1. 1 On the Windfall of Friendship: Inoculation Strategies on Social Networks Dominic Meier Yvonne Anne Oswald Stefan Schmid Roger Wattenhofer

2. Yvonne Anne Oswald @ EC 2008 Once upon a time...... every node follows protocol malicious nodes selfish/rational nodes (game theory meets computer science) History of behavioural network models still no explanation for behaviour of real networks now

3. Yvonne Anne Oswald @ EC 2008 “Traditional“ game theory: player p i selects strategy a i => strategy profile a actual cost for player p i : cost a (i,a) “Friendly“ game theory: consider cost of friends : F 2 [0,1] Friendship Factor perceived cost c p (i,a) = c a (i,a) + F ¢   c a (j,a) New model: care about your friends’ wellbeing neighbour pj  new equilibria (FNE) Windfall of Friendship WoF(F) = cost(a worstNE ) cost(a worstFNE ) ____________

4. 4Yvonne Anne Oswald @ EC 2008 Case study: virus inoculation game [Aspnes et al., SODA 2005] social networks everywhere: facebook, co-authors, email.... => many connections => fast virus distribution

5. 5Yvonne Anne Oswald @ EC 2008 Virus inoculation game [Aspnes et al., SODA 2005] social networks everywhere: facebook, co-authors, email.... => many connections => fast virus distribution

6. 6Yvonne Anne Oswald @ EC 2008 Virus inoculation game [Aspnes et al., SODA 2005] social networks everywhere: facebook, co-authors, email.... => many connections => fast virus distribution Solution: invest in protection but $$$ if all neighbours are protected no need for get protected as well.. invest if expected damage > cost

7. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game - Example

8. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game - Example

9. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game – Example

10. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game - Example

11. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game - Example

12. Yvonne Anne Oswald @ EC 2008 Virus Inoculation Game - Example

13. Yvonne Anne Oswald @ EC 2008 network of n devices owner of node decides whether to protect it or not inoculation cost: C infection cost: L virus infection at 1 arbitrary initial node virus propagation over paths of insecure devices Model [Aspnes et al., SODA 2005]

14. Yvonne Anne Oswald @ EC 2008 strategies of p i Actual cost [Aspnes et al., SODA 2005] a i = 0 : device is not protected a i = 1 : device is protected actual cost: (per node) C if a i = 1 c a (i,a) = L¢ k i /n if a i = 0 k i = size of attack component of p i social cost (network) cost(a) =  p i c a (i,a)

15. Yvonne Anne Oswald @ EC 2008 Previous results [Aspnes et al., SODA 2005] pure equilibria always exist attack components of size Cn/L PoA (price of anarchy) linear in n [Moscibroda et al., PODC 2006] Malicious nodes: lie about their strategies

16. Yvonne Anne Oswald @ EC 2008 Introducing friendship Windfall of Friendship WoF(F) = cost(a worstNE ) cost(a worstFNE ) ____________ F 2 [0,1] Friendship Factor perceived cost: (per node) c p (i,a) = c a (i,a) + F¢  p j neighbour c a (j,a) cost(a) =  p i c a (i,a)

17. Yvonne Anne Oswald @ EC 2008 General graphs Results attack components size depends on topology WoF(F) ≥ 1 WoF(F) ≤ PoA WoF(F) is not monotonically increasing in F computing worst/best FNE is NP -complete PoA ≤ n [Aspnes et al., SODA’05] Example

18. Yvonne Anne Oswald @ EC 2008 n = 13 C = 1 L = 4 WoF(F) is NOT monotonically increasing in F total cost = 4.69 social optimum

19. Yvonne Anne Oswald @ EC 2008 n = 13 C = 1 L = 4 WoF(F) is NOT monotonically increasing in F total cost = 12.76 selfish setting PoA = 2.73

20. Yvonne Anne Oswald @ EC 2008 n = 13 C = 1 L = 4 F = 0.9 WoF(F) is NOT monotonically increasing in F total cost = 12.23 friendly setting WoF(0.9) = 1.04

21. Yvonne Anne Oswald @ EC 2008 WoF(F) is NOT monotonically increasing in F total cost = 4.69 n = 13 C = 1 L = 4 F = 0.1 friendly setting WoF(0.1) = 2.73

22. Yvonne Anne Oswald @ EC 2008 General graphs Results WoF(F) ≥ 1 WoF(F) ≤ PoA WoF(F) is not monotonically increasing in F computing worst/best FNE is NP -complete Reduction from vertex cover and independent dominating set

23. Yvonne Anne Oswald @ EC 2008 Results complete graph a FNE always exists, fast convergence WoF(F) ≤ 4/3 (tight) star graph a FNE always exists, fast convergence sometimes the best FNE is the only FNE 1 FNE => WoF(F) can reach n more than 1 FNE => WoF(F) = O(1) Special graphs  

24. Yvonne Anne Oswald @ EC 2008 Results complete graph a FNE always exists, fast convergence WoF(F) ≤ 4/3 (tight) star graph a FNE always exists, fast convergence sometimes the best FNE is the only FNE 1 FNE => WoF(F) linear in n more than 1 FNE => WoF(F) = O(1) Special graphs  

25. Yvonne Anne Oswald @ EC 2008 Future directions... analyze more complex graphs and real social networks variations of virus game - more than 1 virus - more strategies - other propagation models analyze other games on networks generalize model - include k-hop neigbours - weighted graph: F i,j -...

26. Yvonne Anne Oswald @ EC 2008 There is nothing bad in being social, even for computer scientists and economists... Moral of the story

27. 27Yvonne Anne Oswald @ EC 2008 The End! Thank you! Questions? Comments?