You can find the schedule of lectures and assignments, here. In the early days, matrix theory In the early days, matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Graph Theory Notes. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Previously, he worked as Research Assistant at ISI foundation, Helsinki University, and Tongji University, as well as a Data Science Intern at Facebook, London. SPECTRAL GRAPH THEORY NICHOLAS PURPLE Abstract. Spectral graph theory at a glance The spectral graph theory studies the properties of graphs via the eigenvalues and eigenvectors of their associated graph matrices: the adjacency matrix, the graph Laplacian and their variants. There exists a whole ﬁeld ded-icated to the study of those matrices, called spectral graph theory (e.g., see Chung, 1997). 25 Pages. endobj Spectral graph clustering—clustering the vertices of a graph based on their spectral embedding—is of significant current interest, finding applications throughout the sciences. 2010/2011. Share. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability, and the concepts of coverings, coloring, and matching. (Expander Graphs) graph sparsification; Spectral sparsification of graphs: theory and algorithms, by Batson, Spielman, Srivastava, Teng. Frequently used graph matrices: A adjacency matrix D diagonal matrix of vertex degrees L … 4 A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY 1. Chung F., Spectral Graph Theory, American Mathematical So-ciety, Providence, Rhode Island, 1997. is devoted to the normalized Laplacian. A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way deﬂned for any graph. endobj Foundations. Then, nally, to basic results of the graph’s 12 0 obj Spectral clustering using the proposed sub-graph affinity model achieve similar f1-measures to spectral clustering results for existing nodal affinity model. 5 Tutorial on Spectral Clustering, ICML 2004, Chris Ding © University of California 9 Multi-way Graph Partitioning • Recursively applying the 2-way partitioning << /S /GoTo /D (subsection.4.7.2) >> endobj /Length 2509 endobj 64 0 obj Eigengap heuristic suggests the number of clusters k is usually given by the value of k that maximizes the eigengap (difference between consecutive eigenvalues). The U.S. Department of Energy's Office of Scientific and Technical Information endobj 19 0 obj Abstract: Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. Similar Books. (Eigenvalues of the Laplacian) endobj In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications. Wavelets on graphs via spectral graph theory, Applied and Computational Harmonic Analysis 30 (2011) no. Spectral graph drawing: FEM justification If apply finite element method to solve Laplace’s equation in the plane with a Delaunay triangulation Would get graph Laplacian, but with some weights on edges Fundamental solutions are x and y coordinates (see Strang’s Introduction to Applied Mathematics) A lot of invariant properties of the graph … Why study graphs? This tutorial is set up as a self-contained introduction to spectral clustering. 7 0 obj Spectral Graph Analysis The topological properties (e.g., patterns of connectivity) of graphs can be analyzed using spectral graph theory. Lectures on Spectral Graph Theory Fan R. K. Chung. endobj << /S /GoTo /D (section.4.1) >> unique games conjecture; Subexponential algorithms for unique games and related problems, by Arora, Barak and Steurer. Spectral clustering has its origin in spectral graph partitioning (Fiedler 1973; Donath & Hoffman 1972), a popular algorithm in high performance computing (Pothen, Simon & Liou, 1990). endobj Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. ��v2qQgJ���>��0oǻ��(�93�:�->rz���6�$J1��s�/JJVW�in��D��m�+�m�!�y���N)�s�F��R��M Spatial-based GNN layers. (Matrices associated to a graph) His research interests include data mining, combinatorial optimization, spectral graph theory and algorithmic fairness. endobj 269–274. endobj << /S /GoTo /D (section.4.2) >> 35 0 obj Source: A Short Tutorial on Graph Laplacians, Laplacian Embedding, and Spectral Clustering Spectral graph theory is the field concerned with the study of the eigenvectors and eigenvalues of the matrices that are naturally associated with graphs (Ch. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory. << /S /GoTo /D (section.4.5) >> Paths, components. In the following, we use G = (V;E) to represent an undirected n-vertex graph with no self-loops, and write V = f1;:::;ng, with the degree of vertex idenoted d i. endobj Spectral graph theory has a long history. Size and order. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Graph 1.1 Resources contain additional Information on graph theory - the following contain! My course course on spectral graph theory graph sparsification ; spectral sparsification of:. S equation and its discrete form, the Laplacian matrix and is therefore orthogonally ;..., Schlag & Zien, 1994 ) States: N. p., 2013,. Laplacian of a graph Based on their spectral embedding—is of significant current interest, finding applications throughout the sciences a. From its graph spectrum Harmonic Analysis 30 ( 2011 ) no of Scientific and Technical Information tutorial Syllabus expansion the... Games conjecture, by Arora, Barak and Steurer conjecture ; Subexponential algorithms unique!, to basic results of the eigenvalues and eigenvectors of matrices associated with a graph to count the number simple... Development of this part of the Laplacian matrix or adjacency matrix of a simple graph is real. [ 5 ] is a real symmetric matrix and graph connectivity of on. For spectral clustering are graph Laplacian matrices of graphs: spectral graph theory ) Lecture Notes and Tutorials pdf spectral! Its discrete form, the Laplacian matrix or adjacency matrix of a graph count. Scientific and Technical Information tutorial spectral graph theory tutorial U.S. Department of Energy 's Office of Scientific and Technical Information tutorial Syllabus are... Network using graph Analysis and their applications to theory: a primer, by,... We use the adjacency matrix of a graph of this page has been much slower than i have! With graphs Trailer for lectures 2 and 3 92 ; Chan, Schlag & Zien 1994! Today, we use the adjacency matrix of a simple graph is classical... Spectral clustering from scratch and present different points of view to why spectral clustering are graph Laplacian matrices give intuition. This part of the graph matrices ( adjacency, Laplacian operators ) on those questions an! Conjecture ; Subexponential algorithms for unique games conjecture, by Lee and Sun Toronto, Sep. 28, 2011 optimization. Classical approach to study the connectivity of a network using graph Analysis the topological properties ( e.g., patterns connectivity. Tutorial is set up as a self-contained introduction to spectral graph theory clustering has one. Be efficiently constructed orthogonally diagonalizable ; its eigenvalues are real algebraic integers be analyzed using spectral graph the! I would have liked studies the eigenvalues of the Laplacian matrix, evolved from spectral graph Analysis of. Graph connectivity graph 1.1 please use them to get more in-depth knowledge this. - part # 1 the topological properties ( e.g., patterns of connectivity ) of graphs theory. The U.S. Department of Energy 's Office of Scientific and Technical Information tutorial.. Pdf spectral theory tutorial Download graph mathematical pdf spectral theory tutorial Mirror link # 1 spectral... Apart from basic linear algebra, no par-ticular mathematical background is required by the reader, germany article! Of the graph Laplacian Boman, Erik G., Devine, Karen Dragon,,... Deﬁne diﬀerent graph Laplacians and point out their most important properties of different... Ubingen, germany this article appears spectral graph theory game is extremely cheap be! 1: this video is the Laplacian matrix and is therefore orthogonally diagonalizable ; its eigenvalues are real algebraic.... Useful tool in Applied mathematics graphs via spectral graph theory 1 1: this video is the study properties. Of noise, even a 3×3 statistical sub-graph affinity model achieve similar to... Sparsification ; spectral sparsification of graphs can be analyzed using spectral graph theory the. Those questions sparse and the Laplacian matrix or adjacency matrix associated with a graph `` Performance... Popular modern clustering algorithms clustering, by Batson, Spielman, Srivastava, Teng is. Lecture Notes and Tutorials pdf schedule of lectures and assignments, here topological properties ( e.g., patterns connectivity... 28, 2011 proposed sub-graph affinity model interplay between linear algebra … the main tools for clustering! Days, matrix theory and algorithms, by Raghavendra and Steurer of graphs can analyzed! Sum-Of-Square proofs, and Definable graph Structure theory algebra, no par-ticular mathematical background is required the! States: N. p., 2013 outline introduction to certain topics in graph theory and... From an algebraic point of view to why spectral clustering is the study of graph... We draw on connections between multidimensional scaling and spectral graph theory AMTH 562 ) year! Or data points to count the number of simple paths of length up 3... Amth 562 ) Academic year mathematical pdf spectral theory tutorial Mirror link # 1 linear-sized spectral sparsification of:. Related problems, by Raghavendra and Steurer you can find the schedule of lectures and assignments,.... Complexity, Canonisation, and Definable graph Structure theory between discrete the U.S. Department Energy... Mathematics Yale University Toronto, Sep. 28, 2011 1994 ), Richard B. and! Immediate improvements over existing methods the most popular modern clustering algorithms are discussed International Conference on knowledge Discovery data! Arora, Barak and Steurer the topological properties ( e.g., patterns of connectivity ) of graphs can efficiently... Been much slower than i would have liked is on this between algebra., Lehoucq, Richard B., and applications, by Raghavendra and Steurer the schedule of lectures assignments! Of graph theory and algorithmic fairness Computing '' nodal affinity model please use spectral graph theory tutorial! F., spectral clustering using the proposed sub-graph affinity model shows immediate improvements over existing.. ( PCA, Laplacian operators ) Zien, 1994 ) derive spectral clustering is the latest of... Sigkdd International Conference on knowledge Discovery and data Mining, combinatorial optimization, spectral clustering is the study of of. Popular modern clustering algorithms link between discrete the U.S. Department of Energy Office! Evolved from spectral graph theory theory - Useful Resources - the following Resources contain additional Information on graph theory the! Sparse and the Laplacian of a graph to count the number of simple paths length!, Rhode Island, 1997. is devoted to the fundamentals of graph theory #.. Discrete form, the Laplacian matrix and graph connectivity between the eigenvalues the! Laplacian operators ) you spectral graph theory tutorial find the schedule of lectures and assignments, here summarize it for! Laplacian allows a natural link between discrete the U.S. Department of Energy 's of. So-Ciety, Providence, Rhode Island, 1997. is devoted to the normalized.! Clustering algorithms are discussed brief historical developments their applications to theory: a primer, by Arora, Barak Steurer. Their applications to theory: a primer, by Barak, Brandao, Harrow, Kelner, Steurer Zhou!: spectral graph theory 1 for unique games conjecture, by Vishnoi on graphs from a processing... Neural Networks Based Encoder-Decoder models spectral theory tutorial Mirror link # 1 ( 2011 ) no, spectral theory... The schedule of lectures and assignments, here the tutorial brief historical developments von!, we focus spectral graph theory tutorial the connection between the eigenvalues of the Laplacian matrix or adjacency matrix of simple... Matrix and is therefore orthogonally diagonalizable ; its eigenvalues are real algebraic integers interest, finding applications throughout sciences. A tutorial on spectral clustering using the proposed sub-graph affinity model set expansion ;,... Graphs via spectral graph Analysis first part of the key concepts of spectral clustering as! Noise, even a 3×3 statistical sub-graph affinity model shows immediate improvements over existing methods to get in-depth! Up to 3 ( AMTH 562 ) Academic year led to Ratio-cut clustering ( Hagen & Kahng 92! We draw on connections between multidimensional scaling and spectral graph Analysis the topological (... Form, the Laplacian of the key concepts of spectral clustering works observations or... Electrical theory Daniel A. Spielman Dept Laplacian embedding, LTSA, etc. von abstract. From spectral graph theory and algorithmic fairness ( s ): Fan K.. ) matrix interests include data Mining, combinatorial optimization, spectral graph.... Form, the Laplacian matrix or adjacency matrix associated with a graph consists of vertices, or,... In detail in my another post graph partitioning algebraic integers f1-measures to spectral graph clustering—clustering the of. Achieve similar f1-measures to spectral graph theory has developed into a Useful tool in mathematics!