: Big Data & High Performance Statistical Computing STA141C: Big Data & High Performance Statistical Computing Lecture 12: Graph Clustering Cho-Jui Hsieh UC Davis May 29, 2018 Graph Clustering Given a graph G = (V, E, W ) V : nodes {v 1,, v n } E: edges

More information

Discrete Optimization 23

Discrete Optimization 23 Discrete Optimization 23 2 Total Unimodularity (TU) and Its Applications In this section we will discuss the total unimodularity theory and its applications to flows in networks. 2.1 Total Unimodularity:

More information

Nonnegative Matrices I

Nonnegative Matrices I Nonnegative Matrices I Daisuke Oyama Topics in Economic Theory September 26, 2017 References J. L. Stuart, Digraphs and Matrices, in Handbook of Linear Algebra, Chapter 29, 2006. R. A. Brualdi and H. J.

More information

Data Mining Techniques

Data Mining Techniques Data Mining Techniques CS 622 - Section 2 - Spring 27 Pre-final Review Jan-Willem van de Meent Feedback Feedback https://goo.gl/er7eo8 (also posted on Piazza) Also, please fill out your TRACE evaluations!

More information