Complex Systems 535/Physics 508, Fall 2012: Network Theory

Instructor: Mark Newman
Office: 322 West Hall
Office hours: Wednesdays 1:30-3:30pm
Email: mejn@umich.edu

Description:

This course will introduce and develop the mathematical theory of networks, particularly social and technological networks, with applications to network-driven phenomena in the Internet, search engines, network resilience, epidemiology, and many other areas.

Topics covered will include experimental studies of social networks, the world wide web, information and biological networks; methods and computer algorithms for the analysis and interpretation of network data; graph theory; models of networks including random graphs and preferential attachment models; network dynamics.

Requirements

Students should have studied calculus and linear algebra before taking the course, and should in particular be comfortable with the solution of linear differential equations and with the calculation and properties of eigenvalues and eigenvectors of matrices. In addition, a moderate portion of the course, perhaps three weeks, will deal with computer methods for analyzing networks. Although students will not be required to write computer programs, some experience with computer programming will be a great help in understanding this part of the course.

Coursework

There will be weekly graded problem sets, consisting of questions on both theory and applications. There will be two mid-term exams but no final. The mid-terms will be in class at the usual time on October 18 and December 11.

There will be reading assignments for each lecture. The assignments are listed on the schedule below. Students are expected to do the reading for each lecture in a timely manner.

Books

Textbook (required): Networks: An Introduction, M. E. J. Newman, Oxford University Press, Oxford (2010)

In addition to this required text, a list of other useful books is given below. None of them is required, but you may find them useful if you want a second opinion or more detail on certain topics.

General books on networks:

• Reuven Cohen and Shlomo Havlin, Complex Networks: Structure, Robustness and Function, Cambridge University Press, Cambridge (2010). Quite a short book, but it covers most of the topics of the course, at least to some extent, and some others that are not in the book by Newman.
• S. N. Dorogovtsev, Lectures on Complex Networks, Oxford University Press, Oxford (2010). Very short – genuinely a set of lecture notes, rather than a full text.

Books on specific networky topics:

• R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice Hall, Upper Saddle River, NJ (1993)
• A. Barrat, M. Barthelemy, and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, Cambridge (2008)
• G. Caldarelli, Scale-Free Networks: Complex Webs in Nature and Technology, Oxford University Press, Oxford (2007)
• S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks, Oxford University Press, Oxford (2003)
• A. Degenne and M. Forse, Introducing Social Networks, Sage, London (1999)
• D. Easley and J. Kleinberg, Networks, Crowds, and Markets, Cambridge University Press, Cambridge (2010)
• F. Harary, Graph Theory, Perseus, Cambridge, MA (1995)
• M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ (2008)
• C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, PA (2000)
• M. E. J. Newman, A.-L. Barabasi, and D. J. Watts, The Structure and Dynamics of Networks, Princeton University Press (2006)
• R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet, Cambridge University Press, Cambridge (2007)
• J. Scott, Social Network Analysis: A Handbook, 2nd edition, Sage, London (2000)
• S. Wasserman and K. Faust, Social Network Analysis, Cambridge University Press, Cambridge (1994)
• D. J. Watts, Six Degrees: The Science of a Connected Age, Norton, New York (2003)
• D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ (1996)
• R. J. Wilson, Introduction to Graph Theory, 4th edition, Addison-Wesley, Reading, MA (1997)

Problem sets:

• Homework 1 - Fundamental mathematics of networks
• Homework 2 - Measures and metrics
• Homework 3 - Paths, degrees, clustering
• Homework 4 - Algorithms
• Homework 5 - Random graphs
• Homework 6 - The configuration model
• Homework 7 - Generating functions
• Homework 8 - Generative models
• Homework 9 - Percolation and epidemics

Syllabus

Date	Topic	Reading	On-line resources	Notes
Tuesday, Sept. 4	Introduction	Chapter 1
Thursday, Sept. 6	Technological and social networks	Chapters 2 and 3
Tuesday, Sept. 11	Information and biological networks	Chapters 4 and 5
Thursday, Sept. 13	Basic mathematics of networks	6.1-6.11	Homework 1	Homework 1 handed out
Tuesday, Sept. 18	Centrality	7.1-7.7
Thursday, Sept. 20	Transitivity, reciprocity, assortativity	7.8-7.13	Homework 2	Homework 1 due, Homework 2 handed out
Tuesday, Sept. 25	Network structure and degree distributions	8.1-8.6
Thursday, Sept. 27	Computer algorithms and complexity	Chapter 9	Homework 3	Homework 2 due, Homework 3 handed out
Tuesday, Oct. 2	Shortest paths	10.1-10.4
Thursday, Oct. 4	Maximum flows and minimum cuts	6.12 and 10.5	Homework 4, Data file	Homework 3 due, Homework 4 handed out
Tuesday, Oct. 9	Matrix algorithms and graph partitioning	Chapter 11
Thursday, Oct. 11	Maximum likelihood methods			Homework 4 due, no new homework this week
Tuesday, Oct. 16	No class			Fall Break
Thursday, Oct. 18	Mid-term 1			In class, usual time and place
Tuesday, Oct. 23	Random graphs 1	12.1-12.5
Thursday, Oct. 25	Random graphs 2	12.6-12.8	Homework 5	Homework 5 handed out
Tuesday, Oct. 30	Configuration models 1	13.1-13.4
Thursday, Nov. 1	Configuration models 2	13.5-13.8	Homework 6	Homework 5 due, Homework 6 handed out
Tuesday, Nov. 6	Configuration models 3	13.9-13.11
Thursday, Nov. 8	Network spectra		Homework 7	Homework 6 due, Homework 7 handed out
Tuesday, Nov. 13	Generative models 1	14.1-14.2
Thursday, Nov. 15	Generative models 2	14.3	Homework 8	Homework 7 due, Homework 8 handed out, due Nov. 29
Tuesday, Nov. 20	Generative models 3	14.4-14.5
Thursday, Nov. 22	No class			Thanksgiving
Tuesday, Nov. 27	Percolation	Chapter 16
Thursday, Nov. 29	Epidemics on networks	17.1-17.8	Homework 9	Homework 8 due, Homework 9 handed out
Tuesday, Dec. 4	Network dynamics	Chapter 18
Thursday, Dec. 6	Network search	Chapter 19		Homework 9 due
Tuesday, Dec. 11	Mid-term 2			In class, usual time and place