Explaining Power Laws by Trade-Offs

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Explaining Power Laws by
Trade-Offs

• Alex Fabrikant, Elias Koutsoupias,
  Milena Mihail, Christos Papadimitriou

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Powerlaws in the Internet

• Faloutsos3 1999 the degrees of the Internet
  topology are power law distributed
• Both autonomous systems graph and
  router graph
• Hop distances ditto
• Eigenvalues ditto (!??!)
• Model?

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The world according to Zipf

• Power laws, Zipfs law, heavy tails,
• the signature of human activity
• i-th largest is i-a (cities, words a 1)
• Equivalently probX greater than x x -b
• (compare with law of large numbers)

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Models predicting power laws

• Size-independent growth (the rich get richer)
• Preferential attachment
• Brownian motion in log
• Exponential arrival exponential growth
• Copying (web graph)
• Carlson and Doyle 1999 Highly optimized
  tolerance (HOT)

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Our model
minj lt i ? ? dij hopj
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hopj

• Average hop distance from other nodes
• Maximum hop distance from other nodes
• Distance from center (first node)
• NB Resulting graph is a tree

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Theorem

• if ? lt const, then graph is a star
• degree n -1
• if ? gt ?n, then there is exponential
  concentration of degrees
• prob(degree gt x) lt exp(-ax)
• otherwise, if const lt ? lt ?n, heavy tail
• prob(degree gt x) gt x -a

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Also why are files on the Internet power-law
distributed?

• Suppose each data item i has popularity ai
• Partition data items in files to minimize total
  cost
• Cost of each file
• total popularity size overhead C
• Notice trade-off!
• From CD99

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Files (continued)

• Suppose further that popularities of items are
  iid from distribution f
• Result File sizes are power law distributed for
  any reasonable distribution f (exponential,
  Gaussian, uniform, power law, etc.)
• (CD99 observe it for a few distributions)

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Heuristically optimized tradeoffs

• Power law distributions seem to also come from
  tradeoffs between objectives (a
  signature of human activity?)
• Generalizes CD99 (the other objective need not
  be reliability)
• cf Mandelbrot 1954 Power Laws in language
  are due to a tradeoff between information and
  communication costs

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PS Eigenvalues of the Internet may be a
corollary of the degrees phenomenon
Theorem If a graph has largest degrees d1,
d2,, dk and o(dk ) more edges, then with high
probability its largest eigenvalues are within
(1 o(1)) of ?d1, ?d2,, ?dk (NB The
eigenvalue exponent observed in Faloutsos3 is
about ½ of the degree exponent!)