Superstar Model: ReTweets, Lady Gaga and Surgery on a Branching Process J. Michael Steele Simons Conference on Random Graph Processes Austin, Texas 2016 J.M. Steele (U Penn, Wharton) May 2016 1 / 30
Empirical Observations on the Retweet Graph Passage from the Retweet Graph to the Superstar Model Joint work with Shankar Bhamidi (UNC) and Tauhid Zaman (MIT) — genuine members of the Twitter generation! Retweet graph: Given a topic and a time frame — form all the (undirected) retweet arcs and look at the giant component of the graph you get. Black Entertainment Television (BET) Awards 2010 J.M. Steele (U Penn, Wharton) May 2016 3 / 30
Empirical Observations on the Retweet Graph Reading the Message from Some Empirical Retweet Graphs Retweet graphs were constructed for 13 different public events 1 ◮ Sports, breaking news stories, and entertainment events ◮ Time range for each topic was between 4-6 hours Empirically the graphs are very tree-like (almost no cycles) A) Federer, N = 505 B) England, N = 1024 Empirically the graphs each have one giant component — this is what we model The graphs are taken as undirected — and the the degrees tell the whole story C) BET Awards, N = 1724 D) World Cup, N = 2847 1 Data courtesy of Microsoft Research, Cambridge, MA J.M. Steele (U Penn, Wharton) May 2016 4 / 30
Empirical Observations on the Retweet Graph BET 2010 Data — with Labels J.M. Steele (U Penn, Wharton) May 2016 5 / 30
Where Preferential Attachment Fails What Goes Wrong with Plain Vanilla Preferential Attachment? One finds Max degree in empirically observed retweet graphs have the order of the graph size, i.e. MaxDeg ∼ pn Preferential attachment would predict sub-linear max degree 1000 Maximum degree 800 600 400 200 √ n (preferential attachement) 0 0 2000 4000 6000 8000 Number of vertices (n) J.M. Steele (U Penn, Wharton) May 2016 7 / 30
The Super Star Model: Just One Parameter The Superstar Model — It’s Completely Determined by p v 0 (superstar) Attach to superstar with probability p p Else with probability 1 − p attach to one of the G 2 non-superstar vertices. Non-SS Attachment Rule: probability proportional to v 2 its degree (preferential attachment rule) v 1 (1 − p )deg( v 1 , G 2 ) The only model parameter is p : The super star parameter This is a very simple model: But (1) it has empirical benefits and (2) it is tractable — though not particularly easy. J.M. Steele (U Penn, Wharton) May 2016 9 / 30
Predictions of the Superstar Model The Degree of the Superstar Under the Superstar Model Remark (Built-In Easy Fact) Let deg( v 0 , G n ) be the degree of the superstar in G n . We then have that deg( v 0 , G n ) → p with probability 1 as n → ∞ n Empirically the Superstar degree is Θ( n ) and the Superstar Model “Bakes this into the Cake” But that is ALL that is baked in... The value of p predicts other features of the graph The Superstar Model is TESTABLE . J.M. Steele (U Penn, Wharton) May 2016 11 / 30
Predictions of the Superstar Model The Most Starry of the Non-Superstars Theorem Let deg max ( G n ) be the maximal non-superstar degree in G n , i.e. deg max ( G n ) = max 1 ≤ i ≤ n deg( v i , G n ) .. If we set γ = 1 − p 2 − p . then here is a non-degenerate, strictly positive, random variable ∆ ∗ such that n − γ deg max ( G n )) → ∆ ∗ with probability 1 as n → ∞ Maximal non-superstar degree is little-oh of the degree of the Superstar The Super Star Model makes an explicit prediction for the growth rate of maximum degree of a non-superstar. J.M. Steele (U Penn, Wharton) May 2016 12 / 30
Predictions of the Superstar Model Realized Degree Distribution in the Superstar Model Theorem Let F ( k , G n ) be the realized degree distribution of G n under the Superstar model, F ( k , G n ) = n − 1 |{ 1 ≤ j ≤ n : deg( v j , G n ) = k }| and introduce the superstar model probability mass function � � − 1 k � f SSM ( k , p ) = 2 − p i + 2 − p 1 − p ( k − 1)! . 1 − p i =1 We then have F ( k , G n ) → f SSM ( k , p ) with probability 1 as n → ∞ KEY POINT: The degree distribution scales like k − β , where β = 3 + p / (1 − p ) This contrasts with the preferential attachment model which scales like k − 3 J.M. Steele (U Penn, Wharton) May 2016 13 / 30
Comparison with Preferential Attachment Model Superstar Model vs Preferential Attachment Superstar Preferential Model Model Attachment Superstar Degree ∼ pn NA 1 − p Maximal non-superstar 1 degree exponent 2 − p 2 p Degree distribution 3 + 3 power-law exponent 1 − p J.M. Steele (U Penn, Wharton) May 2016 15 / 30
Comparison with Preferential Attachment Model Superstar Model Predictions Use actual data � G n to fit the superstar degree and predict the degree distribution Consider the observed degree distribution for each empirical retweet graph: G n ) = n − 1 |{ 1 ≤ j ≤ n : deg( v j , G n ) = k }| F ( k , � Consider the theoretical asymptotic degree distribution under the Superstar Model � � − 1 k � f SSM ( k , p ) = 2 − p i + 2 − p 1 − p ( k − 1)! . 1 − p i =1 Bottom Line: We get a pretty impressive fit “observed vs predicted” p = observed superstar degree F ( k , � G n ) ≈ f SM ( k , ˆ p ) where ˆ n Basis for Tests: Preferential Attachment always predicts... 4 f PA ( k ) = k ( k + 1)( k + 2) J.M. Steele (U Penn, Wharton) May 2016 16 / 30
Comparison with Preferential Attachment Model Degree distribution Lebron, p’=0.09 Brazil Portugal, p’=0.28 0 0 10 10 f(k,G n ) f SM (k,p’) −1 −1 10 10 f PA (k) −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 0 5 10 15 20 0 5 10 15 20 k k BET Awards, p’ = 0.58 Federer, p’=0.37 0 0 10 10 −1 10 −1 10 −2 10 −2 10 −3 10 −3 10 −4 10 −5 −4 10 10 0 5 10 15 20 0 5 10 15 20 k k J.M. Steele (U Penn, Wharton) May 2016 17 / 30
Comparison with Preferential Attachment Model Degree distribution Comparison Compare relative error of the Superstar Model and Preferential Attachment for different degrees k Superstar Preferential Model Model Attachment | f ( k , G n ) − f SM ( k , p ′ ) | | f ( k , G n ) − f SM ( k , p ′ ) | Relative Error f SM ( k , p ′ ) f SM ( k , p ′ ) J.M. Steele (U Penn, Wharton) May 2016 18 / 30
Comparison with Preferential Attachment Model Degree Distribution Comparison Preferential degree = 2 degree = 1 Attachment 1 1 Relative Error Relative Error Superstar Model 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 3 5 7 9 11 13 1 3 5 7 9 11 13 Retweet Graph Retweet Graph degree = 3 degree = 4 1 1 Relative Error Relative Error 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 1 3 5 7 9 11 13 1 3 5 7 9 11 13 Retweet Graph Retweet Graph J.M. Steele (U Penn, Wharton) May 2016 19 / 30
Comparison with Preferential Attachment Model The Superstar Model and the Realized Degree Distribution: Bottom Line The Superstar Model implies a mathematical link between the superstar degree and the degree distribution of the non-superstars. When we look at Twitter data for actual events, we see (1) a superstar and (2) a degree distribution of non-superstars that is more compatible with the superstar model than with the preferential attachment model. The first property was “baked” into our model, but the second was not. It’s an honest discovery. Next: How Can one Analyze the Superstar Model? J.M. Steele (U Penn, Wharton) May 2016 20 / 30
Superstar Model: Tools for Analysis Basic Link: Branching Processes Proto-Idea: Branching processes have a natural role almost anytime one considers a stochastically evolving tree. More Concrete Observation: If the birth rates depend on the number of children, the arithmetic of the Poisson process relates lovingly to the arithmetic of preferential attachment — this is sweet. Creating the Superstar: Yule processes don’t come with a superstar. Still, it is not terribly hard to move to multi-type branching processes. In a world with multiple types, you have the possibility of doing some surgery that let you build a super star. Realistic Expectations: The paper is a reasonably dense 35 pages. Some of the branching process theory is drawn from the dark well of experts; it’s not off-the-shelf stuff. Still, if you want the deeper parts of the theory (e.g. the distribution of the maximum degree of the non-superstars) then you have to pay the piper. News You Can Use? One can see the benefits of using multi-type branching processes. One can see that the connection between the Yule process and preferential attachment is natural. This is enough to get you rolling in a variety of applied probability models (social net works are a good start — but they are not the only game.) J.M. Steele (U Penn, Wharton) May 2016 22 / 30
Superstar Model: Tools for Analysis Introduction of a Special Branching Process Two types of vertices: red and blue Each vertex gives birth to vertices according to a non-homogeneous Poisson process that has rate proportional to (1+ number of blue children) c B ( v , t ) = number of blue children of v at t time units after the birth of v At birth vertex is painted red with probability p and painted blue with probability 1 − p c B ( v 1 , t ) = 1 F ( t ) = Branching process at time t v 1 τ n = inf { t : |F ( t ) | = n } c B ( v 3 , t − τ 3 ) = 0 v 4 v 2 v 3 v 5 v 6 J.M. Steele (U Penn, Wharton) May 2016 23 / 30
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