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

Yvonne Anne 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

Yvonne Anne 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 ) ____________

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

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

6Yvonne Anne EC 2008 Virus inoculation game [Aspnes et al., SODA 2005] social networks everywhere: facebook, co-authors, => 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 EC 2008 Virus Inoculation Game - Example

Yvonne Anne EC 2008 Virus Inoculation Game - Example

Yvonne Anne EC 2008 Virus Inoculation Game – Example

Yvonne Anne EC 2008 Virus Inoculation Game - Example

Yvonne Anne EC 2008 Virus Inoculation Game - Example

Yvonne Anne EC 2008 Virus Inoculation Game - Example

Yvonne Anne 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]

Yvonne Anne 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)

Yvonne Anne 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

Yvonne Anne 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)

Yvonne Anne 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

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

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

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

Yvonne Anne 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

Yvonne Anne 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

Yvonne Anne 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  

Yvonne Anne 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  

Yvonne Anne 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 -...

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

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