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Various names, kinds and forms: Long tail, Heavy tail, Zipf’s law, Pareto’s law Heavy tailed distributions: P(x) is proportional to: [Clauset-Shalizi-Newman 2007] 11/1/2011 Jure Leskovec, <strong>Stanford</strong> CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 20
What is the normalizing constant? P(x) = z x -α z=?
Page 1 and 2: CS224W: Social and Information Netw
Page 3 and 4: Expected based on G np Found in dat
Page 5 and 6: Probability: P(k) = P(X=k) Plot th
Page 7 and 8: [Broder et al., 2000] 11/1/2011 Jur
Page 9 and 10: f(x) f ( x) = c 1 0.6 0.2 − x f (
Page 11 and 12: 11/1/2011 Jure Leskovec, Stanford C
Page 13 and 14: Definition: Networks with a power l
Page 15 and 16: Many other quantities follow heavy-
Page 17: CMU grad-students at the G20 meetin
Page 22 and 23: What’s the expectation of a power
Page 24 and 25: BAD! Estimating α from data: 1. F
Page 26 and 27: OK Estimating α from data: 3. Use
Page 28 and 29: What is the expected maximum degree
Page 30 and 31: Can not arise from sums of independ
Page 33 and 34: Preferential attachment [Price ‘6
Page 35 and 36: We will analyze the following model
Page 37 and 38: Consider deterministic and continuo
Page 39 and 40: Initial condition: d i(t)=0, when
Page 41 and 42: What is the constant A? We know:
Page 43 and 44: What is the fraction of nodes with
Page 45 and 46: Preferential attachment gives power
Page 47 and 48: Preferential attachment is not so g
Page 49 and 50: We observe how the connectivity (le
Page 51: Mean path length Real-world networ

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