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

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.

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.

Summer 2015:

Classes: MTWTF 1:15pm-3:30pm, NDSU Morrill Hall Room 101
Office hours: By appointment (Minard 408E22)
Syllabus

Lectures:

Mathematica:

Erdős-Rényi random graph (notebook)
Simple configuration model

Some other useful sources:

Introductory level books (no prerequisites):
- Graph Theory and Complex Networks: An Introduction by Maarten van Steen
- A First Course in Network Theory by Ernesto Estrada and Philip A. Knight
- Complex Networks: Structure, Robustness and Function by Reuven Cohen and Shlomo Havlin
Mathematical books:
- Lecture notes on Random Graphs and Complex Networks by Remco van der Hofstad
- Random Graph Dynamics by Rick Durrett
- Performance Analysis of Complex Networks and Systems by Piet van Mieghem
- Graph Spectra for Complex Networks by Piet van Mieghem
- Spectral Graph Theory by Fan R. K. Chung
- Complex Graphs and Networks by Linyuan Lu and Fan Chung
- A Course on the Web Graph (Graduate Studies in Mathematics) by Anthony Bonato
- Probability on Graphs: Random Processes on Graphs and Lattices by Geoffrey Grimmett
- Networks out of Control by Matthias Grossglauser and Patrick Thiran
- Random Geometric Graphs by Mathew Penrose
- Introduction to Random Graphs, by Alan Frieze and Michal Karonski
- Random Graphs by Bela Bollobas
- Random Graphs by Svante Janson, Tomasz Luczak, and Andrzej Rucinski
- Random Graphs by V. F. Kolchin
- Graphical Evolution by Edgar M. Palmer
- The Strange Logic of Random Graphs by Joel Spencer
- Probability on Trees and Networks by Russell Lyons with Yuval Peres
- Percolation by Bela Bollobas and Oliver Riordan
- Percolation by Geoffrey R. Grimmett
- Random Trees: An Interplay between Combinatorics and Probability by Michael Drmota
Interdisciplinary books:
- Networks: An introduction by Mark Newman
- Social and Economic Networks by Matthew O. Jackson
- Dynamical Processes on Complex Networks by Alain Barrat, Marc Barthelemy, and Alessandro Vespignani
- Lectures on Complex Networks by Sergey Dorogovtsev
- Networks, Crowds, and Markets: Reasoning About a Highly Connected World by David Easley and Jon Kleinberg
- Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools by Didier Sornette
Mathematical epidemiology and complex networks:
- Epidemics and Rumours in Complex Networks by Moez Draief, Laurent Massoulié
- Mathematical Tools for Understanding Infectious Disease Dynamics by Odo Diekmann, Hans Heesterbeek, Tom Britton
- Modeling Infectious Diseases in Humans and Animals by Matt J. Keeling, Pejman Rohani
Popular books:
- Linked: How Everything Is Connected to Everything Else and What It Means by Albert-Laszlo Barabasi
- Small Worlds by Duncan J. Watts
- Six Degrees: The Science of a Connected Age by Duncan J. Watts
- Nexus: Small Worlds and the Groundbreaking Science of Networks by Mark Buchanan
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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