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Networks

Networks. Complex Networks. FIAS Summer School 6th August 2008 raulvicente@mpih-frankfurt.mpg.de. 1. Overview. Introduction Three structural metrics Four structural models Structural case studies Node dynamics and self-organization Visualization Bibliography. 2. Introduction.

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Networks

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  1. Networks Complex Networks FIAS Summer School 6th August 2008 raulvicente@mpih-frankfurt.mpg.de 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? ● Acomplex network is a network with non-trivial topological features (features that do not occur in simple networks such as lattices or random graphs) • degree dist. • clustering • assortativity • comunity • hierarchical struct. Lattice Random ● Natural complex systems often exhibit such topologies. 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 * Measures how quickly info can flow through the network 13

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

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

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

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