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Unformatted text preview: Click to edit Master subtitle style Dan Jurafsky Lecture 19: Networks part II: Small Worlds, Fat Tails, and Weak Ties Slides mainly from Lada Adanic, some also from James Moody Slide from Chris Manning Outline Sketch of some real networks Fat Tails Small Worlds Weak Ties Slide from Chris Manning High school dating Peter S. Bearman, James Moody and Katherine Stovel Chains of affection: The structure of adolescent romantic and sexual netw American Journal of Sociology 110 4491 (2004) Image drawn by Mark Newman Slide from Drago Radev Slide from Chris Manning Slide from Chris Manning More networks http://oracleofbacon.org/ Slide from Chris Manning 5 million edges Graph Colors: Asia Pacific Red Europe/Middl e East/Central Asia/Africa Green North America http://www.opte.org/m aps/ Slide from Drago Radev Slide from Chris Manning Degree of nodes Many nodes on the internet have low degree One or two connections A few (hubs) have very high degree The number P(k) of nodes with degree k follows a power law: Where alpha for the internet is about 2.1 P ( k ) k a Slide from Chris Manning What is a heavy tailed Unlike a normal distribution, powerlaw distributions have no scale (no characteristic value) Right skew normal distribution (not heavy tailed) e.g. heights of human males: centered around 180cm (511) Zipfs or powerlaw distribution (heavy tailed) e.g. city population sizes: NYC 8 million, but many, many small towns High ratio of max to min human heights tallest man: 272cm (811), shortest man: (110) ratio: 4.8 from the Guinness Book of world records city sizes Slide from Lada Adamic Slide from Chris Manning Normal (Gaussian) distribution Slide from Lada Adamic average value close to most typical distribution close to symmetric around average value Slide from Chris Manning Powerlaw distribution linear scale Slide from Lada Adamic n loglog scale n high skew (asymmetry) n straight line on a loglog plot Slide from Chris Manning Logarithmic axes powers of a number will be uniformly spaced Slide from Lada Adamic 1 2 3 1 2 3 1 2 n 20= 1, 21= 2, 22= 4, 23= 8, 24= 16, 25= 32, 26= 64,. Slide from Chris Manning Power laws are seemingly everywhere note: these are cumulative distributions Slide from Lada Adamic Moby Dick scientific papers 19811997 AOL users visiting sites 97 bestsellers AT&T California 1910 Slide from Chris Manning Yet more power laws Slide from Lada Adamic Mo on Solar flares wars (1816 1980) richest US family US cities Slide from Chris Manning Power law distribution Straight line on a loglog plot Exponentiate both sides to get that p(x) , the probability of observing an item of size x is given by Slide from Lada Adamic  = Cx x p ) ( ) ln( )) ( ln( x c x p  = normalization constant (probabilities over all x must sum to 1) power law exponent Slide from Chris Manning What does it mean to be scale free?What does it mean to be scale free?...
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 Winter '09

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