objects/preferential-attach-zipf.html

Preferential Attachment and Zipf's Law

general problem

• cities need to connect to each other
• parts of a network need to make some connections

  

mechanism of growth and development through "preferential attachment"

• new nodes added to a network
  • metaphor - airline traffic through hubs
  • attach (with high probability) to nodes that already have high flows
  • or might attach to the nearest node (in some constraint space)
  • result is a fractal network
  • resilient to many types of stresses
  • can fail with rare impact - removal of a major hub
• nodes within a network are disturbed and realigned
  • range of disturbance levels
  • authentic disturbances that relate to environmental parameters and network function
  • you can train medium sized networks
    • to respond to authentic stresses ahead of time (preparedness drills)
    • to respond to phony stresses (poor administration)
    • to suit your own purposes ("breaking in a help desk")

usually results in a "Zipf's Law" relationship

• log of the frequency of cities of a particular size vs. log of the population size
  • there is a high frequency of little cities
  • there are fewer medium cities
  • a large city is rare

  

This relationship suggests a self-organizing process

• other processes that show a similar relationship
  • landslides on a sandpile
  • earthquake size vs. frequency
  • word frequency vs. rank in your text book
• because these other processes have been studied and shown to have self-organizing characteristic it suggests that city size is also self-orgainizing