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
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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 doesnt make good predictions
• Conclusion Network connections are not random!

Average Path Length
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Measured Network Values
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Measured Network Values
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Comparison
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Evolution of a SF Network
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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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Continuum Theory
Avoid isolated links
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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 -gt 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 ?lt3, 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.