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