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

1

On the Windfall of Friendship: Inoculation Strategies on Social Networks

Dominic Meier Yvonne Anne Oswald

Stefan SchmidRoger Wattenhofer

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


“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)

New model: care about your friends’ wellbeing

neighbour pj

new equilibria (FNE)

Windfall of Friendship WoF(F) = cost(aworstNE)

cost(aworstFNE)

____________

4 Yvonne 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


Virus inoculation game [Aspnes et al., SODA 2005]


=> many connections



Virus inoculation game [Aspnes et al., SODA 2005]


=> many connections


Solution:

invest in protection

but

• $$$

• if all neighbours are protected no need for get protected as well..

invest if expected damage > cost


Virus Inoculation Game - Example




Virus Inoculation Game – Example








• 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]


strategies of pi

Actual cost [Aspnes et al., SODA 2005]

ai = 0 : device is not protected ai = 1 : device is protected

actual cost:

(per node)

C if ai = 1ca(i,a) =

L¢ ki/n if ai = 0

ki = size of attack component of pi

social cost

(network)

cost(a) = pi ca(i,a)


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


Introducing friendship

Windfall of Friendship WoF(F) = cost(aworstNE)

cost(aworstFNE)

____________

F 2 [0,1] Friendship Factor

perceived cost:

(per node)

cp(i,a) = ca(i,a) + F¢ pj neighbour ca(j,a)

cost(a) = pi ca(i,a)


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


n = 13C = 1L = 4

WoF(F) is NOT monotonically increasing in F

total cost = 4.69social optimum


n = 13C = 1L = 4


total cost = 12.76selfish setting

PoA = 2.73


n = 13C = 1L = 4F = 0.9


total cost = 12.23friendly setting

WoF(0.9) = 1.04



total cost = 4.69n = 13C = 1L = 4F = 0.1

friendly setting

WoF(0.1) = 2.73


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


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

011)L/Cn1(F41F2

1F1L/Cn


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


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

- ...


There is nothing bad in being social, even for

computer scientists

and economists ...

Moral of the story


The End!

Thank you!

Questions? Comments?

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

Documents

inoculation cost

woff poa woff

ftotal cost

cost of friends

inoculation strategies

protectedactual cost

topology woff

c infection cost