1. Networks FIAS Summer School 6th August 2008 raulvicente@mpih-frankfurt.mpg.de Complex Networks 1

2. Overview Introduction Three structural metrics Four structural models Structural case studies Node dynamics and self-organization Visualization Bibliography 2

3. Introduction What is a network? What is a complex network? Networks in the real world Elementary features Motivations 3

4. What is a network? ● A network is a set of items (vertices or nodes) with connections between them called edges. Mathematicians call them “graphs”. ● Need not to be physical connections: nodes can be any type of entities and edges any type of abstract relationships. ● Ex.:nodes can be the channels of any multirecording device (EEG, MEG, multielectrode arrays, etc...) while edges can be defined by the relationship (are two channels synchronous or not?). 4

5. What is a network? ● Edges can be undirected or directed (arcs). ● Graphs can allow (friendship networks) or disallow loops (citation networks), parallel edges,... ● Different types of networks: different types of vertices or edges, weighted networks, digraphs, bipartite graphs, evolving networks,... 5

6. What is a complex network? ● A complex network is a network with non-trivial topological features (features that do not occur in simple networks such as lattices or random graphs) LatticeRandom ● Natural complex systems often exhibit such topologies. degree dist. clustering assortativity comunity hierarchical struct. 6

7. Networks in the real world: examples of complex networks Social, information, technological, biological,... 7

8. Elementary features: node diversity and dynamics 8

9. Elementary features: edge diversity and dynamics 9

10. Elementary features: Network Evolution 10

11. 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

12. Three structural metrics Average path length Degree distribution (connectivity) Clustering coefficient 12

13. Structural metrics: Average path length 13 * Measures how quickly info can flow through the network

14. Structural Metrics: Degree distribution (connectivity) 14 * Divided in ‘in-degree’ and ‘out-degree’ for directed systems * High-degree nodes → ‘hubs’

15. Structural Metrics: Clustering coefficient 15 * How likely is that the friend of your friend is also your friend?

16. Four structural models Regular networks Random networks Small-world networks Scale-free networks 16

17. Regular networks – fully connected 17

18. Regular networks – Lattice 18

19. Regular networks – Lattice: ring world 19

20. Random networks 20

21. Random Networks 21

22. Small-world networks 22

23. Small-world networks 23

24. Small-world networks 24

25. Small-world networks 25

26. Scale-free networks 26

27. Scale-free networks 27

28. Scale-free networks 28

29. Scale-free networks 29

30. Scale-free networks 30

31. Case studies Internet World Wide Web Actors & scientists 31

32. The Internet 32

33. The Internet 33

34. The Internet 34

35. The World Wide Web 35

36. World Wide Web 36

37. World Wide Web 37

38. Actors 38

39. Mathematicians & Computer Scientists 39

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 40

41. Node dynamics: individual node 41

42. Node dynamics: coupled nodes 42

43. Node dynamics and self-organization 43

44. Node dynamics and self-organization 44

45. Node dynamics and self-organization 45

46. Node dynamics and self-organization 46

47. Node dynamics and self-organization 47

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

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

50. Visualization & analysis 50 http://vlado.fmf.uni-lj.si/pub/networks/pajek/ ● Program for large networks analysis : Pajek ● Free ● Windows (on Linux too but not so smooth) *Vertices 3 1 “Source” 2 “Sink” 3 “Destination” *Arcs *Edges 1 2 1 2 3 1

51. Based on… Eileen Kramer & Kai Willadsen 51

52. 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.Scale-free networks  Strogatz, S. H. (2001) Exploring complex networks. Nature, 410(6825): 268-276.Exploring complex networks  Wang, X. F. (2002) Complex networks: topology, dynamics and synchronization. International Journal of Bifurcation and Chaos, 12(5): 885-916.  Newman M. E. J. (2003) The structure and function of complex networks. arXiv:cond-mat/0303516v1 52