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  1. Topology and Dynamics ofComplex Networks FRES1010 Complex Adaptive Systems Eileen Kraemer Fall 2005

  2. Based on … • Strogatz (2001), • Barabási & Bonabeau (2003), • http://powerlaws.media.mit.edu/papers/barabasi03.pdf • Wang, X. F. (2002)

  3. Topology and Dynamics ofComplex Networks • Introduction • Three structural metrics • Four structural models • Structural case studies • Node dynamics and self-organization • Bibliography

  4. Introduction • Examples of complex networks • Elementary features • Motivations

  5. Examples of complex networks: geometric, regular

  6. Examples of complex networks: semi-geometric, irregular

  7. Elementary features:node diversity and dynamics

  8. Elementary features:edge diversity and dynamics

  9. Elementary features:Network Evolution

  10. Motivations • complex networks are the backbone of complex systems • every complex system is a network of interaction among numerous smaller elements • some networks are geometric or regular in 2-D or 3-D space • other contain “long-range” connections or are not spatial at all • understanding a complex system = break down into parts + reassemble • network anatomy is important to characterize because structure affects function (and vice-versa) • ex: structure of social networks • prevent spread of diseases • control spread of information (marketing, fads, rumors, etc.) • ex: structure of power grid / Internet • understand robustness and stability of power / data transmission

  11. Three structural metrics • Average path length • Degree distribution(connectivity) • Clustering coefficient

  12. Structural metrics: Average path length

  13. Structural Metrics:Degree distribution(connectivity)

  14. Structural Metrics:Clustering coefficient

  15. Four structural models • Regular networks • Random networks • Small-world networks • Scale-free networks

  16. Regular networks –fully connected

  17. Regular networks –Lattice

  18. Regular networks –Lattice: ring world

  19. Random networks

  20. Random Networks

  21. Small-world networks

  22. Small-world networks

  23. Small-world networks

  24. Small-world networks

  25. Scale-free networks

  26. Scale-free networks

  27. Scale-free networks

  28. Scale-free networks

  29. Scale-free networks

  30. Scale-free networks

  31. Case studies • Internet • World Wide Web • Actors & scientists • Neural networks • Cellular metabolism

  32. The Internet

  33. The Internet

  34. The Internet

  35. The World Wide Web

  36. World Wide Web

  37. World Wide Web

  38. Actors

  39. Mathematicians &Computer Scientists

  40. Node dynamics and self-organization • Node dynamics • Attractors in full & lattice networks • Synchronization in full networks • Waves in lattice networks • Epidemics in complex networks

  41. Node dynamics: individual node

  42. Node dynamics:coupled nodes

  43. Node dynamics and self-organization

  44. Node dynamics and self-organization

  45. Node dynamics and self-organization

  46. Node dynamics and self-organization

  47. Node dynamics and self-organization

  48. Node dynamics and self-organization:Epidemics in complex networks

  49. Node dynamics and self-organization:Epidemics in complex networks

  50. Bibliography • Reviews • Barabási, A.-L. (2002) Linked: The New Science of Networks.Perseus Books. • Barabási, A.-L. and Bonabeau, E. (2003) Scale-free networks. Scientific American, 288: 60-69. • Strogatz, S. H. (2001) Exploring complex networks. Nature, 410(6825): 268-276. • Wang, X. F. (2002) Complex networks: topology, dynamics and synchronization. International Journal of Bifurcation and Chaos, 12(5): 885-916.