124.11.lec19

124.11.lec19 - CS 124/LINGUIST 180: From Languages to...

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Dan Jurafsky Lecture 19: Networks part II: Small Worlds, Fat Tails, and Weak Ties CS 124/LINGUIST 180: From Languages to Information Slides mainly from Lada Adanic, some also from James Moody
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Outline Sketch of some real networks Fat Tails Small Worlds Weak Ties
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High school dating Peter S. Bearman, James Moody and Katherine Stovel Chains of affection: The structure of adolescent romantic and sexual networks American Journal of Sociology 110 44-91 (2004) Image drawn by Mark Newman Slide from Drago Radev
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More networks hBp://oracleoDacon.org/
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5 million edges Graph Colors: Asia Pacific – Red Europe/Middle East/ Central Asia/Africa – Green North America – Blue Latin American and Caribbean – Yellow RFC1918 IP Addresses – Cyan Unknown - White http://www.opte.org/maps/ Slide from Drago Radev
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Degree of nodes Many nodes on the internet have low degree One or two connecJons 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 −α
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What is a heavy tailed-distribution? Unlike a normal distribuJon, power-law distribuJons have no “scale” (no characterisJc value) Right skew normal distribuJon (not heavy tailed) e.g. heights of human males: centered around 180cm (5’11’’) Zipf’s or power-law distribuJon (heavy tailed) e.g. city populaJon sizes: NYC 8 million, but many, many small towns High raJo of max to min human heights tallest man: 272cm (8’11”), shortest man: (1’10”) ra#o: 4.8 from the Guinness Book of world records city sizes NYC: pop. 8 million, Duffield, Virginia pop. 52, ra#o: 150,000 Slide from Lada Adamic
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Normal (Gaussian) distribution of human heights Slide from Lada Adamic average value close to most typical distribution close to symmetric around average value
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Power-law distribution linear scale Slide from Lada Adamic log-log scale high skew (asymmetry) straight line on a log-log plot
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Logarithmic axes powers of a number will be uniformly spaced Slide from Lada Adamic 1 2 3 10 20 30 100 200 2 0 =1, 2 1 =2, 2 2 =4, 2 3 =8, 2 4 =16, 2 5 =32, 2 6 =64,….
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Power laws are seemingly everywhere note: these are cumulative distributions Slide from Lada Adamic Moby Dick scientific papers 1981-1997 AOL users visiting sites ‘97 bestsellers 1895-1965 AT&T customers on 1 day California 1910-1992
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Yet more power laws Slide from Lada Adamic Moon Solar flares wars (1816-1980) richest individuals 2003 US family names 1990 US cities 2003
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Power law distribution Straight line on a log-log plot ExponenJate both sides to get that p(x) , the probability of observing an item of size ‘x’ is given by Slide from Lada Adamic normalization constant (probabilities over all x must sum to 1) power law exponent α
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