CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

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1 CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

2 Setting from the last class: AB-A : gets a AB-B : gets b AB-AB : gets max(a, b) Also: Cost c for maintaining both strategies 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? -c -c a a max(a,b) A A AB AB b B Payoff 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 2

3 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 3 Path graph: Start with all Bs, a > b (A is better) One node switches to A what happens? With just A, B: A spreads if a > b With A, B, AB: Does A spread? Example: a=3, b=2, c=1 a=3 A a=3 A 0 b=2 b=2 B B B A a=3 A a=3 b=2 b=2 AB B B B Cascade stops -1 So even if A is better, not everyone adopts it

4 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 4 Example: a=5, b=3, c=1 A a=5 A 0 b=3 b=3 B B B A a=5 A a=5 b=3 b=3 AB B B B A a=5 A -1 a=5 a=5 b=3 AB B AB B B -1-1 A a=5 A a=5 a=5 b=3 A AB B AB B Cascade never stops! -1-1

5 Infinite path, start with all Bs Payoffs for w: A:a, B:1, AB:a+1-c What does node w in A-w-B do? c B B vs A A A A AB vs B a+1-c=1 w B 1 B AB AB 1 a AB vs A a+1-c=a 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 5

6 Infinite path, start with all Bs Payoffs for w: A:a, B:1, AB:a+1-c What does node w in A-w-B do? c B Since a<1, c>1 B vs A A a is big c is big A A AB vs B a+1-c=1 w B 1 B AB 1 a is high c <1, AB is opt AB 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 6 a AB vs A a+1-c=a

7 Same reward structure as before but now payoffs for w change: A:a, B:1+1, AB:a+1-c Notice: Now also AB spreads AB w What does node w in AB-w-B do? c B B vs A A A AB vs B B 1 B AB AB AB vs A 1 2 a 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 7

8 Same reward structure as before but now payoffs for w change: A:a, B:1+1, AB:a+1-c Notice: Now also AB spreads AB w What does node w in AB-w-B do? c B a<2, c>1 then 2b > 2a B vs A A a is big c >1 A AB vs B B 1 B AB AB c <1, then a+1-c > a AB is opt AB vs A 1 2 a 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 8

9 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 9 Joining the two pictures: c B A 1 AB B AB A 1 2 a

10 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 10 B is the default throughout the network until new/better A comes along. What happens? 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 c B stays B AB A spreads B A B AB A a

10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.

11 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 11 So far: Decision Based Models Utility based Deterministic Node centric: A node observes decisions of its neighbors and makes its own decision Require us to know too much about the data Next: Probabilistic Models Let s you do things by observing data We lose why people do things

12 Simple probabilistic model of cascades where we will learn about the reproductive number

13 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 13 Epidemic Model based on Random Trees (a variant of branching processes) A patient meets d other people With probability q > 0 infects each of them Q: For which values of d and q does the epidemic run forever? Run forever: At least 1infected lim P > h node at depth h Die out: = 0 Root node, patient 0 Start of epidemic 0 d subtrees

14 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 14 pp hh = prob. there is an infected node at depth hh We need: lim h pp h =? (based on qq and dd) Need recurrence for pp hh pp h = 1 1 qq pp h 1 dd llllll No infected node at depth h from the root pp hh = result of iterating hh f x = 1 1 qq xx dd Starting at xx = 1 (since pp 1 = 1) d subtrees

15 f(x) y=x=1 Going to first fixed point What do we know about f(x)? ff 0 = 0 ff 1 = 1 1 qq dd < 1 ff xx = qq dd 1 qqqq dd 1 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 15 x 1 y = f x When is this going to 0? ff 0 = qq dd ssss ff (xx) is monotone decreasing on [0,1]!

16 f(x) y=x y = f x Reproductive number RR 00 = qq dd: There is an epidemic if RR For the epidemic to die out we need f(x) to be below y=x! So: ff 00 = qq dd < 11 lim pp h = 0 wwheeee qq dd < 11 h qq dd = expected # of people at we infect 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, x

17 We will learn about the epidemic threshold

18 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 18 Virus Propagation: 2 Parameters: (Virus) Birth rate β: probability than an infected neighbor attacks (Virus) Death rate δ: Probability that an infected node heals Prob. β Prob. δ N 2 Healthy N 1 Infected N N 3

19 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 19 General scheme for epidemic models: Each node can go through phases: Transition probs. are governed by the model parameters S susceptible E exposed I infected R recovered Z immune

SIR model: Node goes through phases ββ δδ Susceptible Infected

never get infected again Assuming perfect mixing (The network

nodes I(t) R(t) time 10/15/2014 Jure Leskovec, Stanford CS224W:

20 SIR model: Node goes through phases ββ δδ Susceptible Infected Recovered Models chickenpox or plague: Once you heal, you can never get infected again Assuming perfect mixing (The network is a complete graph) the S(t) model dynamics is: Number of nodes I(t) R(t) time 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 20

21 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 21 Susceptible-Infective-Susceptible (SIS) model Cured nodes immediately become susceptible Virus strength : s = β / δ Node state transition diagram: Infected by neighbor with prob. β Susceptible Cured with prob. δ Infective

22 Number of nodes Susceptible time Infected Models flu: Susceptible node becomes infected The node then heals and become susceptible again Assuming perfect mixing (complete graph): ds dt di dt = βsi δi 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, I(t) S(t) = β SI +δi 22

23 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 23 SIS Model: Epidemic threshold of an arbitrary graph G is τ, such that: If virus strength s = β / δ < τ the epidemic can not happen (it eventually dies out) Given a graph what is its epidemic threshold?

24 [Wang et al. 2003] 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 24 We have no epidemic if: (Virus) Death rate Epidemic threshold β/δ < τ = 1/ λ 1,A (Virus) Birth rate largest eigenvalue of adj. matrix A λ 1,A alone captures the property of the graph!

25 [Wang et al. 2003] 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 25 Number of Infected Nodes Oregon β = ,900 nodes and 31,180 edges s=β/δ > τ (above threshold) s=β/δ = τ (at the threshold) Time δ: s=β/δ < τ (below threshold)

26 Does it matter how many people are initially infected? 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 26

27 [Gomes et al., Assessing the International Spreading Risk Associated with the 2014 West African Ebola Outbreak, PLOS Current Outbreaks, 2014] 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 27

28 Gomes et al., 2014] 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 28

29 Gomes et al., 2014] 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 29

31 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 31 Initially some nodes S are active Each edge (u,v) has probability (weight) p uv b c f a g e 0.3 d i 0.3 h When node u becomes active/infected: It activates each out-neighbor v with prob. p uv Activations spread through the network!

32 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 32 Independent cascade model is simple but requires many parameters! Estimating them from data is very hard [Goyal et al. 2010] Solution: Make all edges have the same weight (which brings us back to the SIR model) 0.4 Simple, but too simple Can we do something better? b c f a g e 0.3 d i 0.3 h

33 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, [KDD 12] 33 From exposures to adoptions Exposure: Node s neighbor exposes the node to the contagion Adoption: The node acts on the contagion

34 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 34 Exposure curve: Probability of adopting new behavior depends on the total number of friends who have already adopted What s the dependence? adopters Prob. of adoption k = number of friends adopting Diminishing returns: Viruses, Information Prob. of adoption k = number of friends adopting Critical mass: Decision making

35 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, [KDD 12] 35 From exposures to adoptions Exposure: Node s neighbor exposes the node to information Adoption: The node acts on the information Adoption curve: Prob(Infection) Probability of infection ever increases Nodes build resistance # exposures

36 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 36 Marketing agency would like you to adopt/buy product X They estimate the adoption curve Should they expose you to X three times? Or, is it better to expose you X, then Y and then X again? 3

37 [Leskovec et al., TWEB 07] 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 37 Senders and followers of recommendations receive discounts on products 10% credit 10% off Data: Incentivized Viral Marketing program 16 million recommendations 4 million people, 500k products [Leskovec-Adamic-Huberman, 2007]

10/15/2014 [Leskovec et al., TWEB 07] Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.

38 10/15/2014 [Leskovec et al., TWEB 07] Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 38 Books Probability of purchasing # recommendations received DVD recommendations (8.2 million observations)

39 10/15/2014 [Backstrom et al. KDD 06] Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 40 Group memberships spread over the network: Red circles represent existing group members Yellow squares may join Question: How does prob. of joining a group depend on the number of friends already in the group?

40 [Backstrom et al., KDD 06] 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 41 LiveJournal group membership Prob. of joining k (number of friends in the group)

41 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 42 Twitter [Romero et al. 11] Aug 09 to Jan 10, 3B tweets, 60M users Avg. exposure curve for the top 500 hashtags What are the most important aspects of the shape of exposure curves? Curve reaches peak fast, decreases after!

42 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 43 Persistence of P is the ratio of the area under the curve P and the area of the rectangle of length max(p), width max(d(p)) D(P) is the domain of P Persistence measures the decay of exposure curves Stickiness of P is max(p) Stickiness is the probability of usage at the most effective exposure

43 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 44 Manually identify 8 broad categories with at least 20 HTs in each Persistence True Rnd. subset Idioms and Music have lower persistence than that of a random subset of hashtags of the same size Politics and Sports have higher persistence than that of a random subset of hashtags of the same size

44 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 45 Technology and Movies have lower stickiness than that of a random subset of hashtags Music has higher stickiness than that of a random subset of hashtags (of the same size)

edu 48 So far we considered pieces of information as independently propagating.

46 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 48 So far we considered pieces of information as independently propagating. Do pieces of information interact? Did 1st cat video decrease adoption probability of 2nd cat video? ` Did cat videos increase adoption probability of dog video?

47 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 49 Goal: Model interaction between many pieces of information Some pieces of information may help each other in adoption Other may compete for attention

48 c2 c0 c1 c3 Neighbors The User P(adopt P(adopt c3 c2 P(adopt exposed c1 exposed c0) to to c2 to c1, c1, c0), c0) 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 50

49 You are reading posts on Twitter: You examine posts one by one Currently you are examining X How does your probability of reposting X depend on what you have seen in the past? Contagions adopted by neighbors: Y2 Y1 X c5 c4 c3 c2 Time Adopt? 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 51

50 Time 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 52 We assume K most recent exposures effect a user s adoption: P(adopt X=c 0 exposed Y 1 =c 1, Y 2 =c 2,..., Y K =c k ) Contagion the user is viewing now. Contagions adopted by neighbors: Contagions the user previously viewed. Y1 Y2 X c5 c4 c3 c2 c1 Adopt?

51 Time 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 53 We assume K most recent exposures effect a user s adoption: P(adopt X=c 0 exposed Y 1 =c 1, Y 2 =c 2,..., Y K =c k ) Contagion the user is viewing now. Contagions adopted by neighbors: Contagions the user previously viewed. Y1 Y2 X c5 c4 c3 c2 c1 c0 Adopt?

52 How many parameters? KK ww 22 Too many! KK history size ww number of contagions 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 54 Imagine we want to estimate: P(X Y 1, Y 5 ) What s the problem? What s the size of probability table P(X Y 1, Y 5 )? = (Num. Contagions) 5 1.9x10 21 Simplification: Assume Y i is independent of Y j PP XX YY 11,, YY KK = KK 11 PP XX KK 11 PP(XX YY kk) kk=11

53 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 55 Goal: Model P(adopt X Y 1,, Y K ) First, assume: Next, assume topics : Prior infection prob. Interaction term (still has w 2 entries!)

54 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 56 Details Goal: Model P(adopt X Y 1,, Y K ) First, assume: Next, assume topics : Prior infection prob. Interaction term (still has w 2 entries!) Each contagion uu ii has a vector MM ii Entry MM iiss models how much uu ii belongs to topic ss kk ΔΔ cccccccccc ss, tt models the change in infection prob. given that uu ii is on topic ss and exposure k-steps ago was on topic tt

55 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 57 Details Mj,a cj ci Mi,d (k) a b c d cluster a cluster b cluster c cluster d Memberships to clusters Interactions between clusters

56 10/16/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 58 Details Model parameters: Δ kk topic interaction matrix MM ii,tt... topic membership vector PP(XX)... Prior infection prob. Maximize data likelihood: arg max PP XX XX, YY 1 YY KK 1 PP XX XX, YY 1 YY KK PP xx,mm,δ XX RR XX RR RR contagions X that resulted in infections Solve using stochastic coordinate ascent: Alternate between optimizing Δ and MM

57 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 59 Data from Twitter Complete data from Jan 2011: 3 billion tweets All URLs tweeted by at least 50 users: 191k Task: Predict whether a user will post URL X What do we learn from the model?

58 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 60 How P(post u 2 exp. u 1 ) changes if u 2 and u 1 are similar/different in the content? u 1 is highly viral? Observations: If u 1 is not viral, this boost u 2 If u 1 is highly viral, this kills u 2 BUT: Only if u 1 and u 2 are of low content similarity (LCS) else, u 1 helps u 2 Relative change in infection prob.

59 10/15/2014 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, 61 Modeling contagion interactions 71% of the adoption probability comes from the topic interactions! Modeling user bias does not matter

CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University

CS224W: Social and Information Network Analysis Jure Leskovec Stanford University Jure Leskovec, Stanford University http://cs224w.stanford.edu [Morris 2000] Based on 2 player coordination game 2 players