1 / 27

Scale - free networks

Scale - free networks. Péter Kómár Statistical physics seminar 07/10/2008. Non-regular graph. Elements of graph theory I. A graph consists: vertices edges Edges can be: directed/undirected weighted/non-weighted self loops multiple edges. Elements of graph theory II.

gauri
Download Presentation

Scale - free networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

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

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

  8. actors www 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

  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. pER = 6∙10-4 N=104 10-3 1.5∙10-3 Erdős-Rényi graph (ER) (1960. P. Erdős, A. Rényi) • Construction: • N vertices • probability of each edge: pER • Properties: • pER ≥ 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

  11. ER graph example

  12. Small World graph (WS) (1998. D.J. Watts, S.H. Strogatz) • Construction: • N vertices in sequence • 1st and 2nd neighbor edges • rewiring probability: pWS • Properties: • pWS = 0 → clustered, • 0 < pWS < 0.01 → clustered → small-world propery • pWS = 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

  13. WS graph example

  14. ER graph - WS graph WS ER

  15. m0= 3 m = 2 Barabási-Albert graph (BA) • New aspects: • Continuous growing • Preferential attachment • Construction: • m0 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

  16. 7 5 3 1 = m0 =m N = 300 000 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

  17. BA graph example

  18. ER graph – BA graph

  19. ki(t) ~ t 1/2 t ti Mean-field approximation I. • Time dependence of ki (continuous): • solution: probability of anedge to ith vertex time of occurrence of the ith vertex A.-L. Barabási, R. Albert, H. Jeong (1999) ‘Mean-field theory for scale-free random networks’, Physica A, Vol. 272 p.173-187

  20. Mean-field approximation II. • Distribution of degrees: • Distribution of ti : • 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. p(k) exponential scale-free k 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

  22. t = N N=10 000 5N 40N Without growth • Construction: • Constant # of vertices • + new edges with preferential attachment • Properties: • At early stages → power-law scaling • After t ≈ N2 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

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

  25. Thank you for the attention!

  26. ER – WS – BA

More Related