and Internet Computing Amin Saberi Stanford University

Transcription

1 Epidemics Algorithmic on Social Game Networks Theory and Internet Computing Amin Saberi Stanford University

2 Social Networks as Graphs Nodes: individuals, webpages (blogs), profiles, PC s Edges: friendship, link from one page to the other, exchange of messages, s recent hire worked a while old timer

3 Outline Mathematical models Structure of social networks (power-law networks) Spread of infections (SIS model) SIS Model on Power-law Networks The epidemic threshold Controlling Epidemics on a Network

4 A very important structural property Power-law degree distribution: Prob(vertex has degree k) k γ Observed on the WWW, AS Internet, Social networks It is also referred to as long-tail distribution or scale-free distribution

5 Example I: The Web Graph P(k) ~ k -γ R. Albert, H. Jeong, A-L Barabasi, Nature 99

6 Example II: The Sex Web Lilijeros et al. 01

7 Model: Preferential Attachment Add one vertex at a time New vertex attaches to existing vertices Yule 25, Simon 55, Barabasi-Albert 99, Bollobas-Riordan 00, Bollobas et al 03.

8 Some Properties of Preferential Attachment Graphs Power-law degree distribution: Prob(vertex has degree k) = k -3 Small diameter: O(log n) (n is the number of nodes) This talk: the dynamics of an epidemic on these graphs

9 Modeling Epidemics SIS model: Susceptible-Infected-Susceptible infected healthy at rate 1 #infected neighbors healthy infected at rate ( ) Also known as the Contact Process Studied in probability theory, physics, epidemiology Also in CS for modeling the spread of computer viruses (Kephart-White 93)

10 Epidemic Phase Transition The SIS model on an infinite bounded-degree graph has a phase transition. infection dies with Prob 1 On finite graphs: infection survives polynomial time λ c λ c infection survives with Prob > 0 infection survives exponential time

11 Epidemic Phase Transition The SIS model on an infinite bounded-degree graph has a phase transition. infection dies with Prob 1 On finite graphs: infection survives polynomial time λ c λ c infection survives with Prob > 0 infection survives exponential time What is the epidemic threshold of power-law graphs?

12 Outline Mathematical models The structure of social networks The spread of epidemics SIS Model on Power-law Networks The epidemic threshold Controlling Epidemics over a Network

13 Epidemic Threshold in Scale-Free Network In power-law networks the threshold is zero asymptotically almost surely, i.e. λ c = 0 a.s. Observed by Pastor-Satorras, Vespignani Rigorous proof: Berger, Borgs, Chayes, S. (gives detailed estimates on the survival probability as a function of λ)

14 Main Theorem Consider the SIS model on a preferential attachment graph of size n If the infection starts from a uniformly random vertex x, then with probability 1 O(λ 2 ), x is such that the infection survives a super-polynomial length of time with probability of order λ C log (1/λ) log log (1/λ) (> 0 for all λ > 0) λ c = 0

15 Key Elements of the Proof SIS model: the infection lives for a long time in the nbrhd of a vertex of high degree (lives for exp (kλ 2 ) time in the nbrhd of a degree k node)

16 Key Elements of the Proof v SIS model: the infection lives for a long time in the nbrhd of a vertex of high degree (lives for exp (kλ 2 ) time in the nbrhd of a degree k node) Power-law networks: the infection can find its way to a vertex of a high degree ( e.g. n 0.5 where n is # of nodes) w z

17 Outline Mathematical models The structure of social networks The spread of epidemics SIS Model on Power-law Networks The epidemic threshold Controlling Epidemics

18 Controlling Epidemics Question: What is the best way to distribute a fixed amount of antidote to contain the epidemic, i.e. how do you raise the epidemic threshold of the SIS process on preferential attachment (and more general) graphs? Borgs, Chayes, Ganesh, S., Wilson 06

19 A Variation of the SIS Varying recovery rates: infected healthy healthy at rate ρ x infected at rate λ {# infected nbrs} SIS model: ρ x =1 for all x Our model: distribute the same amount of curing rates (i.e. ρ x = n) and minimize the survival time

20 Two methods for containing the epidemic Method I (site monitoring): Cure Degree Set ρ x degree (x) Method II (contact tracing): Cure Infected degree Set ρ x ρ + ρ i x where i x is the number of infected neighbors of x

21 Our Result Theorem: If ρ x degree (x) for every node, then for any infection with λ < 1/(avg degree), the survival time is O(log n) almost surely. Main implication: site monitoring controls epidemics on all graphs with bounded average degree, including preferential attachment graphs Open problem: combine the advantages of contact tracing with site monitoring

22 Summary In power-law networks the epidemic threshold is zero I.e. infections with any positive rate of transmission, have a positive chance of becoming epidemic. Curing proportional to degree does control the epidemic in the sense that it makes the threshold > 0. The importance of network structure

23 Afterthought: Spreading epidemics Problem: Maximize the expected survival time Infect the optimum set of seed nodes (constant factor approximation algorithms Kempe, Kleinberg, Tardos 06, S. 08) Problem: implementation on a social network platform (what is the right bidding language?)