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

Spreading Through Networks Cascading behavior Diffusion of innovations Network effects Epidemics Behaviors that cascade from node to node like an epidemic Examples: Biological: Diseases via contagion Technological: Cascading failures Spread of information Social: Rumors, news, new technology Viral marketing 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Information Diffusion 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Diffusion in Viral Marketing Product adoption: Senders and followers of recommendations 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Spread of Diseases 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Network Cascades Behavior/contagion spreads over the edges of the network It creates a propagation tree, i.e., cascade Cascade (propagation graph) Network Terminology: Stuff that spreads: Contagion “Infection” event: Adoption, infection, activation We have: Infected/active nodes, adoptors 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

How to Model Diffusion? Probabilistic models: Decision based models: 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 Decision based models: Models of product adoption, decision making A node observes decisions of its neighbors and makes its own decision You join demonstrations if k of your friends do so too 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Decision Based Model of Diffusion

Decision Based Models Two ingredients: Payoffs: Utility of making a particular choice Signals: Public information: What your network neighbors have done (Sometimes also) Private information: Something you know Your belief Now you want to make the optimal decision 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Decision Based Models Collective Action [Granovetter, ‘78] Model where everyone sees everyone else’s behavior 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 the number of people participating in a given activity with network effects would grow or shrink over time? 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Collective Action: The Model n people – everyone observes all actions Each person i has a threshold ti Node i will adopt the behavior iff at least ti other people are adopters: Small ti: early adopter Large ti: late adopter The population is described by {t1,…,tn} F(x) … fraction of people with threshold ti  x 1 P(adoption) ti 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Collective Action: Dynamics Here we really need to split into 2 slides and explain the axes and show in text and in animation why the amount of people grows Think of the step-by-step change in number of people adopting the behavior: F(x) … fraction of people with threshold  x s(t) … number of participants at time t Easy to simulate: s(0) = 0 s(1) = F(0) s(2) = F(s(1)) = F(F(0)) s(t+1) = F(s(t)) = Ft+1(0) Fixed point: F(x)=x Updates to s(t) to converge to a stable fixed point There could be other fixed points but starting from 0 we never reach them Threshold, x F(x) y=x Frac. of population Iterating to y=F(x). Fixed point. F(0) 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Starting Elsewhere What if we start the process somewhere else? Show lines how the behavior will change if we start at blue vs red dot. What if we start the process somewhere else? We move up/down to the next fixed point How is market going to change? Threshold, x Frac. of pop. y=x F(x) If we start here we move up to the next fixed point If we start here we move down to the next fixed point 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Fragile vs. Robust Fixed Point Show lines how the behavior will change if we start at blue vs red dot. Threshold, x Frac. of pop. y=x Fragile fixed point Robust fixed point 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Discontinuous Transition Each threshold ti is drawn independently from some distribution F(x) = Pr[thresh  x] Suppose: Normal with =n/2, variance  Small : Large : 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Discontinuous Transition Small  Medium  F(x) F(x) Fixed point is low No cascades! Small cascades Bigger variance let’s you build a bridge from early adopters to mainstream 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Discontinuous Transition Big  Huge  Fixed point is high! Fixed point gets lower! Big cascades! But if we increase the variance even more we move the higher fixed point lower 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Weaknesses of the Model It does not take into account: 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 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

How should we organize a revolt? Work on these slides to show how the network structure matters! You live in an oppressive society You know of a demonstration against the government planned for tomorrow If a lot of people show up, the government will fall If only a few people show up, the demonstrators will be arrested and it would have been better had everyone stayed at home 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Pluralistic ignorance You should do something if you believe you are in the majority! 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 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Organizing the revolt: The model Personal threshold k: “I will show up to the protest if I am sure at least k people in total (including myself) will show up” Each node in the network knows the thresholds of all their friends 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Subtle issues Will uprising occur? No! 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Subtle issues Will uprising occur? No! 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Subtle issues Will uprising occur? Yes! 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Game Theoretic Model of Cascades

Game Theoretic Model of Cascades [Morris 2000] Game Theoretic Model of Cascades 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 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example: BlueRay vs. HD DVD 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

The Model for Two Nodes Payoff matrix: In some large network: 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 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Calculation of Node v Let v have d neighbors Explain the threshold better – relative reward matters, not the absolute Threshold: v choses A if p>q Let v have d neighbors Assume fraction p of v’s neighbors adopt A Payoffv = a∙p∙d if v chooses A = b∙(1-p)∙d if v chooses B Thus: v chooses A if: a∙p∙d > b∙(1-p)∙d 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario Scenario: Graph where everyone starts with B. Small set S of early adopters of A Hard wire S – they keep using A no matter what payoffs tell them to do Assume payoffs are set in such a way that nodes say: If more than 50% of my friends are red I’ll be red (this means: a = b-ε and q>1/2) 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario u v If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario u v If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario u v If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario u v If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Scenario u v If more than 50% of my friends are red I’ll be red 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Monotonic Spreading Observation: Why? Proof sketch: The use of A spreads monotonically (Nodes only switch from B to A, but never back to B) Why? Proof sketch: Nodes keep switching from B to A: BA Now, suppose some node switched back from AB, consider the first node v to do so (say at time t) Earlier at time t’ (t’<t) the same node v switched BA So at time t’ v was above threshold for A But up to time t no node switched back to B, so node v could only had more neighbors who used A at time t compared to t’. There was no reason for v to switch. !! Contradiction !! 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Infinite Graphs Consider infinite graph G v choses A if p>q Consider infinite graph G (but each node has finite number of neighbors) We say that a finite set S causes a cascade in G with threshold q if, when S adopts A, eventually every node adopts A Example: Path If q<1/2 then cascade occurs S 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Infinite Graphs Infinite Tree: Infinite Grid: If q<1/3 then cascade occurs S If q<1/4 then cascade occurs S 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Cascade Capacity Def: Fact: Proof idea: The cascade capacity of a graph G is the largest q for which some finite set S can cause a cascade Fact: There is no G where cascade capacity > ½ Proof idea: Suppose such G exists: q>½, finite S causes cascade Show contradiction: Argue that nodes stop switching after a finite # of steps X 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Cascade Capacity Fact: There is no G where cascade capacity > ½ Proof sketch: Suppose such G exists: q>½, finite S causes cascade Contradiction: Switching stops after a finite # of steps Define “potential energy” Argue that it starts finite (non-negative) and strictly decreases at every step “Energy”: = |dout(X)| |dout(X)| := # of outgoing edges of active set X The only nodes that switch have a strict majority of its neighbors in S |dout(X)| strictly decreases It can do so only a finite number of steps X 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Stopping Cascades What prevents cascades from spreading? Def: Cluster of density ρ is a set of nodes C where each node in the set has at least ρ fraction of edges in C. ρ=3/5 ρ=2/3 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Stopping Cascades Let S be an initial set of adopters of A All nodes apply threshold q to decide whether to switch to A Two facts: 1) If G\S contains a cluster of density >(1-q) then S can not cause a cascade 2) If S fails to create a cascade, then there is a cluster of density >(1-q) in G\S S ρ=3/5 No cascade if q>2/5 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

END Next year: Points to ask students: Move the herding slides back into lecture 1 Then we will have 2 lectures: Herding and the collective action model And the 2 game-theoretic models Points to ask students: What are Collective action model deficiencies? What are other examples of cascading behavior? Ask them how to complete the proof, what is the argument 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Extending the model: Allow people to adopt A and B

Cascades & Compatibility So far: Behaviors A and B compete Can only get utility from neighbors of same behavior: A-A get a, B-B get b, A-B get 0 Let’s add extra strategy “A-B” AB-A: gets a AB-B: gets b AB-AB: gets max(a, b) Also: Some cost c for the effort of maintaining both strategies (summed over all interactions) 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Cascades & Compatibility: Model Every node in an infinite network starts with B Then a finite set S initially adopts A Run the model for t=1,2,3,… Each node selects behavior that will optimize payoff (given what its neighbors did in at time t-1) How will nodes switch from B to A or AB? b A B a A a AB a+b-c AB Edge payoff 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Path: Start with all Bs, a>b (A is better) One node switches to A – what happens? With just A, B: A spreads if b  a With A, B, AB: Does A spread? Assume a=2, b=3, c=1 a=2 b=3 b=3 A A B B B B A a=2 b=3 AB -1 Cascade stops 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Example Let a=5, b=3, c=1 A A B B B B A AB B A AB A B AB a=5 b=3 b=3 b=3 b=3 A A B B B B A a=5 b=3 AB -1 B A a=5 b=3 AB -1 A a=5 B b=3 AB -1 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

For what pairs (c,a) does A spread? Infinite path, start with all Bs Payoffs: A:a, B:1, AB:a+1-c What does node w in A-w-B do? w A B B vs A AB vs B a c a+1-c=1 A B A 1 AB vs A B a+1-c=a AB AB 1 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

For what pairs (c,a) does A spread? Payoffs: A:a, B:2, AB:a+2-c Notice: Now also AB spreads What does node w in AB-w-B do? w AB B B vs A AB vs B a c A B A 1 AB vs A B AB AB 1 2 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

For what pairs (c,a) does A spread? Joining the two pictures: a c A B 1 AB B→AB → A 1 2 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Lesson You manufacture default B and new/better A comes along: Infiltration: If B is too compatible then people will take on both and then drop the worse one (B) Direct conquest: If A makes itself not compatible – people on the border must choose. They pick the better one (A) Buffer zone: If you choose an optimal level then you keep a static “buffer” between A and B a c A spreads B → A B stays B→AB→A B→AB 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Decision Based Model: Herding [Banerjee ‘92] Decision Based Model: Herding Influence of actions of others Model where everyone sees everyone else’s behavior Sequential decision making Example: Picking a restaurant Consider you are choosing a restaurant in an unfamiliar town Based on Yelp reviews you intend to go to restaurant A But then you arrive there is no one eating at A but the next door restaurant B is nearly full What will you do? Information that you can infer from other’s choices may be more powerful than your own 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Herding: Structure Herding: There is a decision to be made People make the decision sequentially Each person has some private information that helps guide the decision You can’t directly observe private information of the others but can see what they do You can make inferences about the private information of others 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Herding: Simple Experiment Consider an urn with 3 marbles. It can be either: Majority-blue: 2 blue, 1 red, or Majority-red: 1 blue, 2 red Each person wants to best guess whether the urn is majority-blue or majority-red Guess red if P(majority-red | what she has seen or heard) > ½ Experiment: One by one each person: Draws a marble Privately looks are the color and puts the marble back Publicly guesses whether the urn is majority-red or majority-blue You see all the guesses beforehand. How should you make your guess? 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Herding: What Happens? Informally, What happens? [Banerjee ‘92] Herding: What Happens? See ch. 16 of Easley-Kleinberg for formal analysis Informally, What happens? #1 person: Guess the color you draw from the urn. #2 person: Guess the color you draw from the urn. Why? If same color as 1st, then go with it If different, break the tie by doing with your own color #3 person: If the two before made different guesses, go with your color Else, go with their guess (regardless your color) – cascade starts! #4 person: Suppose the first two guesses were R, you go with R Since 3rd person always guesses R Everyone else guesses R (regardless of their draw) 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Herding: Three Ingredients State of the world: Whether the urn is MR or MB Payoffs: Utility of making a correct guess Signals: Models private information: The color of the marble that you just draw Models public information: The MR vs MB guesses of people before you 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Sequential Decision Making Decision: Guess MR if 𝑃 𝑴𝑹 𝑝𝑎𝑠𝑡 𝑎𝑐𝑡𝑖𝑜𝑛𝑠 > 1 2 Analysis (Bayes rule): #1 follows her own color (private signal)! Why? #2 guesses her own color (private signal)! #2 knows #1 revealed her color. So, #2 gets 2 colors. If they are the same, decision is easy. If not, break the tie in favor of her own color 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Sequential Decision Making #3 follows majority signal! Knows #1, #2 acted on their colors. So, #3 gets 3 signals. If #1 and #2 made opposite decisions, #3 goes with her own color. Future people will know #3 revealed its signal If #1 and #2 made same choice, #3’s decision conveyed no info. Cascade has started! How does this unfold? You are N-th person #MB = #MR : you guess your color |#MB - #MR|=1 : your color makes you indifferent, or reinforces you guess |#MB - #MR| ≥ 2 : Ignore your signal. Go with majority. 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Sequential Decision Making Cascade begins when the difference between the number of blue and red guesses reaches 2 Guess B Guess B Guess B Guess R Guess B #MB – #MR guesses Guess R 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Herding: Observations Easy to occur given the right structural conditions Can lead to bizarre patterns of decisions Non-optimal outcomes With prob. ⅓⅓=⅟9 first two see the wrong color, from then on the whole population guesses wrong Can be very fragile Suppose first two guess blue People 100 and 101 draw red and cheat by showing their marbles Person 102 now has 4 pieces of information, she guesses based on her own color Cascade is broken 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Classical Studies of Diffusion Spread of new agricultural practices [Ryan-Gross ‘43] Adoption of a new hybrid-corn between the 259 farmers in Iowa Interpersonal network plays important role in adoption  Diffusion is a social process Spread of new medical practices [Coleman et al. ‘66] Adoption of a new drug between doctors in Illinois Clinical studies and scientific evaluations were not sufficient to convince the doctors It was the social power of peers that led to adoption 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Modeling Diffusion: Curves Basis for models: Probability of adopting new behavior depends on the number of friends who have adopted What’s the dependence? k = number of friends adopting Prob. of adoption k = number of friends adopting Prob. of adoption Diminishing returns: Viruses, Information Critical mass: Decision making 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

Connectivity of Directed Graphs Strongly connected directed graph has a path from each node to every other node and vice versa (e.g., A-B path and B-A path) Weakly connected directed graph is connected if we disregard the edge directions E F B A Graph on the left is not strongly connected. D C G 4/17/2019 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu