http://cs224w.stanford.edu In decision-based models nodes make - PowerPoint PPT Presentation

CS224W: Machine Learning with Graphs Jure Leskovec, Stanford University http://cs224w.stanford.edu

¡ In decision-based models nodes make decisions based on pay-off benefits of adopting one strategy or the other ¡ In epidemic spreading: § Lack of decision making § Process of contagion is complex and unobservable § In some cases it involves (or can be modeled as) randomness Recap 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 2

11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 3

¡ Epidemic Model based on Random Trees § (a variant of branching processes) Root node, § A patient meets d new people “patient 0” Start of epidemic § With probability q > 0 she infects each of them d subtrees ¡ Q: For which values of d and q does the epidemic run forever? $→& 𝑸 𝒃 𝒐𝒑𝒆𝒇 𝒃𝒖 𝒆𝒇𝒒𝒖𝒊 𝒊 § Run forever: lim > 𝟏 𝒋𝒕 𝒋𝒐𝒈𝒇𝒅𝒖𝒇𝒆 $→& 𝑸 𝒃 𝒐𝒑𝒆𝒇 𝒃𝒖 𝒆𝒇𝒒𝒖𝒊 𝒊 § Die out: lim = 𝟏 𝒋𝒕 𝒋𝒐𝒈𝒇𝒅𝒖𝒇𝒆 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 4

¡ 𝒒 𝒊 = prob. a node at depth 𝒊 is infected ¡ We need: lim $→& 𝑞 $ = ? (based on 𝑟 and 𝑒 ) § We are reasoning about a behavior at the root of the tree. Once we get a level out, we are left with identical problem of depth ℎ − 1 . ¡ Need recurrence for 𝒒 𝒊 A 𝑞 $ = 1 − 1 − 𝑟 ⋅ 𝑞 $?@ d subtrees No infected node at depth h from the root ¡ 𝒎𝒋𝒏 𝒊→& 𝒒 𝒊 = result of iterating We iterate: f x = 1 − 1 − 𝑟 ⋅ 𝑦 A x 1 =f(1) x 2 =f(x 1 ) § Starting at the root: 𝑦 = 1 (since 𝑞 @ = 1 ) x 3 =f(x 2 ) 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 5

x … prob. a node y=x=1 f(x) at level h-1 is infected . We start at x=1 Fixed point: because p 1 =1. 𝑔(𝑦) = 𝑦 f(x) … prob. a node This means that at level h is infected prob. there is an y = f x q … infection prob. infected node at depth d … degree ℎ is constant (>0) Going to the first fixed point We iterate: x 1 =f(1) x 2 =f(x 1 ) x 3 =f(x 2 ) x 1 If we want to epidemic to die out, then iterating 𝑔(𝑦) must go to zero. So , 𝑔(𝑦) must be below 𝑧 = 𝑦 . ¡ What’s the shape of 𝒈(𝒚) ? 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 6

y=x=1 f(x) x … prob. a node at level h-1 is infected . We start at x=1 because p 1 =1. f(x) … prob. a node y = f x at level h is infected q … infection prob. d … degree Going to the first fixed point x 1 What do we know about the shape of 𝒈(𝒚) ? • 𝑔 0 = 0 f’(x) is monotone: If g’(y)>0 for all y then g(y) is monotone. • 𝑔 1 = 1 − 1 − 𝑟 A < 1 In our case, 0≤x,q≤1, d>1 so f’(x)>0, so f(x) is monotone. f’(x) non-increasing : since term (1-qx) d-1 in f’(x) is • 𝑔 P 𝑦 = 𝑟 ⋅ 𝑒 1 − 𝑟𝑦 A?@ decreasing as x decreases. • 𝑔 P 0 = 𝑟 ⋅ 𝑒 𝒈′(𝒚) is monotone non-increasing on [0,1]! 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 7

y=x f(x) Reproductive number 𝑺 𝟏 = 𝒓 ⋅ 𝒆: There is an epidemic if y = f x 𝑺 𝟏 ³ 𝟐 1 x For the epidemic to die out we need 𝒈(𝒚) to be below 𝒛 = 𝒚 ! So: 𝒈 P 𝟏 = 𝒓 ⋅ 𝒆 < 𝟐 $→& 𝑞 $ = 0 𝑥ℎ𝑓𝑜 𝒓 ⋅ 𝒆 < 𝟐 lim 𝒓 ⋅ 𝒆 = expected # of people that get infected 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 8

¡ Reproductive number 𝑺 𝟏 = 𝒓 ⋅ 𝒆: § It determines if the disease will spread or die out. ¡ There is an epidemic if 𝑺 𝟏 ≥ 𝟐 ¡ Only R 0 matters: § 𝑺 𝟏 ≥ 𝟐 : epidemic never dies and the number of infected people increases exponentially § 𝑺 𝟏 < 𝟐 : Epidemic dies out exponentially quickly 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 9

¡ When R 0 is close 1, slightly changing 𝒓 or 𝒆 can result in epidemics dying out or happening § Quarantining people/nodes [reducing 𝒆 ] § Encouraging better sanitary practices reduces germs spreading [reducing 𝒓 ] § HIV has an R 0 between 2 and 5 § Measles has an R 0 between 12 and 18 § Ebola has an R 0 between 1.5 and 2 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 10

Characterizing social cascades in Flickr. Cha et al. ACM WOSN 2008

¡ Flickr social network: § Users are connected to other users via friend links § A user can “like/favorite” a photo ¡ Data: § 100 days of photo likes § Number of users: 2 million § 34,734,221 likes on 11,267,320 photos 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 12

¡ Users can be exposed to a photo via social influence (cascade) or external links ¡ Did a particular like spread through social links? § No , if a user likes a photo and if none of his friends have previously liked the photo § Yes, if a user likes a photo after at least one of her friends liked the photo à Social cascade ¡ Example social cascade: A à B and A à C à E 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 13

¡ Recall: 𝑆 0 = 𝑟 ∗ 𝑒 ¡ Estimate of 𝑆 0 : § Estimating 𝒓 : Given an infected node count the proportion of its neighbors subsequently infected and average 𝑒 … avg degree 𝑒 c …degree of node 𝑗 b ) ^_`(A a § Then: 𝑆 ] = 𝑟 ∗ 𝑒 ∗ ^_` A a b Correction factor due to skewed ¡ Empirical 𝑆 0 : degree distribution of the network § Given start node of a cascade, count the fraction of directly infected nodes and proclaim that to be 𝑆 0 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 14

¡ Data from top 1,000 photo cascades ¡ Each + is one cascade 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 15

¡ The basic reproduction number of popular photos is between 1 and 190 ¡ This is much higher than very infectious diseases like measles, indicating that social networks are efficient transmission media and online content can be very infectious. 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 16

Virus Propagation: 2 Parameters: ¡ (Virus) Birth rate β: § probability that an infected neighbor attacks ¡ (Virus) Death rate δ: § Probability that an infected node heals Healthy Prob. δ N 2 Prob. β N 1 P N r o b . β Infected N 3 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 18

¡ 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 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 19

¡ SIR model: Node goes through phases 𝜀 𝛾 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) model dynamics are: R(t) Number of nodes dS dR dt = − β SI dt = δ I I(t) dI dt = β SI − δ I time 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 20

¡ Susceptible-Infective-Susceptible (SIS) model ¡ Cured nodes immediately become susceptible ¡ Virus “strength”: 𝒕 = 𝜸 / 𝜺 ¡ Node state transition diagram: Infected by neighbor with prob. β Susceptible Infective Cured with prob. δ 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 21

¡ Models flu: § Susceptible node I(t) becomes infected Number of nodes § The node then heals and become susceptible again ¡ Assuming perfect mixing (a complete S(t) graph): dS = - b + d SI I dt time dI S usceptible I nfected = b - d SI I dt 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 22

¡ 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? 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 23

[Wang et al. 2003] ¡ Fact: 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 of G ► λ 1, A alone captures the property of the graph! 11/5/19 Jure Leskovec, Stanford CS224W: Machine Learning with Graphs, http://cs224w.stanford.edu 24