Inoculation Strategies for Victims of Viruses and the Sum-of-Squares Partition Problem James Apnes, Kevin Change, and Aleksandr Yampolskiy

Problem: Network of computers all connected to each other Antivirus software that prevents computer from becoming infected Virus introduced and spreads throughout Becoming infected and installing software both have costs

Goal: Install antivirus software on certain number of computers that minimizes cost 2 Ways to approach problem: –Individual Selection –Dictator Selection

Individual Selection Each user of computer makes own decision on whether or not to install software Example: cost of software, C, outweighs cost caused by virus, L Nash Equilibrium must be reached

Game Set-Up Graph G (V, E) where V = {0, 1, 2, …, n-1} All n nodes start vulnerable

Strategies For every i in G, the strategy of i is represented by a i a i is the probability of i installing software So we can also create vector ā that represents all strategies in G I a = set of nodes that are secure G a is “attack graph” G a = G - I a

Attack Model Virus is started at some random node Propagates through network by attacking unprotected nodes

Individual Costs Cost of C to install software Cost of L if node becomes infected Cost of node i is cost i (ā) = a i C + (1-a i )L*p i (ā) p i (ā) is the probability of node getting infected based on its reachability In pure strategies: p i (ā) = k i /n

Social Cost Equal to the sum of individual costs in network We remove m nodes at a cost of C per node and create so many connected components in the graph Each node in the i th connected component has a k i /n chance of getting infected

Social Cost So… Social cost = Cm + Σ(Lk i (k i /n)) = Cm + (L/n)Σ(k i 2 ) Where k i is the number of nodes in all connected components created by removing the m nodes that are in I a

Nash Equilibrium Computers make decision based on their local environment Reached when each node cannot change strategy to minimize cost

Nash Equilibrium Can find a threshold for the decisions made by the computers –t = Cn/L Example: t = 1.5 Nash Equilibrium can be found in O(n 3 ) time

Price of Anarchy How far the Nash equilibrium is from the optimal social cost For this system price of anarchy is Θ(n)

Dictator Selection A centralized solution that makes choices for the computer in the best interest of social cost Can be reduced to sum-of-squares partition problem when removing m nodes Approximation algorithm that allows us to find O(log 1.5 n)m nodes that get us in within a constant of the optimal solution

Sum-of-Squares We see social cost is Cm + (L/n)Σ(k i 2 ) Sum-of-Squares partition gives us approximately best answer for removing m nodes Repeat to make m 1,2, …, n

Sum-of-Squares Sum-of-Squares Partition Problem: –Given a graph G =(V, E), remove a set of nodes F, of at most m, in order to partition the graph into disconnected components H 1, H 2, …, H l such that Σ i (|H i | 2 ) is minimized

Main Goal: Theorem 12: There exists a polynomial time (O(log 1.5 n), O(1))-bicriterion approximation algorithm for the sum-of- squares partition problem Corollary 13: If OPT is the cost of the optimum solution for the inoculation problem, there exists a polynomial time approximation algorithm that finds a solution with cost at most O(log 1.5 n)OPT

Arora-Rao-Vazirani Algorithm Used to find the sparsest edge cut Can be extended to be used for node cuts Takes a set of nodes H and splits them into 3 sets –V 1 and V 2 –R

Arora-Rao-Vazirani Algorithm Approach is similar to greedy set cover algorithm Consequence of ARV algo paper: –There exists some O(log 1/2 n)-approximation for the following problem: given graph G, find a node cut that partitions the nodes of G into three sets: two set defining disconnected subgraphs with nodes sets V 1 and V 2 and a set of removed nodes R, such that the quantity (|V 1 |+(|R|/2))(|V 2 |+(|R|/2)) |R|is maximized

Cost-Effectiveness We have connected subgraph H with k nodes that is split into sets V 1, V 2, and R then the cut’s cost-effectiveness is: k 2 - |V 1 | 2 - |V 2 | 2 |R|

Cost-Effectiveness vs. Sparsity Sparsity: Cost Effectiveness:

Lemma 15 Let H be a graph with k nodes, if α* is the maximum cost effectiveness of all node cuts of H, the Arora-Rao-Vazirani sparse cut algorithm will find a cut with cost effectiveness at least α*/(clog 1/2 k)

Algorithm Input: A graph G and integer m > 0 Initialize: G 1  G, F  , l  0 1.Use sparse cut algorithm to find an approximate most cost effective cut in each connected component in G

Algorithm 2. Let H 1, H 2, …, H k be the components of G l in which the sparse cut algorithm found a cut that removes at most (20clog 1/2 n)m nodes If no such component exists, then halt operation and output the partition of G that results from removing all nodes in set F

Algorithm 3. Choose the component H j from among those considered in the previous step for which the cost-effectiveness is highest. So we have R as the set of nodes that partitioned H j into V 1 and V 2

Algorithm 4. Add all of the nodes in R to the set F. Now let G l +1 be the residual graph induced by removing R from G l. If |F| > (36clog 1.5 n)m, then halt and output the partition of G that results from having removed all nodes in F 5. Otherwise, l = l+1 and repeat

Aspnes, James, Kevin Chang, and Aleksandr Yampolskiy. "Inoculation Strategies for Victims of Viruses and the Sum-of-squares Partition Problem." Journal of Computer and System Sciences 72.6 (2006): Web. S. Arora, S. Rao, U. Vazirani, Expander flows, geometric embeddings and graph partitioning, in: Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222– 231. T. Leighton, S. Rao, Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms, J. Assoc. Comput. Machinery 46 (6) (1999) 787–832.