In Fall 2013 I am scheduled to give a graduate course Math 767: Topics in Applied Mathematics, and the topic I chose is Mathematics of Networks.
Course theme and course description:
Networks are the language that is used today in numerous interdisciplinary studies that unite mathematicians, physicists, biologists, engineers, computer scientists, economists, etc. The major emphasis in the course will be put on the mathematical aspects of the complex network analysis. In particular, the course will tentatively include:
- Analysis of the random graphs of Erdös and Rényi: What a tractable mathematical null model of a network is, what its properties and peculiarities, including the threshold phenomena (such as appearance of the giant component).
- Statistical properties of real-world graphs: Degree distributions, diameter, clustering and other statistics of networks.
- More realistic random graph models: How to build a random graph with a given degree distribution. Analysis of configuration model.
- Scale-free distributions and power laws: Mathematical models to generate power law distributions.
- Mathematical models of network formation: Preferential attachment model and its modifications; the small-world network.
- Processes on random networks, including percolation and epidemics.
The prerequisites will be kept to a minimum and include basic knowledge of probability theory (discrete random variables) and ordinary differential equations. The course will rely in part on the material from Networks: An introduction by Mark Newman. The course web site is https://www.ndsu.edu/pubweb/~novozhil/Teaching/mathematics_of_networks.html.
If you need some projects for your students or some “applications of networks” then I recommend introducing them to the Ohtsuki-Nowak transform from EGT (after you teach them the Erdös-Rényi model). I’ve had a lot of fun playing with it, and it is a pretty easy paper to understand (plus there are more recent refinements if students want to go deeper).
Thanks. I’ll give it a look.