Fall 2003

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

**Website:** http://www.cse.buffalo.edu/~hungngo/classes/2003/Random_Graphs

**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.

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.

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

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

- About 20 pages of dense reading per week (i.e. no bedtime reading).
- Each student is expected to do a presentation at some point during the semester,
based on:
- A survey done by the student on a particular topic to be suggested by the instructor or picked by the student but approved by the instructor.
- A (few) very dense mathematical paper.
- An open problem solved by the student.

- Find learning new things fun.
- Find learning mathematics fun.
- Find solving mathematical puzzles fun.
- Find the instructor funny.

- OK, the real deals are: rudimentary knowledge on linear algebra, algorithms, probability theory. They are not entirely essential to follow things presented in the seminars. Related background materials shall be provided in the forms of small tutorials/notes. You'd have to read them though.