1. Scale-free networks Péter Kómár Statistical physics seminar 07/10/2008

2. 2 Elements of graph theory I.  A graph consists:  vertices  edges  Edges can be:  directed/undirected  weighted/non-weighted  self loops  multiple edges Non-regular graph

3. 3 Elements of graph theory II.  Degree of a vertex:  the number of edges going in and/or out  Diameter of a graph:  distance between the farthest vertices  Density of a graph:  sparse  dense

4. 4 Networks around us I.  Internet:  routers  cables  WWW:  HTML pages  hyperlinks  Social networks:  people  social relationship

5. 5 Networks around us II.  Transportation systems:  stations / routes  routes / stations  Nervous system:  neurons  axons and dendrites  Biochemical pathways:  chemical substances  reactions

6. 6 Real networks  Properties:  Self-organized structure  Evolution in time (growing and varying)  Large number of vertices  Moderate density  Relatively small diameter (Small World phenomenon)  Highly centralized subnetworks

7. 7 Random networks  Measuring real networks:  Relevant state-parameters  Evolution in time  Creating models:  Analytical formulas  Growing phenomenon  Checking:  ‘Raising’ random networks  Measuring

8. 8 Scale-free property  1999. A.-L. Barabási, R. Albert  measured the vertex degree distribution → power-law tail:  movie actors:  www:  US power grid: A.-L. Barabási, R. Albert (1999) ‘Emergence of Scaling in Random Networks’, Science Vol. 286 actors www

9. 9 Small diameter  2000. A.-L. Barabási, R. Albert  measured the diameter of a HTML graph  325 729 documents, 1 469 680 links  found logarithmic dependence:  ‘small world’ A.-L. Barabási, R. Albert, H. Jeong (2000) ‘Scale-free characteristics of random networks: the topology of the wold-wide web’, Physica A, Vol. 281 p. 69-77

10. 10 Erdős-Rényi graph (ER)  Construction:  N vertices  probability of each edge: p ER  Properties:  p ER ≥ 1/N → → Asympt. connected  degree distribution: Poisson (short tail)  not centralized  small diameter A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 (1960. P. Erdős, A. Rényi) N=10 4 p ER = 6∙10 -4 10 -3 1.5 ∙ 10 -3

11. 11 ER graph example

12. 12 Small World graph (WS)  Construction:  N vertices in sequence  1 st and 2 nd neighbor edges  rewiring probability: p WS  Properties:  p WS = 0 → clustered,  0 < p WS < 0.01 → clustered → small-world propery  p WS = 1 → not clustered, A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 (1998. D.J. Watts, S.H. Strogatz)

13. 13 WS graph example

14. 14 ER graph - WS graph WSER

15. 15 Barabási-Albert graph (BA)  New aspects:  Continuous growing  Preferential attachment  Construction:  m 0 initial vertices  in every step: +1 vertex with m edges  P (edge to vertex i ) ~ degree of i A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 m 0 = 3 m = 2

16. 16 Barabási-Albert graph II.  Properties:  Power-law distribution of degrees:  Stationary scale-free state  Very high clustering  Small diameter A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 1 = m 0 = m 3 5 7 N = 300 000

17. 17 BA graph example

18. 18 ER graph – BA graph

19. 19 Mean-field approximation I.  Time dependence of k i (continuous):  solution: probability of an edge to i th vertex time of occurrence of the i th vertex A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 k i (t) t titititi ~ t 1/2

20. 20 Mean-field approximation II.  Distribution of degrees:  Distribution of t i :  Probability density: A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187

21. 21 Without preferential attachment  Uniform growth:  Exponential degree distribution: A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 p(k)p(k)p(k)p(k) k scale-free exponential

22. 22 Without growth  Construction:  Constant # of vertices  + new edges with preferential attachment  Properties:  At early stages → power-law scaling  After t ≈ N 2 steps → dense graph A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187 t = N 5N 40N N=10 000

23. 23 Conclusion  Power-law = Growth + Pref. Attach.  Varieties  Non-linear attachment probability: → affects the power-law scaling  Parallel adding of new edges →  Continuously adding edges (eg. actors) → may result complete graph  Continuous reconnecting (preferentially) → may result ripened state A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187

24. 24 Network research today A.-L. Barabási, R. Albert, ‘Statistical Mechanics of Complex Networks’, arXiv:cond-mat/0106096 v1 6 Jun 2001 Centrality Adjacency matrix Spectral density Attack tolerance

25. Thank you for the attention!

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27. 27 ER – WS – BA