Random Graphs CS224W

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1 Random Graphs CS224W

2 Network models Why model? simple representation of complex network can derive properties mathematically predict properties and outcomes Also: to have a strawman In what ways is your real-world network different from hypothesized model? What insights can be gleaned from this?

3 Erdös and Rényi

4 Erdös-Renyi: simplest network model Assumptions nodes connect at random network is undirected Key parameter (besides number of nodes N) : p or M p = probability that any two nodes share and edge M = total number of edges in the graph

5 what they look like after spring layout

6 Degree distribution (N,p)-model: For each potential edge we flip a biased coin with probability p we add the edge with probability (1-p) we don t Alternate notation: G np

7 Quiz Q: As the size of the network increases, if you keep p, the probability of any two nodes being connected, the same, what happens to the average degree a) stays the same b) increases c) decreases

8

9 Degree distribution What is the probability that a node has 0,1,2,3 edges? Probabilities sum to 1

10 How many edges per node? Each node has (N 1) tries to get edges Each try is a success with probability p The binomial distribution gives us the probability that a node has degree k: B(N 1;k; p) = " $ # N 1 k % ' p k (1 p) N 1 k &

11 Quiz Q: The maximum degree of a node in a simple (no multiple edges between the same two nodes) N node graph is a) N b) N - 1 c) N / 2

12 Explaining the binomial distribution 8 node graph, probability p of any two nodes sharing an edge What is the probability that a given node has degree 4? A B C D E F G

13 binomial coefficient explained number of ways of choosing k items out of (n-1) number of ways of arranging n-1 items = (# of ways to arrange k things)*(# ways to arrange n-1-k things) n-1! = k! (n-1-k)! Note that the binomial coefficient is symmetric there are the same number of ways of choosing k or n-1-k things out of n-1

14 Quiz Q: What is the number of ways of choosing 2 items out of 5?

15 Now the distribution p = probability of having edge to node (blue) (1-p) = probability of not having edge (white) The probability that you connect to 4 of the 7 nodes in some particular order (two white followed by 3 blues, followed by a white followed by a blue) is P(white)*P(white)*P(blue)*P(blue)*P(blue)*P(white)*P(blue) = p 4 *(1-p) 3

16 Binomial distribution If order doesn t matter, need to multiply probability of any given arrangement by number of such arrangements: B(7;4; p) =! # " 7 4 $ & p 4 (1 p) 3 % +.

17 if p = 0.5

18 p = 0.1

19 What is the mean? Average degree <k>= z = (n-1)*p in general µ = E(X) = Σx p(x) probabilities that sum to 1 0 * + 1 * + 2 * + 3 * + 4 * + 5 * + 6 * + 7 * µ = 3.5

20 Quiz Q: What is the average degree of a graph with 10 nodes and probability p = 1/3 of an edge existing between any two nodes?

21 Approximations p k p p k = = k σ n 1 p k = 1 z k e k! z 2π e k (1 (k z) 2 2σ 2 p) n 1 k Binomial limit p small Poisson limit large n Normal

22 Poisson distribution Poisson distribution

23 What insights does this yield? No hubs You don t expect large hubs in the network

24 Insights Previously: degree distribution / absence of hubs Emergence of giant component Average shortest path

25 Emergence of the giant component (standard model in NetLogo library) models/giantcomponent

26 Quiz Q: What is the average degree z at which the giant component starts to emerge? 0 1 3/2 3

27 Percolation on a 2D lattice

28 Quiz Q: What is the percolation threshold of a 2D lattice: fraction of sites that need to be occupied in order for a giant connected component to emerge? 0 ¼ 1/3 1/2

29 Percolation threshold size of giant component Percolation threshold: how many edges need to be added before the giant component appears? As the average degree increases to z = 1, a giant component suddenly appears average degree av deg = 0.99 av deg = 1.18 av deg = 3.96

30 Evolution of the G np What happens to G np when we vary p?

31 Back to Node Degrees of G np Remember, expected degree E[ X v ] = ( n 1) p If want E[X v ] be independent of n let: p=c/(n-1)

32 Probability of a node being isolated Observation: If we build random graph G np with p=c/(n-1) we have many isolated nodes Why? 38 c n n n e n c p v P = = ) (1 has degree 0] [ c c x x c x n n e x x n c = = = lim 1 lim = n c x Use substitution e (by definition)

33 No Isolated Nodes How big do we have to make p before we are likely to have no isolated nodes? We know: P[v has degree 0] = e -c Event we are asking about is: I = some node is isolated I = v N I v where I v is the event that v is isolated We have: P I = P v ( ) I ( ) v P Iv N v N = ne c Union bound A i A i Ai i i 39

34 No Isolated Nodes We just learned: P(I) = n e -c Let s try: c = ln n then: n e -c = n e -ln n =n 1/n= 1 c = 2 ln n then: n e -2 ln n = n 1/n 2 = 1/n So if: p = ln n then: P(I) = 1 p = 2 ln n then: P(I) = 1/n 0 as n Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 40

35 Evolution of a Random Graph Graph structure of G np as p changes: p Giant component Avg. deg const. Fewer isolated No isolated nodes. 0 1 Empty graph 1/(n-1) appears c/(n-1) Lots of isolated nodes. log(n)/(n-1) nodes. 2*log(n)/(n-1) Complete graph Emergence of a Giant Component: avg. degree k=2e/n or p=k/(n-1) k=1-ε: all components are of size Ω(log n) k=1+ε: 1 component of size Ω(n), others have size Ω(log n) Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 41

36 Giant component another angle How many other friends besides you does each of your friends have? By property of degree distribution the average degree of your friends, you excluded, is z so at z = 1, each of your friends is expected to have another friend, who in turn have another friend, etc. the giant component emerges

37 Why just one giant component? What if you had 2, how long could they be sustained as the network densifies?

38 Quiz Q: If you have 2 large-components each occupying roughly 1/2 of the graph, how long does it typically take for the addition of random edges to join them into one giant component 1-4 edge additions 5-20 edge additions over 20 edge additions

39 Average shortest path How many hops on average between each pair of nodes? again, each of your friends has z = avg. degree friends besides you ignoring loops, the number of people you have at distance l is z l

40 Average shortest path

41 friends at distance l N l =z l scaling: average shortest path l av l av ~ log N log z

42 What this means in practice Erdös-Renyi networks can grow to be very large but nodes will be just a few hops apart average shortest path num nodes

43 Logarithmic axes powers of a number will be uniformly spaced n 2 0 =1, 2 1 =2, 2 2 =4, 2 3 =8, 2 4 =16, 2 5 =32, 2 6 =64,.

44 Erdös-Renyi avg. shortest path average shortest path num nodes

45 Quiz Q: If the size of an Erdös-Renyi network increases 100 fold (e.g. from 100 to 10,000 nodes), how will the average shortest path change it will be 100 times as long it will be 10 times as long it will be twice as long it will be the same it will be 1/2 as long

46 Realism Consider alternative mechanisms of constructing a network that are also fairly random. How do they stack up against Erdös- Renyi? NetLogo/RandomGraphs.nlogo

47 Introduction model Prob-link is the p (probability of any two nodes sharing an edge) that we are used to But, with probability prob-intro the other node is selected among one of our friends friends and not completely at random

48 Introduction model

49 Quiz Q: Relative to ER, the introduction model has: more edges more closed triads longer average shortest path more uneven degree smaller giant component at low p

50 Static Geographical model Each node connects to num-neighbors of its closest neighbors use the num-neighbors slider, and for comparison, switch PROB-OR-NUM to off to have the ER model aim for numneighbors as well turn off the layout algorithm while this is running, you can apply it at the end

51 static geo

52 Quiz Q: Relative to ER, the static geographical model has : longer average shortest path shorter average shortest path narrower degree distribution broader degree distribution smaller giant component at a low number of neighbors larger giant component at a low number of neighbors

53 Growth model Instead of starting out with a fixed number of nodes, nodes are added over time use the num-neighbors slider, and for comparison, switch PROB-OR-NUM to off to have the ER model aim for numneighbors as well

54 growth model

55 Quiz Q: Relative to ER, the growth model has : more hubs fewer hubs smaller giant component at a low number of neighbors larger giant component at a low number of neighbors

56 other models in some instances the ER model is plausible if dynamics are different, ER model may be a poor fit

57 Small world networks CS 224W

58 Small world phenomenon: Milgram s experiment MA NE

59 Milgram s experiment Instructions: Given a target individual (stockbroker in Boston), pass the message to a person you correspond with who is closest to the target. Outcome: 20% of initiated chains reached target average chain length = 6.5 Six degrees of separation

60 Milgram s experiment repeated experiment Dodds, Muhamad, Watts, Science 301, (2003) (optional reading) 18 targets 13 different countries 60,000+ participants 24,163 message chains 384 reached their targets average path length 4.0 Source: NASA, U.S. Government;;

61 Interpreting Milgram s experiment n Is 6 is a surprising number? n In the 1960s? Today? Why? n Pool and Kochen in (1978 established that the average person has between 500 and 1500 acquaintances)

62 Quiz Q: Ignore for the time being the fact that many of your friends friends are your friends as well. If everyone has 500 friends, the average person would have how many friends of friends? 500 1,000 5, ,000

63 Quiz Q: With an average degree of 500, a node in a random network would have this many friends-of-friends-of-friends (3 rd degree neighbors): 5, ,000 1,000, ,000,000

64 Interpreting Milgram s experiment n Is 6 is a surprising number? n In the 1960s? Today? Why? n If social networks were random? n Pool and Kochen (1978) - ~ acquaintances/person n ~ 500 choices 1 st link n ~ = 250,000 potential 2 nd degree neighbors n ~ = 125,000,000 potential 3 rd degree neighbors n If networks are completely cliquish? n all my friends friends are my friends n what would happen?

65 Quiz Q: If the network were completely cliquish, that is all of your friends of friends were also directly your friends, what would be true: (a) None of your friendship edges would be part of a triangle (closed triad) (b) It would be impossible to reach any node outside the clique by following directed edges (c) Your shortest path to your friends friends would be 2

66 complete cliquishness If all your friends of friends were also your friends, you would be part of an isolated clique.

67 Uncompleted chains and distance n Is 6 an accurate number? n What bias is introduced by uncompleted chains? n are longer or shorter chains more likely to be completed?

68 Attrition probability of passing on message average position in chain 95 % confidence interval Source: An Experimental Study of Search in Global Social Networks: Peter Sheridan Dodds, Roby Muhamad, and Duncan J. Watts (8 August 2003); Science 301 (5634), 827.

69 Quiz Q: n if each intermediate person in the chain has 0.5 probability of passing the letter on, what is the likelihood of a chain being completed n of length 2? n of length 5? sends for sure receives chain of length 2 passes on with probability 0.5

70 Quiz Q: n if each intermediate person in the chain has 0.5 probability of passing the letter on, what is the likelihood of a chain of length 5 being completed (a) ½ (b) ¼ (c) 1/8 (d) 1/16

71 Estimating the true distance observed chain lengths recovered histogram of path lengths inter-country intra-country Source: An Experimental Study of Search in Global Social Networks: Peter Sheridan Dodds, Roby Muhamad, and Duncan J. Watts (8 August 2003); Science 301 (5634), 827.

72 Navigation and accuracy Is 6 an accurate number? Do people find the shortest paths? Killworth, McCarty,Bernard, & House (2005): less than optimal choice for next link in chain is made ½ of the time

73 Small worlds & networking What does it mean to be 1, 2, 3 hops apart on Facebook, Twitter, LinkedIn, Google Plus?

74 Transitivity, triadic closure, clustering Transitivity: if A is connected to B and B is connected to C what is the probability that A is connected to C? my friends friends are likely to be my friends A? C B

75 Clustering Global clustering coefficient 3 x number of triangles in the graph number of connected triples of vertices C = 3 x number of triangles in the graph number of connected triples

76 Local clustering coefficient (Watts&Strogatz 1998) For a vertex i The fraction pairs of neighbors of the node that are themselves connected Let n i be the number of neighbors of vertex i C i = # of connections between i s neighbors max # of possible connections between i s neighbors Ci directed = # directed connections between i s neighbors n i * (n i -1) Ci undirected = # undirected connections between i s neighbors n i * (n i -1)/2

77 Local clustering coefficient (Watts&Strogatz 1998) Average over all n vertices C = 1 n i C i i n i = 4 max number of connections: 4*3/2 = 6 3 connections present C i = 3/6 = 0.5 link present link absent

78 Quiz Q: The clustering coefficient for vertex i is: i (a)0 (b)1/3 (c)1/2 (d)2/3

79 Explanation n i = 3 there are 2 connections present out of max of 3 possible C i = 2/3 i

80 beyond social networks Small world phenomenon: high clustering C network >> C random graph low average shortest path l network ln( N) what other networks can you think of with these characteristics?

81 Comparison with random graph used to determine whether real-world network is small world Network size av. shortest path Shortest path in fitted random graph Clustering (averaged over vertices) Clustering in random graph Film actors 225, MEDLINE coauthorship 1,520, x 10-4 E.Coli substrate graph C.Elegans

82 Small world phenomenon: Watts/Strogatz model Reconciling two observations: High clustering: my friends friends tend to be my friends Short average paths Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

83 Watts-Strogatz model: Generating small world graphs Select a fraction p of edges Reposition on of their endpoints Add a fraction p of additional edges leaving underlying lattice intact n As in many network generating algorithms n Disallow self-edges n Disallow multiple edges Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

84 Watts-Strogatz model: Generating small world graphs Each node has K>=4 nearest neighbors (local) tunable: vary the probability p of rewiring any given edge small p: regular lattice large p: classical random graph

85 Quiz question: Which of the following is a result of a higher rewiring probability? (a) Left (b) Right (c) insufficient information

86 What happens in between? Small shortest path means low clustering? Large shortest path means high clustering? Through numerical simulation As we increase p from 0 to 1 Fast decrease of mean distance Slow decrease in clustering

87 Clust coeff. and ASP as rewiring increases 1% of links rewired 10% of links rewired Source: Watts, D.J., Strogatz, S.H.(1998) Collective dynamics of 'small-world' networks. Nature 393:

88 Trying this with NetLogo

89 Power laws and preferential attachment CS224W

90 Online Question & Answer Forums

91 Uneven participation cumulative probability number of people one received replies from degree (k) α = 1.87 fit, R 2 = number of people one replied to answer people may reply to thousands of others question people are also uneven in the number of repliers to their posts, but to a lesser extent

92 Real-world degree distributions Sexual networks Great variation in contact numbers

93 Power-law distribution P(x) linear scale n log-log scale x P(x) x n high skew (asymmetry) n straight line on a log-log plot

94 Poisson distribution P(x) x 1e-64 1e-36 1e-08 linear scale n log-log scale P(x) x n little skew (asymmetry) n curved on a log-log plot

95 Power law distribution Straight line on a log-log plot ln(p(k)) = c α ln(k) Exponentiate both sides to get that p(k), the probability of observing an node of degree k is given by p(k) = Ck α normalization constant (probabilities over all k must sum to 1) power law exponent α

96 Quiz Q: As the exponent α increases, the downward slope of the line on a log-log plot stays the same becomes milder becomes steeper

97 2 ingredients in generating power-law networks nodes appear over time (growth)

98 2 ingredients in generating power-law networks nodes prefer to attach to nodes with many connections (preferential attachment, cumulative advantage)

99 Ingredient # 1: growth over time nodes appear one by one, each selecting m other nodes at random to connect to m = 2

100 random network growth one node is born at each time tick at time t there are t nodes change in degree k i of node i (born at time i, with 0 < i < t) dk ( t) i = dt m t there are m new edges being added per unit time (with 1 new node) the m edges are being distributed among t nodes

101 a node in a randomly grown network how many new edges does a node accumulate since it's birth at time i until time t? integrate from i to t to get dk i ( t) = dt k ( t) = m + m i born with m edges m t t log( i )

102 age and degree on average k ( t) k ( t) i > j if i < j i.e. older nodes on average have more edges

103 growing random networks Let τ(100) be the time at which node with degree e.g. 100 is born. The the fraction of nodes that have degree <= 100 is (t τ)/t k ( t) = m + τ m log( t τ )

104 random growth: degree distribution continuing m m k t = ) log( τ m m k e t = τ The probability that a node has degree k or less is 1-τ/t m m k e k k P = < ' 1 ) ( ' exponential distribution in degree

105 Quiz Q: The degree distribution for a growth model where new nodes attach to old nodes at random will be a curved line on a log-log plot a straight line on a log-log plot

106 2 nd ingredient: preferential attachment Preferential attachment: new nodes prefer to attach to well-connected nodes over less-well connected nodes Process also known as cumulative advantage rich-get-richer Matthew effect

107 Preferential attachment

108 Cumulative advantage: how? copying mechanism visibility

109 Barabasi-Albert model First used to describe skewed degree distribution of the World Wide Web Each node connects to other nodes with probability proportional to their degree the process starts with some initial subgraph each new node comes in with m edges probability of connecting to node i Π(i) = m j ki k j Results in power-law with exponent α = 3

110 Basic BA-model Very simple algorithm to implement start with an initial set of m 0 fully connected nodes e.g. m 0 = 3 3 now add new vertices one by one, each one with exactly m edges each new edge connects to an existing vertex in proportion to the number of edges that vertex already has preferential attachment easiest if you keep track of edge endpoints in one large array and select an element from this array at random 1 2 the probability of selecting any one vertex will be proportional to the number of times it appears in the array which corresponds to its degree

111 generating BA graphs contʼ d To start, each vertex has an equal number of edges (2) the probability of choosing any vertex is 1/ We add a new vertex, and it will have m edges, here take m= draw 2 random elements from the array suppose they are 2 and Now the probabilities of selecting 1,2,3,or 4 are 1/5, 3/10, 3/10, 1/ Add a new vertex, draw a vertex for it to connect from the array etc

112 after a while...

113 contrasting with random (non-preferential) growth m = 2 random preferential

114 Exponential vs. Power-Law Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, /1/15

115 Properties of the BA graph The distribution is power-law with exponent α = 3 P(k) = 2 m 2 /k 3 The graph is connected Every new vertex is born with a link or several links (depending on whether m = 1 or m > 1) It then connects to an older vertex, which itself connected to another vertex when it was introduced And we started from a connected core The older are richer Nodes accumulate links as time goes on, which gives older nodes an advantage since newer nodes are going to attach preferentially and older nodes have a higher degree to tempt them with than some new kid on the block

116 Young vs. old in BA model vertex introduced at time t=5 vertex introduced at time t=95

117 try it yourself

118 Quiz Q: Relative to the random growth model, the degree distribution in the preferential attachment model resembles a power-law distribution less resembles a power-law distribution more

119 Summary: growth models Most networks aren't 'born', they are made. Nodes being added over time means that older nodes can have more time to accumulate edges Preference for attaching to 'popular' nodes further skews the degree distribution toward a power-law

120 Heavy tails: right skew Right skew normal distribution (not heavy tailed) e.g. heights of human males: centered around 180cm (5 11 ) Zipf s or power-law distribution (heavy tailed) e.g. city population sizes: NYC 8 million, but many, many small towns

121 Normal distribution (human heights) average value close to most typical distribution close to symmetric around average value

122 Heavy tails: max to min ratio High ratio of max to min human heights tallest man: 272cm (8 11 ), shortest man: (1 10 ) ratio: 4.8 from the Guinness Book of world records city sizes NYC: pop. 8 million, Duffield, Virginia pop. 52, ratio: 150,000

123 Power-law distribution x^(-2) x x^(-2) linear scale n log-log scale n high skew (asymmetry) n straight line on a log-log plot x

124 Power laws are seemingly everywhere note: these are cumulative distributions, more about this in a bit Moby Dick scientific papers AOL users visiting sites 97 bestsellers AT&T customers on 1 day California Source:MEJ Newman, ʼ Power laws, Pareto distributions and Zipfʼ s lawʼ, Contemporary Physics 46, (2005)

125 Yet more power laws Moo n Solar flares wars ( ) richest individuals US family names US cities Source:MEJ Newman, ʼ Power laws, Pareto distributions and Zipfʼ s lawʼ, Contemporary Physics 46, (2005)

126 Anatomy of the Long Tail [Chris Anderson, Wired, 2004] Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, /1/15

127 Power law distribution Straight line on a log-log plot ln( p( x)) = Exponentiate both sides to get that p(x), the probability of observing an item of size x is given by p ( x) c α ln( x) α = Cx normalization constant (probabilities over all x must sum to 1) power law exponent α

128 What does it mean to be scale-free? A power law looks the same no mater what scale we look at it on (2 to 50 or 200 to 5000) Only true of a power-law distribution! p(bx) = g(b) p(x) Scale-free definition: shape of the distribution is unchanged except for a multiplicative constant p(bx) = (bx) α = b α x α log(p(x)) x b*x log(x)

129 Popular distributions to try and fit

130 Fitting power-law distributions Most common and not very accurate method: Bin the different values of x and create a frequency histogram ln(# of times x occurred) ln(x) is the natural logarithm of x, but any other base of the logarithm will give the same exponent of α because log 10 (x) = ln(x)/ln(10) ln(x) x can represent various quantities, the indegree of a node, the magnitude of an earthquake, the frequency of a word in text

131 Log-log scale plot of simple binning of the data n Same bins, but plotted on a log-log scale here we have tens of thousands of observations when x < frequency Noise in the tail: Here we have 0, 1 or 2 observations of values of x when x > integer value Actually don t see all the zero values because log(0) =

132 Log-log scale plot of straight binning of the data n Fitting a straight line to it via least squares regression will give values of the exponent α that are too low fitted α true α 10 4 frequency integer value

133 What goes wrong with straightforward binning Noise in the tail skews the regression result 10 6 data 10 5 have few bins here α = 1.6 fit have many more bins here

134 First solution: logarithmic binning bin data into exponentially wider bins: 1, 2, 4, 8, 16, 32, normalize by the width of the bin 10 6 data α = 2.41 fit evenly spaced datapoints less noise in the tail of the distribution n disadvantage: binning smoothes out data but also loses information

135 Second solution: cumulative binning No loss of information No need to bin, has value at each observed value of x But now have cumulative distribution i.e. how many of the values of x are at least X The cumulative probability of a power law probability distribution is also power law but with an exponent α - 1 cx α = c 1 α x ( α 1)

136 Fitting via regression to the cumulative distribution fitted exponent (2.43) much closer to actual (2.5) data α-1 = 1.43 fit frequency sample > x x

137 Where to start fitting? some data exhibit a power law only in the tail after binning or taking the cumulative distribution you can fit to the tail so need to select an x min the value of x where you think the power-law starts certainly x min needs to be greater than 0, because x α is infinite at x = 0

138 Example: Distribution of citations to papers power law is evident only in the tail (x min > 100 citations) x min Source:MEJ Newman, ʼ Power laws, Pareto distributions and Zipfʼ s lawʼ, Contemporary Physics 46, (2005)

139 Some exponents for real world data x min frequency of use of words number of citations to papers number of hits on web sites copies of books sold in the US telephone calls received magnitude of earthquakes diameter of moon craters intensity of solar flares intensity of wars net worth of Americans $600m 2.09 frequency of family names population of US cities exponent α

140 Many real world networks are power law exponent α (in/out degree) film actors 2.3 telephone call graph 2.1 networks 1.5/2.0 sexual contacts 3.2 WWW 2.3/2.7 internet 2.5 peer-to-peer 2.1 metabolic network 2.2 protein interactions 2.4

141 Hey, not everything is a power law number of sightings of 591 bird species in the North American Bird survey in cumulative distribution n another example: n size of wildfires (in acres) Source:MEJ Newman, ʼ Power laws, Pareto distributions and Zipfʼ s lawʼ, Contemporary Physics 46, (2005)

142 Not every network is power law distributed reciprocal, frequent communication power grid Roget s thesaurus company directors

143 Example on a real data set: number of AOL visitors to different websites back in 1997 simple binning on a linear scale simple binning on a log-log scale

144 trying to fit directly direct fit is too shallow: α = 1.17

145 Binning the data logarithmically helps select exponentially wider bins 1, 2, 4, 8, 16, 32,.

146 Or we can try fitting the cumulative distribution Shows perhaps 2 separate power-law regimes that were obscured by the exponential binning Power-law tail may be closer to 2.4

147 Another common distribution: power-law with an exponential cutoff p(x) ~ x -a e -x/κ starts out as a power law ends up as an exponential p(x) but could also be a lognormal or double exponential x

148 Wrap up on power-laws Power-laws are cool and intriguing But make sure your data is actually power-law before boasting

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