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. 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 This course is open to students from Statistics and Computer Science as well. in Computer Science and Engineering program at School of Engineering, Amrita Vishwa Vidyapeetham. 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. Salil Vadhan chapter on expander graphs from his Pseudorandomness monograph. Welcome to the homepage for Graph Theory (Math/CSCI 4690/6690)! 'Networks and Spectral Graph Theory' is an elective course offered in M. Tech. The lectures take "place" on Tuesday 9:00-12:00 via this Zoom link. Submissions are in pairs. Office Hours: Friday, 3:00 - 4:00 . 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. Algebraic graph theory by Godsil and Royle. Ta-Shma's explicit construction of codes close to the Gilbert-Varshamov bound. 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 Course: Spectral Graph Theory from Yale. The adjacency matrix of a simple graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. 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. Location: WTS A60. At a very high level the course shows how one can use linear algebra to solve fundamental problems in computer science much more efficiently. Explicit constructions of expander graphs, Unit 12. Yale College juniors and seniors are also welcome. Search. In particular, after a short linear algebra refresher, tentatively, we plan on covering. At a very high level the course shows how one can use linear algebra to solve fundamental problems in computer science much more efficiently. The recitation, by Shir, is in the following hour. Design algorithms to solve large real-world network problems, devise models of network structure to predict the behavior of networked systems. I will post all the homework assignments for the course on this page. 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. 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. 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. Instructor: Dan Spielman. Access. Explicit Almost Ramanujan Graphs, Unit 9. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, 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. to get a sense of what this course is mostly about. Spectral theory of real symmetric matrices, Unit 3. 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). 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. Guido Caldarelli, “Scale-Free Networks”, Oxford University Press, 2007. Please take a minute to review the course policies. In the summer of 2006, the daunting task of revision finally but surely got started. Please take a minute to review the course policies. Then, we will cover recent progress on graph sparsification, Kadison-Singer problem and approximation algorithms for traveling salesman problems. The extreme eigenvalues of the adjacency matrix, Unit 7. 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. View scribe8.pdf from COMS 4995 at Columbia University. (I recall the whole book used to be free, but apparently that has changed) Spectral Graph Theory, Fall 2019 Time: M-W 2:30-3:45. The interconnections between properties of graphs and the eigenvalues of some matrices associated with graphs are studied. 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. 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. 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. All but for the Godsil-Royle book are available, for free, online. in Computer Science and Engineering program at School of Engineering, Amrita Vishwa Vidyapeetham. 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. At the end of the course the students will be able to, Apply Now Visit Us Jobs & Careers Contact Us, M. Tech. SPECTRAL GRAPH THEORY (revised and improved) Fan Chung The book was published by AMS in 1992 with a second printing in 1997. weighted digraph), which is a more general type of graph than the type de ned in Chapter 1 of Spielman’s book. Daniel Spielman's book (incomplete; available online). Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. However, I stress that this is an advanced course of mathematical nature, and so mathematical maturity is essential to follow the course successfully. In this course we will cover the basics of the field as well as applications to theoretical computer science. Alain Barrat, Marc Barthelemy and Alessandro Vespignani, “Dynamical processes on Complex networks”, Cambridge University Press, 2008. Students are expected to do active research on a topic in spectral graph theory. This course is about understanding graphs by using the methods of Linear Algebra. You can find the schedule of lectures and assignments, here. 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. However, substantial revision is clearly needed as the list of errata got longer. In this case, we should Welcome to Spectral Graph Theory. On the other hand, most of our examples are … A two-part video talk by Amnon Ta-Shma on his small-bias sets construction (part 1, part 2). The main objective of spectral graph theory is to relate properties of graphs with the eigenvalues and eigenvectors (spectral properties) of associated matrices. A take-home exam, submitted individually, of course, will determine the remaining part of the grade. COMS 4995-2: Advanced Algorithms (Spring’20) Feb 13, 2020 Lecture 7: Spectral Graph Theory Instructor: Alex Andoni 1 … 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 … Dougles West, “Introduction to Graph Theory”, Second Edition, PHI Learning Private Limited, 2011. Get Free Graph Theory With Applications Courses now and use Graph Theory With Applications Courses immediately to get % off or $ off or free shipping. The theory of linear algebra of symmetric matrices: the Spectral Theorem and the Courant-Fischer Theorem. In this course we will cover the basics of the field as well as applications to theoretical computer science. In particular, after a short linear algebra refresher. 3 Graphs In this course, the word \graph" will refer to a weighted directed graph (a.k.a. We expect to hand out about 5 problem sets throughout the semester that will account for half the grade. Welcome to Spectral Graph Theory. We will start by reviewing classic results relating graph expansion and spectra, random walks, random spanning trees, and their electrical network representation. The requirements include two presentations and a written report. Unit 2. The main course work is the course project. Course projects. Within algorithmic spectral graph theory, both older structural results and recent algorithmic results will be presented. 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. Spectral Graph Theory, Fall 2015 Applied Mathematics 561/ Computer Science 662 . Hall's graph drawing using the Laplacian's eigenvectors. 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. 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. Overview . If M2Cm n 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. The course meets Tuesdays and Thursdays in Rhodes 571 from 10:10-11:25AM. Some background in discrete probability and graph theory is helpful, but the course is self-contained. I suggest you'll watch Spielman's talk Miracles of Algebraic Graph Theory to get a sense of what this course is mostly about. Centre for International ProgramsAmritapuri, Corporate & Industry RelationsAmritapuri | Bengaluru | Coimbatore | Kochi | Mysuru. Several video talks from Simons Berkeley on SGT. Fundamental network algorithms- Graph partitioning, Maximum flows and minimum cuts, Spectral graph partitioning, Community detection, Girvan and Newman Algorithm, Simple modularity maximization, Spectral modularity maximization, Fast methods based on the modularity. If x= a+ibis a complex number, then we let x= a ibdenote its conjugate. 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 … Time: M-W 2:30-3:45. CPSC 462/562 is the latest incarnation of my course course on Spectral Graph Theory. The technical prerequisite is very mild: a first course on linear algebra and the first course on algorithms. 'Networks and Spectral Graph Theory' is an elective course offered in M. Tech. Reuven Cohen and Shlomo Havlin, “Complex Networks: Structure, Robustness and Function”, Cambridge University Press, 2010. 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. Expander graphs - properties, constructions, and applications. We will not follow any particular text but below are resources which we will use. It will also be broadcast to Cornell NYC Tech, Ursa room. . 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. The extreme eigenvalues of the adjacency matrix and the Perron-Frobenius theorem. This course is about applications of linear algebra to graph theory and to graph algorithms. We begin with a brief review of linear algebra. 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. Overview . 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. MATH 674, Extremal Combinatorics: The course is a stand alone introduction to extremal combinatorics. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. 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. The book for the course is on this webpage. You won't need a copy of the latter, so no worries. 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. 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. Measures and metrics- Degree centrality, Eigenvector centrality, Katz centrality, PageRank, Hubs and authorities, Closeness centrality, Betweenness centrality, Transitivity, Reciprocity, Similarity, assortative mixing. Offered by University of California San Diego. On the one hand, there is, of course, the linear algebra that underlies the spectral ideas in graph theory. Newman, “Networks: An Introduction”, Oxford University Press, 2010. in Computer Science & Engineering, Statistical and Probabilistic Modeling in Civil Engineering. For the purely mathematical approach, Reinhard Diestel's book is both excellent and affordably priced, with a free PDF preview available from the author. This is a research-oriented graduate course in spectral graph theory. M.E.J. Graph partitioning and Cheeger's inequality, Unit 9. To connect this to spectral graph theory, one can observe that if a realization is rigid (i.e. 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. Lecture 11: Introduction to Spectral Graph Theory Rajat Mittal IIT Kanpur We will start spectral graph theory from these lecture notes. 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