Mathematics of Networks: Math 867: Topics in Applied Mathematics
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
The prerequisite is the instructor's discretion; the mathematical prerequisites will be kept to a minimum an include basic knowledge
of probability theory (discrete random variables), linear algebra, and ordinary differential equations.
- Analysis of the random graphs of Erdős 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.
- Classes: MTWTF 1:15pm-3:30pm, NDSU Morrill Hall Room 101
- Office hours: By appointment (Minard 408E22)
Some other useful sources:
- Introductory level books (no prerequisites):
- Mathematical books:
- Interdisciplinary books:
- Mathematical epidemiology and complex networks:
- Popular books:
- And everything else:
- Evolutionary games on graphs by Gyorgy Szabo, Gabor Fath
- The Structure and Dynamics of Networks by Mark Newman, Albert-Laszlo Barabasi, Duncan J. Watts (editors)
- Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications, by Dean Lusher, Johan Koskinen, Garry Robins
- Network Analysis: Methodological Foundations by U.Brandes. and T.Erlebach
- Evolution of Networks: From Biological Nets to the Internet and WWW by S. N. Dorogovtsev and J. F. F. Mendes
- Evolution and Structure of the Internet: A Statistical Physics Approach by Romualdo Pastor-Satorras and Alessandro Vespignani
- Scale-Free Networks: Complex Webs in Nature and Technology by Guido Caldarelli
- Statistical Mechanics of Complex Networks by Romualdo Pastor-Satorras, Miguel Rubi, and Albert Diaz-Guilera (Editors)
- Statistical Mechanics of Complex Networks (Albert and Barabasi; 2001)
- Evolution of networks (S.N. Dorogovtsev and J.F.F. Mendes; 2001)
- The structure and function of complex networks. (M. Newman; 2003)
- Mathematical results on scale-free random graphs (B. Bollobas; 2003)
- Power-law distributions in empirical data by Aaron Clauset, Cosma Rohilla Shalizi, M. E. J. Newman
- A Brief History of Generative Models for Power Law and Lognormal Distributions by Michael Mitzenmacher
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