http://cs224w.stanford.edu Last time: Decision Based Models Utility - PowerPoint PPT Presentation

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

 Last time: 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  Today: Probabilistic Models  Let’s you do things by observing data  We loose “why people do things” 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2

 Epidemic Model based on Random Trees  (a variant of branching processes) Root node, “patient 0”  A patient meets d other people Start of epidemic  With probability q > 0 infects each d subtrees of them  Q: For which values of d and q does the epidemic run forever?   infected node  Run forever:  lim  P 0   at depth h    h  Die out: ‐‐ || ‐‐ 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 4

� = prob. there is an infected node at depth   We need: (based on and ) � �→�  Need recurrence for � ��� � � No infected node d subtrees at depth h from the root � = result of iterating  �→� �  Starting at (since � ) 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 5

y=x=1 f(x) y � f x Going to first fixed point When is this going to 0? 1 x What do we know about f(x)? � 0 � 0 � 1 � 1 � 1 � � � � 1 � � � � � ⋅ � 1 � �� ��� � � is monotone decreasing on [0,1]! 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 6

y=x f(x) Reproductive number � There is an epidemic if y � f x �  1 x For the epidemic to die out we need f(x) to be bellow y=x ! So: � � �→� = expected # of people at we infect 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 7

 In this model nodes only go from healthy  infected  We can generalize to allow nodes to alternate between healthy and infected state by: 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 8

Virus Propagation: 2 Parameters:  (Virus) birth rate β :  probability than an infected neighbor attacks  (Virus) death rate δ :  probability that an infected node heals Healthy Prob. δ N 2 Prob. β N 1 N Infected N 3 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10

 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 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 11

 SIR model: Node goes through phases � � S usceptible I nfected R ecovered  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) R(t) model dynamics is: Number of nodes dS dR    SI dt   I I(t) dt dI   SI   I dt time 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 12

 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 Infective Cured internally with prob. δ 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 13

 Models flu:  Susceptible node I(t) becomes infected Number of nodes  The node then heals and become susceptible again  Assuming perfect mixing (complete S(t) graph): dS      SI I dt time dI S usceptible I nfected     SI I dt 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 14

 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? 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 15

[Wang et al. 2003]  We have no epidemic if: Epidemic threshold (Virus) Death rate β / δ < τ = 1/ λ 1, A largest eigenvalue (Virus) Birth rate of adj. matrix A ► λ 1, A alone captures the property of the graph! 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 16

[Wang et al. 2003] 10,900 nodes and 500 Oregon 31,180 edges β = 0.001 Number of Infected Nodes β / δ > τ 400 (above threshold) 300 200 β / δ = τ 100 (at the threshold) 0 β / δ < τ 0 250 500 750 1000 (below threshold) Time δ : 0.05 0.06 0.07 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 17

 Does it matter how many people are initially infected? 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 18

 Initially some nodes S are active  Each edge (u,v) has probability (weight) p uv 0.4 a d 0.4 0.2 0.3 0.3 0.2 0.3 b f f 0.2 e e h 0.4 0.4 0.3 0.2 0.3 0.3 g g i 0.4 c  When node v becomes active:  It activates each out ‐ neighbor v with prob. p uv  Activations spread through the network 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 20

0.4 a  Independent cascade model d 0.4 0.2 0.3 is simple but requires 0.3 0.2 0.3 b f f many parameters! 0.2 e e h 0.4 0.4 0.3  Estimating them from 0.2 0.3 0.3 g g data is very hard i 0.4 c [Goyal et al. 2010]  Solution: Make all edges have the same weight (which brings us back to the SIR model)  Simple, but too simple  Can we do something better? 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 21

[KDD ‘12]  From exposures to adoptions  Exposure: Node’s neighbor exposes the node to the contagion  Adoption: The node acts on the contagion 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 22

 Exposure curve:  Probability of adopting new behavior depends on the number of friends who have already adopted … adopters  What’s the dependence? Prob. of adoption Prob. of adoption k = number of friends adopting k = number of friends adopting Diminishing returns: Critical mass: Viruses, Information Decision making 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 23

[KDD ‘12]  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 Nodes build infection ever resistance increases # exposures 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 24

 Marketing agency would like you to adopt/buy product X  They estimate the adoption curve  Should they expose you to X three times? 3  Or, is it better to expose you X , then Y and then X again? 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 25

[Leskovec et al., TWEB ’07]  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] 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 26

[Leskovec et al., TWEB ’07] 0.1 Probability of purchasing 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 10 20 30 40 0.06 # recommendations received 0.05 Probability of Buying 0.04 DVD recommendations 0.03 Books (8.2 million observations) 0.02 0.01 0 2 4 6 8 10 Incoming Recommendations Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 27

 What is the effectiveness of subsequent recommendations? -3 12x 10 0.07 DVDs BOOKS 0.06 Probability of buying 10 Probability of buying 0.05 8 0.04 6 0.03 4 0.02 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Exchanged recommendations Exchanged recommendations 10/18/2012 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 28