In the late 1990s a number of researchers noticed that networks in biology, sociology, and telecommunications exhibited similar characteristics unlike standard random networks. In particular, researchers found that the cummulative degree distributions of these graphs followed a power law rather than a binomial distribution and that their clustering coefficients tended to a nonzero constant as the number of nodes, n, became large rather than O(1/n). Moreover, these networks shared an important property with traditional random graphs---as n becomes large the average shortest path length scaled with log n. This latter property has been coined the small-world property. When taken together these three properties---small-world, power law, and constant clustering coefficient---describe what are now most commonly referred to as scale-free networks.

TOPIC OUTLINE

My plan for the course is to cover roughly two chapters of the book LINKED each week. You will be given discussion questions for each chapter and will be expected to actively participate in classroom discussion. Coincident with these chapters we will cover basic graph theory, chapters in Caldarelli's book, and portions of Newman's SIAM Review article. There will be group research projects in lieu of a final exam.

1. week 1: Ch. 1-2 of Linked; 1-3,1-4,1-5 (G. Chartrand)
2. week 2: Ch. 3-4 of Linked; 1-6,2-1,2-2 (G. Chartrand)
3. week 3: Ch. 5-6 of Linked; 2-3,2-4 (G.C.) 1 (Caldarelli)
4. week 4: Ch. 7-8 of Linked; 3-1,3-2 (G.C.) 1 (Caldarelli)
5. week 5: Ch. 9-10 of Linked; 2 (Caldarelli) 1-2 (Newman)
6. week 6: Ch. 11-12 of Linked; 3 (Caldarelli) 3-4 (Newman)
7. week 7: Ch. 13-14 of Linked; 4 (Caldarelli) 5-6 (Newman)
8. week 8: Ch. 15-16 of Linked; 5 (Caldarelli) 7-8 (Newman)
9. Exam
10. week 9: Paper by LATDW.
11. week 10: Paper by LATDW. Examples (Caldarelli)
12. week 11: Executive Summary of current papers. Examples (Caldarelli)
13. week 12: Executive Summary of current papers. Examples (Caldarelli)
14. week 13: Group Project meetings with RKK
15. week 14: Project presentations.

REFERENCE TEXTS.

1. Barabasi, A-L., Linked: The New Science of Networks, Perseus Publishing, Cambridge, Massahusetts, 2002.
2. Newman, M.E.J., ``The Structure and Function of Complex Networks,'' (2003) SIAM Review, Vol. 15, No. 2, pp. 167-256.
3. Caldarelli, G., Scale-Free Networks, Oxford University Press, 2007.
4. Chartrand, G., Introductory Graph Theory, Dover Publications Inc., New York, 1977. (Chapters 1-3).
5. Bollobas, B., Random Graphs, Academic Press, New York, 1985.
6. Palmer, E.M., Graphical Evolution: An Introduction to the Theory of Random Graphs, John Wiley and Sons, New York, 1985.
7. Li, L., D. Alderson, R. Tanaka, J.C. Doyle and W. Willinger, ``Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)'', Technical Report CIT-CDS-04-006, Cal Tech, 2005. (arXiv:cond-mat/0501169 v1 9 Jan 2005)