Analysis of Social Information Networks Thursday January 27 th, Lecture 3: Popularity-Power law 1

 The ubiquitous power-law  The three sources of power law −reinforcement −optimization −artefacts of random stopping  How to handle the long tail? 2 Outline

 Distribution: different values of a variable metric −Among individuals (size, income), webpage (#links)...  Large values −P[X>x] is decreasing to 0 as x increases, but how fast? −Light-tailed: ∃ λ such that P[X>x] ≤ exp(-λx) −Heavy-tailed: ∀ λ, P[X>x] > exp(-λx) for large x  Power-law is a special case of heavy tailed −Polynomial decrease: P[X>x] ~ c x -α for large x 3 What is a power-law?

 #1: very large values are rare but not impossible −Ex.: α=2, c=1, P[X>1000] = 1/1,000,000  #2: large values have an impact collectively −Unfairness: the 20% richest own 80% of wealth … and the 5% richest own 75% of wealth −Variance, mean can be infinite  #3: invariance by {conditioning + mult. rescaling} −Exponential : P[X>x+1|X>x] = e -λ for any x −Power law: P[X>2x|X>x]=2 -α for any x 4 What makes power-law special?

What makes this graph different? 5 Power laws, Pareto distributions and Zipf’s law. M. Newman, Cont. Physics (2005)

 Plot CCDF or density using log. x and log. y axis −Is the shape linear? For how many magnitude orders? −What is the slope?  In contrast, light tailed or lognormal decreases fast  Care must be taken to identify a power-law −Density is sensitive to binning, CCDF preferred −Validation: goodness of fit test, alternative −Coefficient Maximum likelyhood 6 How to spot a power law? Power-law distributions in empirical data. A. Clauset, C. Shalizi, M. Newman, SIAM review (2009)

The ubiquitous power law 7 Power laws, Pareto distributions and Zipf’s law. M. Newman, Cont. Physics (2005)

Non-power law exist too! −Length of a relationship −# birds in species −# of addresses kept −Size of forest fire 8 Power laws, Pareto distributions and Zipf’s law. M. Newman, Cont. Physics (2005)

Power law in Internet 9 Graph structure in the web. Broder et al. Comp. Netw. (2000) On power-law relationships of the internet topology Faloutsos, Faloutsos & Faloutsos. SIGCOMM (1999)

Power law & traveled distance 10 Understanding individual human mobility patterns. Gonzalez, Hidalgo, Barabasi, Nature (2008) The scaling laws of human travel. D Brockmann, L Hufnagel, T Geisel, Nature (2006)

Power law in time statistics 11 The origin of bursts and heavy tails in human dynamics. A. Barabasi, Nature (2005)

 Popularity of items: Amazon, YouTube, Flickr  Degree of nodes in large graphs −links to website, connection between Internet Ases −Collaboration between actors, scientists  Across a broad spectrum of natural system −# Species in genus, proteine seq. genome, words  Time and space: characterizes human activities 12 The ubiquitous power law

 The ubiquitous power-law  The three sources of power law −reinforcement −optimization −artefacts of random stopping  How to handle the long tail? 13 Outline