Scale Free Networks Robin Coope April 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

Random Graphs (Erdõs- Rényi )

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

Measured Network Values

Comparison

Evolution of a SF Network Charleton Heston > 150 links Nancy Kerrigan ~ 1 link

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 t i :

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:

Finding P(k)

Finally……. where And for fixed p,m:

Regimes As q -> qmax, distribution gets exponential.

Simulation Results

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.