Dynamics of Complex Networks I: Networks II: Percolation

1. Dynamics of Complex NetworksI: NetworksII: Percolation Panos Argyrakis Department of Physics University of Thessaloniki

2. Characteristics of networks: Structures that are formed by two distinct entities: Nodes Connections, synapses, edges N= 5 nodes, n=4 connections They can be simple structures or very complicated (belonging to the class of complex systems) How did it all start?

3. Complexity Main Entry: 1com·plexFunction: nounEtymology: Late Latin complexus totality, from Latin, embrace, from complectiDate: 16431:a whole made up of complicated or interrelated parts A popular paradigm: Simple systems display complex behavior  non-linear systems  chaos  fractals 3 Body Problem Earth( ) Jupiter ( ) Sun ( ) What is Complexity?

6. Image of the Internet at the AS level Scale-free P(k) ~ k-γ

7. Network Structure

8. 1970 – 10 hosts 1990 – 1.75*105 hosts Now – 1.2*109 hosts 1990 – 1 web site 1996 – 105 web sites Now – 109 web sites Internet and WWW explosive growth

9. Internet and WWW are characteristic complex networks routers nodes domain (AS) Internet communication lines edges nodes web-sites WWW hyperlinks (URL) edges

10. Connectivity between nodes in the Internet • How are the different nodes in the Internet connected? • No real top-down approach ever existed • Has been characterized by a large degree of randomness • What is the probability distribution function (PDF) of connectivities of all nodes? • Naïve approach: Normal distribution (Gaussian) • Experimental result? • Faloutsos, Faloutsos, and Faloutsos, 1997

11. Assuming random connections Degree distribution, P(k): Probability that a node has k links (connections) with other nodes

12. Gaussian distribution

13. Probability distribution P(k) of the No. of connections(degree distribution)

14. INTERNET BACKBONE Nodes: computers, routers Links: physical lines P(k) k=number of connections of a node P(k)= the distribution of k k (Faloutsos, Faloutsos and Faloutsos, 1997) P(k) ~ k-g

15. Assuming random connections

16. Degree distributionin a scale-free network P(k) ~ k -g

17. What did we expect? k ~ 6 P(k=500) ~ 10-99 NWWW ~ 109  N(k=500)~10-90 We find: out= 2.45 in = 2.1 P(k=500) ~ 10-6 NWWW ~ 109  N(k=500) ~ 103 Pout(k) ~ k-out Pin(k) ~ k- in

18. Netwotk with a power law(scale-free network)P(k) ~ k-g

19. World Wide Web Nodes: WWW documents Links: URL links 800 million documents (S. Lawrence, 1999) ROBOT:collects all URL’s found in a document and follows them recursively R. Albert, H. Jeong, A-L Barabasi, Nature, 401 130 (1999)

20. WWW It has apower-law Diameter ofWWW: <d>=0.35 + 2.06 logN For: Ν=8x108<d>=18.59 [compare to a square of same size: edge length=30000]

21. Communication networks The Earth is developing an electronic nervous system, a network with diverse nodes and links are -computers -routers -satellites -phone lines -TV cables -EM waves Communication networks: Many non-identical components with diverseconnections between them.

22. Coauthorship SCIENCE COAUTHORSHIP Nodes: scientist (authors) Links: write paper together (Newman, 2000, H. Jeong et al 2001)

23. Citation 25 2212 SCIENCE CITATION INDEX Nodes: papers Links: citations Witten-Sander PRL 1981 1736 PRL papers (1988) P(k) ~k- ( = 3) (S. Redner, 1998)

24. Actors ACTOR CONNECTIVITIES Nodes: actors Links: cast jointly Days of Thunder (1990) Far and Away (1992) Eyes Wide Shut (1999) N = 212,250 actors k = 28.78 P(k) ~k- =2.3

25. Sex-web Nodes: people (Females; Males) Links: sexual relationships 4781 Swedes; 18-74; 59% response rate. Liljeros et al. Nature 2001

26. Yeast protein network Nodes: proteins Links: physical interactions (binding) P. Uetz, et al.Nature 403, 623-7 (2000).

27. What does it mean? Poisson distribution Power-law distribution Exponential Network Scale-free Network

28. Network Backbone at University of Thessaloniki

29. Connections with probability p p=1/6 N=10 k ~ 1.5 Poisson distribution The model ofErdös-Rényi(1960) Pál Erdös(1913-1996) - Democratic - Random

30. Problem: • construct an Erdos-Renyi Network • use N=1000 • use p=0.1666 • draw a random number (rn) from a uniform distribution • if rn<0.1666 link exists, otherwise it does not • find the complete distribution of links for all N=1000 nodes • Construct the P(k) vs. k diagram

31. TheErdos-Renyi model p=6x10-4 p=10-3

32. Problem: Scale free networks • need a random number generator that produces random numbers with a power-law distribution • P(k)~ k-g • not so simple distribution • check book “Numerical Recipes” by Press et al • construction of network is more cumbersome • can do node-by-node….or • can do link-by-link • many more methods, e.g. thermalization and annealing of links

33. Fill node-by-node 3 2 1 1 ? 3 1 1 5 3 2 2 2 1 1 2 2 1 1 1 1 k k-connectivity node, completed with k links k k-connectivity node, with missing links

34. Link-to-link 3 3 2 2 1 1 1 1 5 5 3 3 2 2 2 2 1 1 2 2 1 1 1 1 k k-connectivity node, completed with k links k k-connectivity node, with missing links

35. Regular network

36. Small world network

37. TheWatts-Strogatz model C(p) : clustering coeff. L(p) : average path length (Watts and Strogatz, Nature 393, 440 (1998))

38. Small world network

39. Small world network

40. Small-world network

41. Most real networks have similar internal structure: Scale-free networks Why? What does it mean?

42. Origins SF (2) The additions are not uniform A node that already has a large number of connections is connected with larger probability than another node. Example: WWW : new topics usually go to well-known sites (CNN, YAHOO, NewYork Times, etc) Citation : papers that have a large number of references are more probable to be refered again Scale-free networks (1) The number of nodes is not pre-determined Τhe networks continuously expand with the addition of new nodes Example: WWW : addition of new pages Citation : publication of new articles

43. BA model Scale-free networks P(k) ~k-3 (1)Growth:At every moment we add a new node withm connections (which is added to the already existing nodes). (2)Preferential Attachment:The probability Π that a new node will be connected to node i dependson the number ki , the number of connections of this node A.-L.Barabási, R. Albert, Science 286, 509 (1999)

44. Network categories: • (1) Random network (Erdos-Renyi) • (2) Network on a regular lattice • (2) Small world network (Strogatz-Watts) • (3) Network with a power-law

45. Society Nodes: individuals Links: social relationship (family/work/friendship/etc.) S. Milgram (1967) John Guare How many (n) connectionsare needed so that an individual is connected with any other person in the world? N=6 billion people Result: n~6 Conclusion:We live in a small world Six Degrees of Separation!!

46. New books:

48. Facebook: • 900,000,000 registered users • 50,000,000 active users • 5,000,000 generate 95% of the traffic • Questions needing answers: • How many people communicate ? • How many connections does one have? • How often does he communicate? • How long does it last?

49. Real-world phenomena related to communication patterns • Crowd behavior: strategies to evacuate people and stop panic. • Search strategies: efficient networks for searching objects and people. • Traffic flow: optimization of collective flow. • Dynamics of collaboration: human relationship networks such as collaboration, opinion propagation and email networks. • Spread of epidemics: efficient immunization strategies. • Patterns in economics and finance: dynamic patterns in other disciplines, such as Economics and Finance, and Environmental networks.

50. Where is George?