Math 76: Topics in Applied Mathematics -- An Introduction to Complex Systems - Fall 2008

Professor D. Rockmore, Kemeny 233, tel 6-3260, email: rockmore at

A picture of the "network" of the NYSE, the epitome of a complex system.

(From "Topological Structures in the Equities Market Network," G. Leibon, S. Pauls, D. Rockmore, and R. Savell, PNAS, December 30, 2008 )

The phrase "complex adaptive systems" refers to a very broad area of research meant to encompass a quantitative approach to the study of "evolved" systems. What these systems generally share are the idea that they can begin to be described as a collection of simple interactions (i.e., rules) between basic elements, but that out of this simple description, something quite complex and surprising may emerge. Examples include markets and economies that arise out of the fundamental interaction between "buyers" and "sellers," the phenomena of thought and consciousness that (presumably) derive from well-understood neuron-to-neuron signaling, ecosystems that arise out a multitude of predator-prey relationships. In addition to potentially simple pairwise dynamics, these systems also share the ability to adapt to changing circumstances - hence the use of the word adaptive. Of course, these phenomena share much more than these simple properties including a multiscale nature, hysteresis, multiple equilibria, network characteristics. The list goes on and on. As the description suggests, there are some basic principles and tools that we can apply to begin to understand these various phenomena. In this class we'll touch on some of these, focusing on the study of networks, but also using and introducing tools from agent-based modeling (ABM), statistical learning, geometric analysis, as well as other disciplines as we see fit. Related to this are aspects of data visualization and analysis (e.g., multidimensional scaling, spectral graph drawing, spectral partitioning, spectral clustering). Other topics may be included as time permits.

Prerequisites include linear algebra (Math 22/24) and some knowledge of probability (Math 20), and general mathematical maturity (we will prove a fact or two). You should also have a reasonable comfort level with doing a little bit of computing, say with Matlab or Mathematica. Some knowledge of basic statistics might be helpful.

Lectures: Kemeny 007, TTh 10:00am -- 11:50am (period 10A), X-hour is 3:00--3:50pm Wed., and I imagine it will be used about half the time. Do not schedule anything regular in this X-hr. I encourage you to come to office hours: M,T 2-3 pm (or by appt.)

  • Materials -- Complex systems is an evolving subject and new enough that there is no standard book. However, there are some materials available on the web as well as some good books on some aspects of modeling.

    Good notes : Professor P. Dodds (UVM) has developed an excellent set of lectures on networks and complex systems and is happy to share them with us. We'll be making good use of them.

    Required book: Complex Adaptive Systems: An Introduction to Computational Models of Social Life, by John H. Miller and Scott E.Page Published by Princeton University Press.

    Required book: Micromotives and Macrobehavior, by Thomas C. Schelling. (Note: it is available at many venues)

    Recommended book: Social and Economic Networks, by Matthew Jackson. Published by Princeton University Press. New text, encyclopedic, good reference.

    Some useful online resources (more to be found on BBD).

  • Materials on BBD
  • Terrific online probability book: Introduction to Probability, by C. Grinstead and J. L. Snell
  • Excellent online R-based statistics book: Introduction to Statistical Thought, by M. Lavine
  • Social networks information
  • Network visualization tools:

    --- yEd network software for any platform (Windows, Linux, Mac)

    --- Pajek network software for Windows (also runs on Linux or Mac using Wine or SoftPC)

    --- GraphViz network software for any platform (Windows, Linux, Mac)

    Honor principle. Homework: group discussion and collaboration on problem techniques is great and helpful. Write-ups/papers must be done individually (i.e., no copying). However, there is the possibility of collaborative work on the final projects and this would need to be approved.

    Electronic Etiquette. When in class, you are expected to be doing classwork. Thus, ABSOLUTELY no websurfing, no emailing, no texting, no phone calls during class.

    Grades: Will be based on Class Participation (you are expected to attend class and participate) 20%, HW 30%, one short presentation (ten minute presentation of an interesting technical paper for class discussion) 10%, one related writing assignment 10% and a final project 30%.

    Special needs: I encourage students with disabilities, including "invisible" disabilities like chronic diseases and learning disabilities, to discuss with us any appropriate accommodations that might be helpful. The official College policy statement is as follows: "Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see me, ideally, before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested." So, let me know asap, certainly in first 2 weeks. Also stop by the Academic Skills Center in 301 Collis to register for support services.