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##### Scale Free Networks

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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**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!**Properties of Networks**• Small World Property • Clustering – “Grade Seven Factor” • Degree – Distribution of # of links**Predictions of Random Graphs**Path Length vs. Theory Clustering vs. Theory**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**2**2 2 7 4 5 2 3 2 7 Nancy Kerrigan ~ 1 link 2 Charleton Heston > 150 links Evolution of a SF Network**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.**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**Combined Equation**Time Dependency of system size and # of links Initial Condition for connectivity of a node added at time ti:**Solution**YOU MANIACS! YOU BLEW IT UP! DAMN YOU! GOD DAMN YOU ALL TO HELL!!**Finding P(k)**Can get analytic solution for P(k) if:**Finally…….**where And for fixed p,m:**Regimes**As q -> qmax, distribution gets exponential.**Experimental Results**93.7% new links for current actors 6.3% new actors**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**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**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.