Inoculation strategies for victims of viruses

Inoculation Strategies for
Victims of Viruses and
the Sum-of-Squares
Partition Problem
James Aspnes, Kevin Chang,
and Aleksandr Yampolskiy
(Yale University)
Copyright (C) 2005 by Aleksandr
Yampolskiy

Outline

Ø Motivation
n Our Model
n Nash Strategies
n Optimal Strategies
n Sum-of-Squares Partition Problem
n Conclusion
Yampolskiy

Question: Will you install anti-virus
software?

Norton AntiVirus 2005 = $49.95

Value of your data = $350.00
Infection probability = 1/10
Expected loss = $35.00
Yampolskiy

Answer: Probably not.

Norton AntiVirus 2005 = $49.95

Value of your data = $350.00
Infection probability = 1/10
Expected loss = $35.00
Yampolskiy

This selfish behavior…
n …fails to achieve the social optimum.

Yampolskiy

What if instead…
n …a benevolent dictator decided which
computers install an anti-virus?

Center node
must install
an anti-virus
or else!
Yampolskiy

Outline

n Motivation
Ø Our Model
n Nash Strategies
n Conclusion
Yampolskiy

Our Model
n The network is an undirected graph
G = (V,E).
n Installing anti-virus software is a single
round non-cooperative game.
n The players are the network nodes:
V = {0,1,…,n-1}.

Yampolskiy

Our Model : Strategies
n Each node has two actions: do nothing or
inoculate itself.
n Strategy profile summarizes
players’ choices.
n ai = probability that node i installs anti-
virus software

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Our Model : Attack Model
n After the nodes choose their strategies,
the adversary picks a starting point for
infection uniformly at random
n Node i gets infected if it has no anti-virus
software installed and if any of its
neighbors become infected.

Yampolskiy

Our Model : Attack Model (cont.)
n Example: Only node 3 installs anti-virus
software. Adversary chooses to infect
node 2.
0 1

2 3

4 5
Yampolskiy

Our Model : Attack Graph

0 1 0 1

2 3 2 3

4 5 4 5

network graph G Copyright (C) 2005 by Aleksandr
Yampolskiy
attack graph Ga= G - Ia

Our Model : Individual Costs
n Anti-virus software costs C. Expected loss
from virus is L.
n Cost of strategy to node i:

n Here, pi(a) = Pr[i is infected | i does not
install an anti-virus]

Yampolskiy

Our Model : Social Cost
n Social cost of is simply a sum of
individual costs:

Yampolskiy

Outline

n Motivation
n Our Model
Ø Nash Strategies
n Conclusion
Yampolskiy

Nash Strategies
n Def: Strategy profile is in Nash
equilibrium if no node can improve its
payoff by switching to a different strategy:
for i = 0,...,n-1 and any x 2 [0,1],

n Fact: Nash strategies do not optimize total
social cost (cf. Prisoner’s Dilemma)
Yampolskiy

Nash Strategies (cont.)

Thm: There is a threshold t=Cn/L such that each
node in a Nash equilibrium
¨ will install an anti-virus if it would otherwise end up in
a component of expected size > t
¨ will not install an anti-virus if it would end up in a
component of expected size < t.
¨ is indifferent between installing and not installing
when the expected size = t.

Yampolskiy

n Corollary: Let t = Cn/L. Then a pure
strategy is a Nash equilibrium if and only
if
¨ Every component in Ga has size · t
¨ Inserting any secure node j and its edges into
Ga yields a component of size ¸ t.

Yampolskiy

n Example: Let C=0.5,L=1 so that t=Cn/L=2.5.
Then is not a Nash equilibrium.

0 1 0 1

2 3 2 3

4 5 4 5
network graph G Yampolskiy attack graph Ga= G - Ia

Thm: It is NP-hard to compute a pure Nash
equilibrium with lowest (resp., highest) cost.
Proof sketch: By reduction to VERTEX COVER
(resp., INDEPENDENT DOMINATING SET) .
¨ Set C, L so that t=Cn/L=1.5.
¨ In a Nash equilibrium, (a) every vulnerable node
has all neighbors secure; (b) every secure node
has an insecure neighbor

Yampolskiy

n If V’µ V is a minimal vertex cover, then
installing software on its nodes satisfies
(a) because V’ is a vertex cover and (b)
because V’ is minimal.
n Conversely, if V’ are secure nodes in a
Nash equilibrium, then V’ is a vertex cover
by (a).

Yampolskiy

n Nash Theorem guarantees our game has
a mixed Nash equilibrium.
n But does it make sense talking about pure
Nash equilibria?

Yampolskiy

Yes, it does!

Thm: If at each step some node with
suboptimal strategy switches its strategy,
the system converges to a pure Nash
equilibrium in · 2n steps.

Yampolskiy

Price of Anarchy [KP99]
n Price of anarchy measures how far away a
Nash equilibrium can be from the social
optimum
n Formally, it is the worst-case ratio between
cost of Nash equilibrium and cost of social
optimum
n For network G and costs C, L, we denote it:

Yampolskiy

Price of Anarchy (cont.)
Lower Bound: For a star graph K1,n,
ρ(G, C, L) = n/2.
Upper Bound: For any graph G and any C, L,
ρ(G, C, L)· n.

Thm: Price of anarchy in our game is
ρ(G, C, L) = Θ(n).

Yampolskiy

Proof for lower bound:
Consider a star graph K1,n.
Let C=L(n-1)/n so that t=Cn/L=n-1.

1
n-1 2

n-2 3
0

…
G = K1,n
Yampolskiy

Then, is an optimum strategy with
cost C+L(n-1)/n.

1 1
n-1 2 n-1 2

n-2 3 n-2 3
0 0

… …
G = K1,n Yampolskiy Ga*

And is worst-cost Nash with
cost C+L(n-1)2/n.

1 1
n-1 2 n-1 2

n-2 3 n-2 3
0 0

… …
G = K1,n Yampolskiy Ga*

n Therefore,

n Proof for upper bound uses similar ideas.
Yampolskiy

Outline

n Motivation
n Our Model
n Nash Strategies
Ø Optimal Strategies
n Conclusion
Yampolskiy

Optimal Strategies
n So, allowing users to selfishly choose
whether or not to install anti-virus software
may be very inefficient
n Instead, let’s have a benevolent dictator
compute and impose a solution
maximizing overall welfare

Yampolskiy

Optimal Strategies (cont.)
n We can show:
Thm: Let t=Cn/L. If is an optimum
strategy, then every component in Ga has
size · max(1, (t+1)/2).

n Unfortunately,
Thm: It is NP-hard to compute an optimal
strategy.

Yampolskiy

Optimum Strategies (cont.)
n Naturally, we consider approximating the
solution.

k1=2
0 1 0 1 secure
nodes
2 3 2 3 Ia

k2=2
4 5 4 5

network graph G attack graph Ga=G - Ia
Yampolskiy

Optimum Strategies (cont.)
n For pure strategy , we have:

we concentrate on
this part
Yampolskiy

Outline

n Motivation
n Our Model
n Nash Strategies
Ø Sum-of-Squares Partition Problem
n Conclusion
Yampolskiy

Sum-of-Squares Partition
n We guess that there are m=|Ia| secure
nodes.
n Problem: By removing a set of at most
m · n nodes, partition the graph into
components H1, …, Hk such that ∑i |Hi|2 is
minimum.

Yampolskiy

Sum-of-Squares Partition (cont.)
Thm: We can find a set of O(log2 n)¢m nodes whose
removal partitions the graph into components
H1,…,Hk such that ∑i |Hi|2 · O(1)¢OPT.
Proof sketch: We use the Leighton-Rao sparse cut
algorithm [LR99]. The approach is similar to greedy
log n approximation algorithm for set cover. We
repeatedly remove the node cut that gives the best
per-node benefit.

Yampolskiy

Outline

n Motivation
n Our Model
n Nash Strategies
Ø Conclusion
Yampolskiy

Conclusion
n We proposed a simple game for modeling
containment of viruses in a network.
n Nash equilibria of our game have a simple
characterization.
n We showed that, in the worst case, they can be
far off from the optimal solution.
n However, a near-optimal deployment of anti-
virus software can be computed by reduction to
the sum-of-squares partition problem.

Yampolskiy

Open Problems
n Introduce a discount (or taxation) mechanism into the
system.
n Suppose nodes can lie about their level of security (or
about who their neighbors are). How do we make truth-
telling a dominant strategy?
n Consider a “smart” adversary who targets the biggest
graph component.
n How do we evaluate what C and L are?
n Is there an algorithm for the sum-of-squares partition
problem with a better approximation ratio?

Yampolskiy

Acknowledgments
Joan Feigenbaum, Hong Jiang, and Yang
Richard Yang

Yampolskiy

Thank you!

Yampolskiy

Inoculation strategies for victims of viruses

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Inoculation strategies for victims of viruses