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Inoculation Strategies for
Victims of Viruses and
the Sum-of-Squares
Partition Problem
Kevin Chang
Joint work with James Aspnes and
Aleksandr Yampolskiy
(Yale University)
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
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
Answer: Probably not.
Norton AntiVirus 2005 = $49.95
Value of your data = $350.00
Infection probability = 1/10
Expected loss = $35.00
This selfish behavior…

…fails to achieve the social optimum.
What if instead…

…a benevolent dictator decided which
computers install an anti-virus?
Center node
must install
an anti-virus
or else!
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
Our Model
The network is an undirected graph
G = (V,E).
 Installing anti-virus software is a single
round non-cooperative game.
 The players are the network nodes:
V = {0,1,…,n-1}.

Our Model : Strategies
Each node has two actions: do nothing or
inoculate itself.
 ai = probability that node i installs antivirus software. Write the strategies of all n
players as vector

Our Model : Attack Model
After the nodes choose their strategies,
the adversary picks a starting point for
infection uniformly at random
 Node i gets infected if it has no anti-virus
software installed and if any of its
neighbors become infected.

Our Model : Attack Model (cont.)

Example: Only node 3 installs anti-virus
software. Adversary chooses to infect
node 2.
0
1
2
3
4
5
Our Model : Attack Graph
Attack Graph: subgraph of G induced by removing nodes
that install antivirus
0
1
0
1
2
3
2
3
4
network graph G
5
4
5
attack graph Ga= G - Ia
Our Model : Individual Costs



Anti-virus software costs C. Loss from infection is
L.
Cost of strategy
to node i:
Here, pi( ) = Pr[i is infected | i does not install an
anti-virus]
Our Model : Social Cost

Social cost of
is simply a sum of
individual costs:
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
Nash Strategies: Quick definition

Def: Strategy profile
is in Nash
equilibrium if no node can improve its
payoff by switching to a different strategy.
Nash Strategies: Some intuition
Some Intuition: Suppose ai=0, but node i
expects that it will lie in component of size
greater than t=Cn/L in attack graph.
 Prob[i infected] = t/n.
 Then its expected loss from infection is
>L·(Cn/L)/n = C and it will switch to ai = 1.

Nash Strategies: Characterization
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 in the attack graph.
 will not install an anti-virus if it would end up in a
component of expected size < t in the attack graph.
 is indifferent between installing and not installing
when the expected size = t in the attack graph.
Nash Strategies (cont.)

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
network graph G
4
5
attack graph Ga= G - Ia
Nash Strategies (cont.)


Thm: It is NP-hard to compute a pure Nash
equilibrium with lowest (resp., highest) cost.
Thm: There exists a pure Nash equilibrium,
which can be achieved by a distributed,
iterative process in 2n steps.
Price of Anarchy
Price of anarchy measures how far away a
Nash equilibrium can be from the social
optimum
 Formally, it is the worst-case ratio between
cost of Nash equilibrium and cost of social
optimum
 For network G and costs C, L, we denote it:

A lower bound
Consider a star graph K1,n.
Let C=L(n-1)/n so that t=n-1.
n-1
1
n-2
2
3
0
…
G = K1,n
Price of Anarchy (cont.)
Then,
cost C+L(n-1)/n.
n-1
1
n-2
is an optimum strategy with
n-1
2
3
1
n-2
3
0
0
…
…
G = K1,n
2
Ga*
Price of Anarchy (cont.)
And
is worst-cost Nash with
cost C+L(n-1)2/n.
n-1
1
2
n-1
n-2
1
2
3
0
n-2
3
0
…
G = K1,n
…
Ga*
Price of Anarchy (cont.)
Lower Bound: For a star graph K1,n,
(G, C, L) = n/2.
Upper Bound: For any graph G and any C, L,
(G, C, L)≤ n.
Thm: Price of anarchy in our game is
(G, C, L) = (n).
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
Optimum Strategies
So, allowing users to selfishly choose
whether or not to install anti-virus software
may be very inefficient
 Instead, let’s have a benevolent dictator
compute and impose a solution
maximizing overall welfare

Optimum Strategies (cont.)


Unfortunately,
Thm: It is NP-hard to compute an optimum
strategy.
Fortunately,
Thm: a strategy with cost at most O(log2 n)
OPT can be computed in polytime.
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
Sum-of-Squares Partition
We can reduce the network security
problem to the following.
 Problem: By removing a set of at most
m nodes, partition the graph into mutually
disconnected components {Hi}, such that
i |Hi|2 is minimum.

Sum-of-Squares Partition (cont.)
Thm: We can find a set of O(log2 n)m nodes whose
removal partitions the graph into components {Hi}
such that i |Hi|2 = O(1)OPT.
Proof sketch:
The approach is similar to greedy log n approximation
algorithm for set cover. We recursively partition the
graph by repeatedly removing the node cut that
gives the best “per-node benefit.”
Sum-of-squares partition (cont.)



Suppose we remove node set R to cut a graph H into
components Hi and Hj.
cost effectiveness of cut R is: (|H|2-|Hi|2- |Hj|2)/|R|
Lemma: Leighton-Rao algorithm for node cuts will find
a cut with cost effectiveness within O(log n) of best cost
effectiveness.
O(log0.5 n) algorithms do not extend to directed cuts, and
thus cannot be used for node cuts.
Rough Outline of Algorithm

In first iteration, partition the graph G into
H1,H2 by removing a sparse node cut.
G
Rough Outline of Algorithm

In first iteration, partition the graph G into
H1,H2 by removing a sparse node cut.
H1
H2
Rough Outline of Algorithm

In subsequent iterations, cut the Hi for
which we find the cut with the best cost
effectiveness.
H1
H2
Rough Outline of Algorithm

Complete details can be found in full version:
http://www.cs.yale.edu/~aspnes/inoculation-abstract.html
H1
H2
H3
Outline
 Motivation
 Our
Model
 Nash Strategies
 Optimum Strategies
 Sum-of-Squares Partition Problem
 Conclusion
Conclusion




We proposed a simple game for modeling
containment of viruses in a network.
Nash equilibria of our game have a simple
characterization.
We showed that, in the worst case, they can be
far off from the optimum solution.
However, a near-optimum deployment of antivirus software can be computed by reduction to
the sum-of-squares partition problem.
Open Problems
Introduce a discount (or taxation)
mechanism into the system.
 Consider a “smart” adversary who targets
the biggest component in the attack graph.
 Is there an approximation algorithm for the
sum-of-squares partition problem that
removes fewer nodes or has a better
approximation ratio?

Questions?