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Scale Free Networks
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  1. Scale Free Networks Robin Coope April 4 2003 Abert-László Barabási, Linked (Perseus, Cambridge, 2002). Réka Albert and AL Barabási,Statistical Mechanics of Complex Networks, Rev. Mod. Phys 74 (1) 2002 Réka Albert and AL Barabási, Topology of Evolving Networks: Local Events and Universality, Phys. Rev. Lett. 85 (24) 2000

  2. Motivation • Many networks, (www links, biochemical & social networks) show P(k) ~ k- scale free behaviour. • Classical theories predict P(k) ~ exp(-k). • Something must be done!

  3. Properties of Networks • Small World Property • Clustering – “Grade Seven Factor” • Degree – Distribution of # of links

  4. Random Graphs (Erdõs-Rényi )

  5. Predictions of Random Graphs Path Length vs. Theory Clustering vs. Theory

  6. What About Scale Free Random Graphs? • Restrict distributions to P(k) ~ k- • Still doesn’t make good predictions • Conclusion: Network connections are not random! Average Path Length

  7. Measured Network Values

  8. Measured Network Values

  9. Comparison

  10. 2 2 2 7 4 5 2 3 2 7 Nancy Kerrigan ~ 1 link 2 Charleton Heston > 150 links Evolution of a SF Network

  11. Assumptions for Scale Free Model • Networks are open – they add and lose nodes, and nodes can be rewired. • Older nodes get more new links. • More popular nodes get more new links • Result: no characteristic nodes – Scale Free • Both growth and rewiring required.

  12. 1. Addition of m new links with prob. p 2. Rewiring of m links with prob. q 3. Add a new node with prob. (1-p-q) Continuum Theory Avoid isolated links

  13. Combined Equation Time Dependency of system size and # of links Initial Condition for connectivity of a node added at time ti:

  14. Solution YOU MANIACS! YOU BLEW IT UP! DAMN YOU! GOD DAMN YOU ALL TO HELL!!

  15. Finding P(k) Can get analytic solution for P(k) if:

  16. Finding P(k)

  17. Finally……. where And for fixed p,m:

  18. Regimes As q -> qmax, distribution gets exponential.

  19. Simulation Results

  20. Experimental Results 93.7% new links for current actors 6.3% new actors

  21. Implications – Attack Tolerance • Robust. For <3, removing nodes does not break network into islands. • Very resistant to random attacks, but attacks targeting key nodes are more dangerous. Max Cluster Size Path Length

  22. Implications • Infections will find connected nodes. • Cascading node failures a problem • Treatment with novel strategies like targeting nodes for treatment - AIDS • Protein hubs critical for cells 60-70% • Biological complexity: # states ~2# of genes

  23. Conclusion • Real world networks show both power law and exponential behaviour. • A model based on a growing network with preferential attachment of new links can describe both regimes. • Scale free networks have important implications for numerous systems.