On the Windfall of Friendship: Inoculation Strategies on Social Networks

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

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

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

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

6. 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 Yvonne Anne Oswald @ EC 2008

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

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

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

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

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

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

13. Yvonne Anne Oswald @ EC 2008 Model [Aspnes et al., SODA 2005] • 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

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

15. 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 Yvonne Anne Oswald @ EC 2008

16. Introducing friendship F 2 [0,1] Friendship Factor cp(i,a) = ca(i,a) + F¢pj neighbour ca(j,a) perceived cost: (per node) cost(aworstNE) cost(aworstFNE) ____________ Windfall of Friendship WoF(F) = cost(a) = pi ca(i,a) Yvonne Anne Oswald @ EC 2008

17. Results PoA ≤ n [Aspnes et al., SODA’05] General graphs • 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 Example Yvonne Anne Oswald @ EC 2008

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

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

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

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

22. Results General graphs • 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 Yvonne Anne Oswald @ EC 2008

23. Results Special graphs • a FNE always exists, fast convergence • WoF(F) ≤ 4/3 (tight) complete 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) star graph Yvonne Anne Oswald @ EC 2008

24. Results Special graphs • a FNE always exists, fast convergence • WoF(F) ≤ 4/3 (tight) complete 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) star graph Yvonne Anne Oswald @ EC 2008

25. 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: Fi,j - ... Yvonne Anne Oswald @ EC 2008

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

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