### CSE 713 - Random Graphs: Theory and Applications Fall 2003

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Instructor: Dr. Hung Q. Ngo
```	Office: 239 Bell Hall
Office Hours: Fridays 1:00-3:00pm
Phone: 645-3180 x 160
Email: hungngo@cse.buffalo.edu ```

Grading: to be done on an S/N (or S/U) basis only.

Time and place: Mondays 4:30-5:45pm, and Wednesdays 2:00-3:15pm at Bell 242.

The first meeting is on Wednesday, Aug 27.

#### Description:

Erdös and Rényi founded the area of random graph theory at around 1960, modeled after a few earlier works of Erdös. Although the ideas of the probabilistic methods in mathematics have been around, probably, since the 1930, Paul Erdös was the first mathematician to show us the full potential of the method.

Imagine a graph which evolves over time. New edges are formed, old edges are leaving, new nodes are coming, old nodes are removed. What can we say, probabilistically, about such graph's properties: connectivity, diameter, maximum degree, average degree, and their relationships? In networking, for example, P2P networks exhibit precisely this kind of evolution. Recently, ad hoc wireless networks present a type of "geometric" random graphs which are very interesting on their own. The structure and evolution of the World Wide Web is yet another perfect example where random graphs prove to be extremely useful.

This seminar aims to skim through the foundation of the theory, hoping that students shall be able to know where/what to look for later on when faced with a random graphs type of problem.

I shall spend roughly half of the semester presenting the foundations of random graphs. Elementary knowledge on probability theory is required. I shall provide reading materials on basic probability theory.
At the second half, each member of the class presents a paper or a topic which uses random graphs. The papers & topics should be in one of three areas: networking, algorithms, or graph theory.

Also, each class member prepares scribe note for at least one lecture, depending on how large the class is. The LATEX template shall be provided. Part of the grading is based on how much effort I have to spend modifying the scribe note. All notes shall be shared to the class.

A list of recommended papers and/or topics shall also be provided.

#### Objectives:

• Have fun learning!
• Learn the foundations of random graph theory
• Learn their applications to networking, algorithms & graph theory

#### Topics (tentative): we shall follow the text "The probabilistic method" by Alon and Spencer. It would be helpful if you have the text, but it is not absolutely required.

• Intro. to the probabilistic methods
• Linearity of expectation
• Alterations
• The second moment
• Local lemma
• Correlation Inequalities
• Martingales and Tight Concentration
• Pseudorandomness