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Long Tails and Navigation Networked Life CIS 112 Spring 2010 Prof. Michael Kearns
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One More ( Structural ) Property… A pro pe rly tune d  α -m o de l c an  simultaneously  explain small diameter high clustering coefficient other models can, too (e.g. cycle+random rewirings) But what about connectors and heavy-tailed degree distributions?   α -model and simple variants will  not  explain this intuitively, no “bias” towards large degree evolves “all vertices are created equal” As usual, we want a “natural” model to explain this
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Quantifying Connectors: Heavy-Tailed Distributions
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Heavy-tailed Distributions Pareto  or  power law  distributions:  for random variables assuming integer values > 0 probability of value x ~ 1/x^ α typically 0 <  α  < 2; smaller  α gives heavier tail here are some  examples sometimes also referred to as being  scale-free For binomial, normal, and Poisson distributions the tail probabilities approach 0  exponentially  fast  Inverse  polynomial  decay vs.  inverse  exponential decay What kind of phenomena does this distribution model? What kind of process would  generate  it?
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Distributions vs. Data All these distributions are  idealized models In practice, we do not see distributions, but  data Thus, there will be some  largest  value we observe Also, can be difficult to “eyeball” data and choose model So how do we distinguish between Poisson, power law, etc? Typical procedure: might restrict our attention to a  range  of values of interest accumulate  counts  of observed data into equal-sized bins look at counts on a  log-log plot note that power law:  log(Pr[value = x]) = log(1/x^ α ) = - α  log(x)  linear, slope – α Normal/Gaussian: log(Pr[value = x]) = log(a exp(-x^2/b)) = log(a) – x^2/b non-linear, concave near mean  Poisson:  log(Pr[value = x]) = log(exp(- λ λ ^x/x!)  also non-linear Let’s look at the paper on  dollar bill migration
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Heavy Tails Recap We plot the distribution or histogram of some “resource” on the x-axis, we put the amount or quantity of this resource (e.g. degrees)
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