http://cs224w.stanford.edu Observations Models Algorithms Small - PowerPoint PPT Presentation

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University http://cs224w.stanford.edu

Observations Models Algorithms Small diameter, Erdös-Renyi model, Decentralized search Edge clustering Small-world model Patterns of signed Structural balance, Models for predicting edge creation Theory of status edge signs Viral Marketing, Blogosphere, Independent cascade model, Influence maximization, Memetracking Game theoretic model Outbreak detection, LIM Preferential attachment, PageRank, Hubs and Scale-Free Copying model authorities Densification power law, Microscopic model of Link prediction, Shrinking diameters evolving networks Supervised random walks Strength of weak ties, Community detection: Kronecker Graphs Core-periphery Girvan-Newman, Modularity 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2

¡ Spreading through ¡ Examples: networks: § Biological: § Cascading behavior § Diseases via contagion § Technological: § Diffusion of innovations § Cascading failures § Network effects § Spread of information § Epidemics § Social: ¡ Behaviors that cascade § Rumors, news, new from node to node like technology an epidemic § Viral marketing 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 3

Obscure tech story Small tech blog Engadget Slashdot Wired BBC NYT CNN 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 4

10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 5

10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 6

¡ Product adoption: § Senders and followers of recommendations 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 7

10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 8

¡ Contagion that spreads over the edges of the network ¡ It creates a propagation tree, i.e., cascade Cascade Network (propagation graph) Terminology: • Stuff that spreads: Contagion • “Infection” event: Adoption, infection, activation • We have: Infected/active nodes, adoptors 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 9

¡ Decision based models (today!): § Models of product adoption, decision making § A node observes decisions of its neighbors and makes its own decision § Example: § You join demonstrations if k of your friends do so too ¡ Probabilistic models (on Thursday): § Models of influence or disease spreading § An infected node tries to “push” the contagion to an uninfected node § Example: § You “catch” a disease with some prob. from each active neighbor in the network 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10

[Granovetter ‘78] ¡ Collective Action [Granovetter, ‘78] § Model where everyone sees everyone else’s behavior (that is, we assume a complete graph) § Examples: § Clapping or getting up and leaving in a theater § Keeping your money or not in a stock market § Neighborhoods in cities changing ethnic composition § Riots, protests, strikes ¡ How does the number of people participating in a given activity grow or shrink over time? 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 12

¡ n people – everyone observes all actions ¡ Each person i has a threshold t i ( 0 ≤ 𝑢 $ ≤ 1 ) § Node i will adopt the behavior iff 1 P(adoption) at least t i fraction of people have already adopted: 0 § Small t i : early adopter t i § Large t i : late adopter § Time moves in discrete steps ¡ The population is described by {t 1 ,…,t n } § F(x) … fraction of people with threshold t i ≤ x § F(x) is given to us. F(x) is a property of the contagion. 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 13

¡ F(x) … fraction of people with threshold t i ≤ x § F(x) is non-decreasing: 𝑮 𝒚 + 𝜻 ≥ 𝑮 𝒚 ¡ The model is dynamic: § Step-by-step change Frac. of people 1 y=x in number of people with threshold ≤ 𝒚 Frac. of population adopting the behavior: F(x) § F(x) … frac. of people with threshold ≤ x § s(t) … frac. of people participating at time t § Simulate: § s(0) = 0 s(1) F(0) § s(1) = F(0) § s(2) = F(s(1)) = F(F(0)) s(0) Threshold, x 0 1 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 14

¡ Step-by-step change in number of people : § F(x) … fraction of people with threshold ≤ x § s(t) … number of participants at time t ¡ Easy to simulate: y=x § s(0) = 0 Frac. of population § s(1) = F(0) F(x) § s(2) = F(s(1)) = F(F(0)) § s(t+1) = F(s(t)) = F t+1 (0) ¡ Fixed point: F(x)=x Iterating to y=F(x). § Updates to s(t) to converge Fixed point. to a stable fixed point F(0) § There could be other fixed points but starting from 0 we only reach the first one Threshold, x 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 15

¡ What if we start the process somewhere else? § We move up/down to the next fixed point § How is market going to change? y=x Frac. of pop. F(x) Note: we are assuming a fully connected graph Threshold, x 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 16

y=x Frac. of pop. Unstable fixed point Stable fixed point Threshold, x 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 17

¡ Each threshold t i is drawn independently from some distribution F(x) = Pr[thresh ≤ x] § Suppose: Truncated normal with µ =45 , variance σ Small σ : Large σ : Normal(45, 10) Normal(45, 27) 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 18

Normal(45, 10) Normal(45, 27) Medium σ Small σ F(x) F(x) Fixed point is low No cascades! Small cascades Bigger variance lets you build a bridge from early adopters to mainstream 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 19

Normal(45, 33) Normal(45, 50) Huge σ Big σ Fixed point is high! Fixed point gets lower! Big cascades! But if we increase the variance the fixed point starts going down! 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 20

¡ No notion of social network: § Some people are more influential § It matters who the early adopters are, not just how many ¡ Models people’s awareness of size of participation not just actual number of people participating § Modeling perceptions of who is adopting the behavior vs. who you believe is adopting § Non-monotone behavior – dropping out if too many people adopt § People get “locked in” to certain choice over a period of time ¡ Modeling thresholds § Richer distributions § Deriving thresholds from more basic assumptions § Game theoretic models 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 21

¡ Dictator tip: Pluralistic ignorance – erroneous estimates about the prevalence of certain opinions in the population § Survey conducted in the U.S. in 1970 showed that while a clear minority of white Americans at that point favored racial segregation, significantly more than 50% believed that it was favored by a majority of white Americans in their region of the country 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 22

[Morris 2000] ¡ Based on 2 player coordination game § 2 players – each chooses technology A or B § Each person can only adopt one “behavior”, A or B § You gain more payoff if your friend has adopted the same behavior as you Local view of the network of node v 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 24

10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 25

10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 26

¡ Payoff matrix: § If both v and w adopt behavior A , they each get payoff a > 0 § If v and w adopt behavior B , they reach get payoff b > 0 § If v and w adopt the opposite behaviors, they each get 0 ¡ In some large network: § Each node v is playing a copy of the game with each of its neighbors § Payoff : sum of node payoffs per game 10/18/16 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 27