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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

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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!

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

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Random Graphs (Erdõs- Rényi )

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

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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

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Measured Network Values

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Comparison

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

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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.

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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

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

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

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

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Finding P(k)

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Finally……. where And for fixed p,m:

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Regimes As q -> qmax, distribution gets exponential.

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Simulation Results

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

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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

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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

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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.

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