Spectral Graph Theory and ApplicationsAdvanced 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: http://www.mpi-inf.mpg.de/departments/d1/teaching/ws11/SGT/index.html

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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.

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Topics

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Random Walks

Cheeger Inequality

Approximation Algorithms

ExpandersPseudorandomness

Max Cut

EigenvaluesComplexity

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).

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INTRODUCTIONLecture 1

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Seven Bridges of Königsberg, 1736

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L. Euler(1707-1783)

From then on...

• Connectivity

• Chromatic number

• Euler Path

• Hamiltonian Path

• Matching

• Graph homomorphism

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About 50 years ago...

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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.

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What is the graph spectrum?

Consider a d-regular graph G:

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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.)

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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

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In Computer Science In Mathematics

Example 1: Super concentrators

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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.

2010 ACM Turing Award Citation

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History

Author Density Year Reference

Valiant 238 1975 STOC

Gabber 271.8 1981 JCSS

Shamir 118 1984 STACS

Alon 60 1987 JACM

Alon 44+o(1) 2003 SODA

Explicit constructions

Existence Proof

Author Density Year Reference

Chung 36 1978 Bell Sys. Tech. J.

Schöning 34 2000 Ran Str. Algo.

Schöning 28 2006 IPL

Only 7 pages for constructions and analysis

Based on Kolmogorov Complexity

Lower Bound [Valiant]: 5-o(1)

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Example 2: Graph Partitioning

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Applications

•Community detection

•Graph partitioning

•Machine learning

Example 3: Ramanujan Graphs

Ramanujan graphs are graphs having the best expansion ratio.

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Ramanujan graphs

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Big Open Problem: Construct Ramanujan graphs with any degree.

Example 4: Random walks

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G. Pólya (1887-1985)

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.

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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)

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Example 5: Randomness Complexity

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From Art to Science

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Example 5: Randomness Complexity

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A.N. Kolmogorov (1903-1987) Andrew Yao (1946- )

From Art to Science

Generate “almost random” sequences using modern computers.

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Example 5: Randomness ComplexityFrom Art to Science