http://cs224w.stanford.edu Probabilistic models of network contagion - PowerPoint PPT Presentation

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

 Probabilistic models of network contagion  Probabilistic models of network contagion  How contagions diffuse in real ‐ life: g  Viral marketing  Blogs Blogs  Group membership 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 2

 How do viruses/rumors propagate? How do viruses/rumors propagate?  Will a flu ‐ like virus linger, or will it become extinct?  (Virus) birth rate β :  (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 2 Prob. β N 1 N Infected N 3 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 3

 General scheme for epidemic models:  General scheme for epidemic models: S…susceptible E…exposed I…infected R…recovered d Z…immune 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 4

 Assuming perfect g p mixing, i.e., a network is a odes complete graph mber of no  The model dynamics: Nu time Susceptible Infected Recovered 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 5

 Susceptible Infective Susceptible (SIS) model  Susceptible ‐ Infective ‐ Susceptible (SIS) model  Cured nodes immediately become susceptible  Virus “strength”: s = β / δ  Virus strength : s = β / δ Infected by neighbor with prob. β Susceptible Infective Cured internally with prob. δ 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 6

 Assuming perfect f mixing (complete nodes graph): graph): Number of n dS       SI I dt N dI       SI SI I I dt time S sceptible Susceptible Infected Infected 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 7

 Representing SIS epidemic an SIR model  Representing SIS epidemic an SIR model 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 8

 Epidemic threshold of a graph is a  Epidemic threshold of a graph is a value of t , such that:  If strength s = β / δ < t epidemic can not  If strength s = β / δ < t epidemic can not happen (it eventually dies out)  Given a graph compute its epidemic threshold 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 9

 What should t depend on?  What should t depend on?  Avg. degree? And/or highest degree?  A d/  And/or variance of degree? i f d ?  And/or third moment of degree?  And/or diameter? A d/ di ? 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 10

[Wang et al. 2003]  We have no epidemic if:  We have no epidemic if: Epidemic threshold (Virus) Death ( ) rate β β / δ < τ = 1/ λ 1 A 1, A largest eigenvalue (Virus) Birth rate of adj. matrix A ► λ A alone captures the property of the graph! ► λ 1, A alone captures the property of the graph! 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 11

[Wang et al. 2003] 10,900 nodes and 500 Oregon 3 , 31,180 edges g β = 0.001 β 0 001 d Nodes β / δ > τ 400 (above threshold) f Infected 300 200 200 umber of β / δ = τ 100 (at the threshold) N 0 β β / δ < τ 0 250 500 750 1000 (below threshold) Time δ : 0.05 0.06 0.07 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 12

 Does it matter how many people are  Does it matter how many people are initially infected? 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 13

 Prob. of adoption depends on the number of  Prob of adoption depends on the number of friends who have adopted [Bass ‘69, Granovetter ’78]  What is the shape? What is the shape?  Distinction has consequences for models and algorithms on on of adoptio of adoptio Prob. o Prob. o k = number of friends adopting k = number of friends adopting k = number of friends adopting k = number of friends adopting Diminishing returns? Critical mass? 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 14

[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 illi l • 500,000 products 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 15

[Backstrom et al., KDD ’06]  Use social networks where people belong to  Use social networks where people belong to explicitly defined groups  Each group defines a behavior that diffuses  Each group defines a behavior that diffuses  Data – LiveJournal:  On ‐ line blogging community with friendship links and user ‐ defined groups p g p  Over a million users update content each month  Over 250,000 groups to join 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 16

[Leskovec et al., TWEB ’07] DVD recommendations DVD recommendations (8.2 million observations) asing 0.1 0.09 of purcha 0.08 0.07 0.06 bability o 0 05 0.05 0.04 0.03 0.02 0 02 Prob 0.01 0 0 0 10 10 20 20 30 30 40 40 # recommendations received 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 17

[Backstrom et al., KDD ’06]  LiveJournal community membership  LiveJournal community membership oining rob. of jo Pr k ( k (number of friends in the community) b f f i d i th it ) 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 18

 For viral marketing:  For viral marketing:  We see that node v receiving the i ‐ th recommendation and then purchased the product p p  For communities:  At time t we see the behavior of node v ’s friends  Questions:  When did v become aware of recommendations or friends’ behavior? f i d ’ b h i ?  When did it translate into a decision by v to act?  How long after this decision did v act?  How long after this decision did v act? 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 19

 Large anonymous online retailer Large anonymous online retailer (June 2001 to May 2003)  15,646,121 recommendations 15 646 121 d ti  3,943,084 distinct customers  548 523 products recommended 548,523 products recommended  Products belonging to 4 product groups:  books  DVDs  music  VHS 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 20

purchase following a recommendation customer recommending a product customer not buying a recommended product Majority of  recommendations do not cause purchases nor cause purchases nor propagation Notice many star ‐ like  patterns patterns Many disconnected  components 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 21

 t < t <  t 1 < t 2 < … < t n < t legend t 3 bought but didn’t receive a discount i di t t 1 bought and received a discount t 2 received a recommendation but didn’t buy t 5 t 4 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 22

 What role does the product category play?  What role does the product category play? buy + get recommenda- buy + no products customers edges tions tions discount discount discount discount Book 103,161 2,863,977 5,741,611 2,097,809 65,344 17,769 DVD 805,285 962,341 19,829 8,180,393 17,232 58,189 Music 393,598 794,148 1,443,847 585,738 7,837 2,739 Video 26,131 239,583 280,270 160,683 909 467 Full F ll 542,719 542 719 3 943 084 3,943,084 15 646 121 15,646,121 3 153 676 3,153,676 91,322 91 322 79 164 79,164 people recommendations high low 10/13/2009 Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 23