# spectral graph theory course

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

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