Course : Data mining Lecture : Spectral graph analysis Aristides Gionis Department of Computer Science Aalto University visiting in Sapienza University of Rome fall 2016. spectral graph theory. in Computer Science and Engineering program at School of Engineering, Amrita Vishwa Vidyapeetham. In this course we will cover the basics of the field as well as applications to theoretical computer science. However, substantial revision is clearly needed as the list of errata got longer. Spectral Graph Theory, Fall 2019 Time: M-W 2:30-3:45. M.E.J. In the summer of 2006, the daunting task of revision finally but surely got started. Networks models - Random graphs, Generalized random graphs, The small-world model, Exponential random graphs, The large-scale structure of networks- small world effect, Degree distributions, Power laws and scale-free networks; Structure of the Internet, Structure of the World Wide Web. Then, we will cover recent progress on graph sparsification, Kadison-Singer problem and approximation algorithms for traveling salesman problems. Models of network formation-Preferential attachment, The model of Barabasi and Albert, Vertex copying models, Network optimization models; Epidemics on networks- Models of the spread of disease, SI model, SIR model, SIS model, SIRS model; Network search-Web search, Searching distributed databases. On the other hand, most of our examples are … In this course, we will study how eigenvalues and eigenvectors of matrices can be used to design and analyze algorithms for combinatorial problems. Unit 2. Graph drawing using the Laplacian, Unit 4. Today, we View scribe8.pdf from COMS 4995 at Columbia University. The course meets Tuesdays and Thursdays in Rhodes 571 from 10:10-11:25AM. 3 Graphs In this course, the word \graph" will refer to a weighted directed graph (a.k.a. The lectures take "place" on Tuesday 9:00-12:00 via this Zoom link. Course projects. The theory of linear algebra of symmetric matrices: the Spectral Theorem and the Courant-Fischer Theorem. This course is about understanding graphs by using the methods of Linear Algebra. Of course, for distance-transitive graphs, this implies that the realization is spectral. . Lecture 4 { Spectral Graph Theory Instructors: Geelon So, Nakul Verma Scribes: Jonathan Terry So far, we have studied k-means clustering for nding nice, convex clusters which conform to the standard notion of what a cluster looks like: separated ball-like congregations in space. Spectral graph theory is the powerful and beautiful theory that arises from the following question: ... Of course, if the graph has a single connected component, the smallest eigenvector v 1 = (1= p n;:::;1= p n), which is not helpful for embedding, as all points have the same value. On the one hand, there is, of course, the linear algebra that underlies the spectral ideas in graph theory. In this course we will cover the basics of the field as well as applications to theoretical computer science. The requirements include two presentations and a written report. Experience working with complex network data sets and implement computer algorithms to solve network problems, use modern network tools to analyze data, Compare different solutions of large network problems in terms of network performance measures, Compare structure of different types of networks. Expander graphs - properties, constructions, and applications. Spectral graph theory is the study of a graph via algebraic properties of matrices associated with the graph, in particular, the corresponding eigenvalues and eigenvectors. Spectral theory of real symmetric matrices, Unit 3. Explicit constructions of expander graphs, The recitation notes may contain Hebrew letters, Recitation 2 - operations on graphs and the resulted spectrum, Recitation 3 - group theory and characters recall; Cayley graphs, Recitation 5 - Hoffman's lower bound on the chromatic number, Recitation 8 - Cont last time; The Gaber-Galil expander, Recitation 9 - Cont the Gaber-Galil construction. Please take a minute to review the course policies. In particular, after a short linear algebra refresher, tentatively, we plan on covering. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. Get Free Graph Theory With Applications Courses now and use Graph Theory With Applications Courses immediately to get % off or $ off or free shipping. Graphs and Networks- Review of basic graph theory, Mathematics of networks- Networks and their representation, Graph spectra, Graph Laplacian, The structure of complex networks, Clustering, Community structures, Social networks - the web graph, the internet graph, citation graphs. This is a research-oriented graduate course in spectral graph theory. Spectral Graph Theory. Course: Spectral Graph Theory from Yale. to get a sense of what this course is mostly about. The recitation, by Shir, is in the following hour. it cannot be deformed without loosing symmetry), and irreducible, then one can show that it satisfies $(*)$ (it is not necessarily spectral, but almost). This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. In this case, we should Course speci fics, motivation, and intro to graph theory (PDF - 1.5MB) 2: Introduction to graph theory (PDF) 3: Strong and weak ties, triadic closure, and homophily (PDF) 4: Centrality measures (PDF) 5: Centrality and web search, spectral graph theory (PDF) 6 : Spectral graph theory, spectral clustering, and community detection I (PDF - 1.3MB) 7 Submissions are in pairs. weighted digraph), which is a more general type of graph than the type de ned in Chapter 1 of Spielman’s book. Overview . (I recall the whole book used to be free, but apparently that has changed) For the purely mathematical approach, Reinhard Diestel's book is both excellent and affordably priced, with a free PDF preview available from the author. Guido Caldarelli, “Scale-Free Networks”, Oxford University Press, 2007. The interconnections between properties of graphs and the eigenvalues of some matrices associated with graphs are studied. The technical prerequisite is very mild: a first course on linear algebra and the first course on algorithms. Location: WTS A60. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, in Computer Science & Engineering, Statistical and Probabilistic Modeling in Civil Engineering. If M2Cm n Several video talks from Simons Berkeley on SGT. Textbooks include: I Spectral and Algebraic Graph Theory (Daniel A. Spielman) I Scalable Algorithms for Data and Network Analysis (Shang-Hua Teng) About the Course 5 Objective of the course: I To explore what eigenvalues and eigenvectors of … COMS 4995-2: Advanced Algorithms (Spring’20) Feb 13, 2020 Lecture 7: Spectral Graph Theory Instructor: Alex Andoni 1 … The adjacency matrix of a simple graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. Alain Barrat, Marc Barthelemy and Alessandro Vespignani, “Dynamical processes on Complex networks”, Cambridge University Press, 2008. Offered by University of California San Diego. This course will consider the interplay between continuous and discrete optimization broadly speaking, but with a focus on algorithmic spectral graph theory and applications of the multiplicative weights update paradigm. These are lecture notes that are based on the lectures from a class I taught on the topic of Spectral Graph Methods at UC Berkeley during the Spring 2015 semester. Describe fundamental tools to study networks, mathematical models of network structure, computer algorithms for network data analysis and the theories of processes taking place on networks. This course is about applications of linear algebra to graph theory and to graph algorithms. Explicit Almost Ramanujan Graphs, Unit 9. 'Networks and Spectral Graph Theory' is an elective course offered in M. Tech. We will not follow any particular text but below are resources which we will use. Instructor: Dan Spielman. Reuven Cohen and Shlomo Havlin, “Complex Networks: Structure, Robustness and Function”, Cambridge University Press, 2010. Hall's graph drawing using the Laplacian's eigenvectors. All but for the Godsil-Royle book are available, for free, online. The book for the course is on this webpage. We expect to hand out about 5 problem sets throughout the semester that will account for half the grade. The extreme eigenvalues of the adjacency matrix and the Perron-Frobenius theorem. CPSC 462/562 is the latest incarnation of my course course on Spectral Graph Theory. Ben-Aroya - Ta-Shma paper on the wide replacement product, Alon-Goldreich-Hastad-Peralta's construction of small-bias sets, Ta-Shma's construction of near-optimal small-bias sets. However, I stress that this is an advanced course of mathematical nature, and so mathematical maturity is essential to follow the course successfully. Access. Dougles West, “Introduction to Graph Theory”, Second Edition, PHI Learning Private Limited, 2011. Course description: Spectral graph methods use eigenvalues and eigenvectors of matrices associated with a graph, e.g., adjacency matrices or Laplacian matrices, in order to understand the properties of the graph. I suggest you'll watch Spielman's talk Miracles of Algebraic Graph Theory to get a sense of what this course is mostly about. This course is open to students from Statistics and Computer Science as well. Welcome to the homepage for Graph Theory (Math/CSCI 4690/6690)! Design algorithms to solve large real-world network problems, devise models of network structure to predict the behavior of networked systems. Spectral theory of real symmetric matrices, The extreme eigenvalues of the adjacency matrix, Graph partitioning and Cheeger's inequality, Explicit constructions of expander graphs, Explicit constructions of expander graphs, The lectures take "place" on Tuesday 9:00-12:00 via. Office Hours: Friday, 3:00 - 4:00 . Lecture notes from a course by Linial and Wigderson with the same title; Lecture notes on expansion and spectral graph theory by Luca Trevisan (also an earlier version) Book on spectral and algebraic graph theory by Dan Spielman The main course work is the course project. Algebraic graph theory by Godsil and Royle. A two-part video talk by Amnon Ta-Shma on his small-bias sets construction (, Daniel Spielman's book (incomplete; available online), Several video talks from Simons Berkeley on SGT, Salil Vadhan chapter on expander graphs from his Pseudorandomness monograph. Newman, “Networks: An Introduction”, Oxford University Press, 2010. A take-home exam, submitted individually, of course, will determine the remaining part of the grade. Welcome to Spectral Graph Theory. Yale College juniors and seniors are also welcome. Introduction to Spectral Graph Theory Spectral graph theory is the study of a graph through the properties of the eigenvalues and eigenvectors of its associated Laplacian matrix. This course will consider connections between the eigenvalues and eigenvectors of graphs and classical questions in graph theory such as cliques, colorings, cuts, flows, paths, and walks. Spectral graph theory-- a book focused on the definition and development of the normalized Laplacian written by Fan Chung, the first four chapters of the revised version are available online. We will start by reviewing classic results relating graph expansion and spectra, random walks, random spanning trees, and their electrical network representation. You won't need a copy of the latter, so no worries. Graph partitioning and Cheeger's inequality, Unit 9. Centre for International ProgramsAmritapuri, Corporate & Industry RelationsAmritapuri | Bengaluru | Coimbatore | Kochi | Mysuru. In particular, after a short linear algebra refresher. A two-part video talk by Amnon Ta-Shma on his small-bias sets construction (part 1, part 2). Time: M-W 2:30-3:45. Spectral methods • Understanding a graph using eigen values and eigen vectors of the matrix • We saw: • Ranks of web pages: components of 1st eigen vector of suitable matrix • Pagerank or HITS are algorithms designed to compute the eigen vector • Today: other ways spectral methods help in … Spectral graph theory studies how the eigenvalues of the adjacency matrix of a graph, which are purely algebraic quantities, relate to combinatorial properties of the graph. At a very high level the course shows how one can use linear algebra to solve fundamental problems in computer science much more efficiently. 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. We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Some background in discrete probability and graph theory is helpful, but the course is self-contained. in Computer Science and Engineering program at School of Engineering, Amrita Vishwa Vidyapeetham. 'Networks and Spectral Graph Theory' is an elective course offered in M. Tech. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. At a very high level the course shows how one can use linear algebra to solve fundamental problems in computer science much more efficiently. Spectral Graph Theory, Fall 2015 Applied Mathematics 561/ Computer Science 662 . You can find the schedule of lectures and assignments, here. The recitation, by Shir, is in the following hour. Please take a minute to review the course policies. Measures and metrics- Degree centrality, Eigenvector centrality, Katz centrality, PageRank, Hubs and authorities, Closeness centrality, Betweenness centrality, Transitivity, Reciprocity, Similarity, assortative mixing. Welcome to Spectral Graph Theory. Let the course sta know if you need help nding other students in the class willing to discuss the course material or collaborate on Problem Set 0. The extreme eigenvalues of the adjacency matrix, Unit 7. To connect this to spectral graph theory, one can observe that if a realization is rigid (i.e. The course aims to bring the students to the forefront of a very active area of research. In this course we will cover the basics of the field as well as applications to theoretical computer science. Ta-Shma's explicit construction of codes close to the Gilbert-Varshamov bound. Students are expected to do active research on a topic in spectral graph theory. We begin with a brief review of linear algebra. It will also be broadcast to Cornell NYC Tech, Ursa room. SPECTRAL GRAPH THEORY (revised and improved) Fan Chung The book was published by AMS in 1992 with a second printing in 1997. The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. In this course, we will study both the mathematics and the algorithmic applications of spectral graph theory, including some results from the past couple of years. At the end of the course the students will be able to, Apply Now Visit Us Jobs & Careers Contact Us, M. Tech. Salil Vadhan chapter on expander graphs from his Pseudorandomness monograph. Explicit constructions of expander graphs, Unit 12. Spectral graph theory is the study of a graph via algebraic properties of matrices associated with the graph, in particular, the corresponding eigenvalues and eigenvectors. Within algorithmic spectral graph theory, both older structural results and recent algorithmic results will be presented. I will post all the homework assignments for the course on this page. Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. This is a research-oriented graduate course. 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