# Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA.

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Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

Blog Posts Week 2

Diseases and Genes Poster: Lalith Polepeddi About: Properties of a network Link: http://tinyurl.com/69e388/

Lecture Nine: Power laws and rich-get-richer phenomena

Numbers Your grades so far in this class The weight of an apple The temperature in Chicago on July 4 th

Numbers Your grades so far in this class. The weight of an apple. The temperature in Chicago on July 4 th. The height of a Dutch man. The speed of a car on I94. Most instances are typical. Seeing a rare number is very surprising.

Typical numbers These numbers are well-characterized by the average and the standard deviation.

Q. What is the largest city in the US? A. New York, population 8,310,212

Q. What is the 2 nd largest city in the US? A. Los Angeles, population 3,834,340

Q. What is the 3 rd largest city in the US? A. Chicago, population 2,836,658

City populations 1.New York8,310,212 2.Los Angeles 3,834,340 3.Chicago2,836,658 4.Houston 2,208,180 5.Phoenix1,552,259 6.Philadelphia1,449,634 7.San Antonio 1,328,984 8.San Diego1,266,731 9.Dallas1,266,372 10.San Jose 939,899

City populations 1.New York8,310,212 2.Los Angeles 3,834,340 3.Chicago2,836,658 230. Cambridge, MA 101,335 240. Gainesville, FL 95,447 250. McKinney, TX 54,369 A few cities with high population Many cities with low population

City populations

Power Law: The number of cities with population at least k is proportional to k -c for a constant c.

Power Law: The number of cities with population > k is proportional to k -c. “fraction of items” “popularity = k”

Power Law: Fraction f(k) of items with popularity k is proportional to k -c. f(k) k -c log [f(k)] log [k -c ] log [f(k)] -c log [k]

A power law is a straight line on a log-log plot.

City populations

Other examples

Why does data exhibit power laws?

Today ImitationPower law

Constructing the web 1.Pages are created in order, named 1, 2, …, N 2.When created, page j links to a page by a)With probability p, picking a page i uniformly at random from 1, …, j-1 b)With probability (1-p), pick page i uniformly at random and link to the page that i links too Imitation

The rich get richer 2 b) With prob. (1-p), pick page i uniformly at random and link to the page that i links too 1/43/4

The rich get richer 2 b) With prob. (1-p), pick page i uniformly at random and link to the page that i links too Equivalently, 2 b)With prob. (1-p), pick a page proportional to its in-degree and link to it

Simulation

Optional material Rich get richer Power law

Why is Harry Potter popular? If we could re-play history, would we still read Harry Potter, or would it be some other book?

Information cascades and the rich Information cascade = so some people get a little bit richer by chance and then rich-get-richer dynamics = the random rich people get a lot richer very fast

Music download site – 8 worlds 1.“Let’s go driving,” Barzin 2.“Silence is sexy,” Einstürzende Neubauten 3.“Go it alone,” Noonday Underground 10.“Picadilly Lilly,” Tiger Lillies 1.“Let’s go driving,” Barzin 2.“Silence is sexy,” Einstürzende Neubauten 3.“Go it alone,” Noonday Underground 10.“Picadilly Lilly,” Tiger Lillies 18 3 47 2 59 7 10 1

Next time TBA

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