Spectral Graph Theory and Applications Advanced Course WS2011/2012 Thomas Sauerwald He Sun Max Planck Institute for Informatics

Course Information Time: Wednesday 2:15PM – 4:00PM Location: Room 024, MPI Building Credit: 5 credit points Lecturers: Thomas Sauerwald, He Sun Office Hour: Wednesday 10:00AM – 11:00AM Prerequisites: Basic knowledge of discrete mathematics and linear algebra Lecture notes: See homepage for weekly update Homepage: 2/25

Course Information (contd.) Grading –Homework (3 problem sets) –You need to collect at least 40% of the homework points to be eligible to take the exam. –The final exam will be based on the homework and lectures. 3/25

Topics 4/25 Random Walks Cheeger Inequality Approximation Algorithms Expanders Pseudorandomness Max Cut Eigenvalues Complexit y The Unified Theory

Why do you need this course? Provides a powerful tool for designing randomized algorithms. Gives the basics of Markov chain theory. Covers some of the most important results in the past decade, e.g. derandomization of log-space complexity class. Nicely combines classical graph theory with modern mathematics (geometry, algebra, etc). 5/25

INTRODUCTION Lecture 1 6/25

Seven Bridges of Königsberg, /25 L. Euler( )

From then on... Connectivity Chromatic number Euler Path Hamiltonian Path Matching Graph homomorphism 8/25

About 50 years ago... 9/25

Magic graphs: Expanders Combinatorically, expanders are highly connected graphs, i.e., to disconnect a large part of the graph, one has to remove many edges. Geometrically, every vertex set has a large boundary. Probabilistically, expanders are graphs whose behavior is “like” random graphs. Algebraically, expanders correspond to real-symmetric matrices whose first positive eigenvalue of the Laplacian matrix is bounded away from zero. 10/25

What is the graph spectrum? Consider a d-regular graph G: 11/25 Adjacency Matrix Laplacian Matrix

If A is a real symmetric matrix, then all the eigenvalues are real. Moreover, if G is a d-regular graph, then. What is the graph spectrum? (contd.) 12/25 We call the spectrum of graph G.

Applications of graph spectrum Pseudorandomness Circuit complexity Network design Approximation algorithms Graph theory Group theory Number theory Algebra 13/25 In Computer ScienceIn Mathematics

Example 1: Super concentrators 14/25 There is a long and still ongoing search for super concentrators with n inputs and output vertices and Kn edges with K as small as possible. This “sport” has motivated quite a few important advances in this area. The current “world record” holders are Alon and Capalbo. S. Hoory et al. In: Bulletin of American Mathematical Society, 2006.

Finally, the super concentrators constructed by Valiant in the context of computational complexity established the fundamental role of expander graphs in computation ACM Turing Award Citation 15/25

History AuthorDensityYearReference Valiant STOC Gabber JCSS Shamir STACS Alon601987JACM Alon44+o(1)2003SODA Explicit constructions Existence Proof AuthorDensityYearReference Chung361978Bell Sys. Tech. J. Schöning342000Ran Str. Algo. Schöning282006IPL Only 7 pages for constructions and analysis Based on Kolmogorov Complexity Lower Bound [Valiant]: 5-o(1) 16/25

Example 2: Graph Partitioning 17/25 Applications Community detection Graph partitioning Machine learning

Example 3: Ramanujan Graphs Ramanujan graphs are graphs having the best expansion ratio. 18/25

Ramanujan graphs 19/25 Big Open Problem: Construct Ramanujan graphs with any degree.

Example 4: Random walks 20/25 G. Pólya ( ) Applications Simulation of physical phenomenon Information spreading on social networks Approximation of counting problems Hardness amplification

Example 4: Random walks Theorem (Pólya, 1921) Consider a random walk on an infinite D-dimensional grid. If D = 2, then with probability 1, the walk returns to the starting point an infinite number of times. If D > 2, then with probability 1, the walk returns to the starting point only a finite number of times. 21/25 A drunk man will eventually return home but a drunk bird will lose its way in space.

What a random walk! Interviewed on his 90th birthday Pólya stated, "I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between." (Alexanderson, 1979) 22/25

Example 5: Randomness Complexity 23/25 From Art to Science

Example 5: Randomness Complexity 24/25 A.N. Kolmogorov ( ) Andrew Yao (1946- ) From Art to Science

Generate “almost random” sequences using modern computers. 25/25 Example 5: Randomness Complexity From Art to Science