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On the Windfall of Friendship:
Inoculation Strategies on Social Networks
Dominic Meier
Yvonne Anne Oswald
Stefan Schmid
Roger Wattenhofer
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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
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(a
worstNE)
____________
Windfall of Friendship WoF(F) =
cost(aworstFNE)
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
Yvonne Anne Oswald @ EC 2008
4
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
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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
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Virus Inoculation Game - Example
Yvonne Anne Oswald @ EC 2008
Virus Inoculation Game - Example
Yvonne Anne Oswald @ EC 2008
Virus Inoculation Game – Example
Yvonne Anne Oswald @ EC 2008
Virus Inoculation Game - Example
Yvonne Anne Oswald @ EC 2008
Virus Inoculation Game - Example
Yvonne Anne Oswald @ EC 2008
Virus Inoculation Game - Example
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
Yvonne Anne Oswald @ EC 2008
Actual cost [Aspnes et al., SODA 2005]
strategies of pi ai = 0 : device is not protected
ai = 1 : device is protected
actual cost:
C
if ai = 1
L¢ ki/n
if ai = 0
ca(i,a) =
(per node)
ki = size of attack component of pi
social cost
cost(a) = pi ca(i,a)
(network)
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
Yvonne Anne Oswald @ EC 2008
Introducing friendship
F 2 [0,1] Friendship Factor
perceived cost: cp(i,a) = ca(i,a) + F¢ pj neighbour ca(j,a)
(per node)
cost(a
worstNE)
____________
Windfall of Friendship WoF(F) =
cost(aworstFNE)
cost(a) = pi ca(i,a)
Yvonne Anne Oswald @ EC 2008
Results
General graphs
PoA ≤ n
[Aspnes et al., SODA’05]
• attack components size depends on topology
• WoF(F) ≥ 1
Example
• WoF(F) ≤ PoA
• WoF(F) is not monotonically increasing in F
• computing worst/best FNE is NP-complete
Yvonne Anne Oswald @ EC 2008
WoF(F) is NOT monotonically increasing in F
social optimum
n = 13
C=1
L=4
total cost = 4.69
Yvonne Anne Oswald @ EC 2008
WoF(F) is NOT monotonically increasing in F
selfish setting
n = 13
C=1
L=4
total cost = 12.76
PoA = 2.73
Yvonne Anne Oswald @ EC 2008
WoF(F) is NOT monotonically increasing in F
friendly setting
n = 13
C=1
L=4
F = 0.9
total cost = 12.23
WoF(0.9) = 1.04
Yvonne Anne Oswald @ EC 2008
WoF(F) is NOT monotonically increasing in F
friendly setting
n = 13
C=1
L=4
F = 0.1
total cost = 4.69
WoF(0.1) = 2.73
Yvonne Anne Oswald @ EC 2008
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
Results
Special graphs
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
 WoF(F) can reach n
• 1 FNE =>
more than 1 FNE =>
 WoF(F) = O(1)
1
Cn / L  1  F   2F

 1 4F(1 Cn / L)  1  1  0
Yvonne Anne Oswald @ EC 2008

Results
Special graphs
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
 WoF(F) linear in n
• 1 FNE =>
more than 1 FNE =>
 WoF(F) = O(1)
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: Fi,j
- ...
Yvonne Anne Oswald @ EC 2008
Moral of the story
There is nothing bad in being social, even for
computer scientists
and economists ...
Yvonne Anne Oswald @ EC 2008
The End!
Thank
you!
Questions? Comments?
Yvonne Anne Oswald @ EC 2008
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