1. Maximization of Network Survivability against Intelligent and Malicious Attacks (Cont’d) Presented by Erion Lin

2. Outline Problem Description Model Solution Approach

3. Problem Description

4. Assume the budget allocation policy is given, we want to know the minimal attack cost for an attacker to compromise a network. The system is survivable if there is at least one available path for each critical OD-pair.

5. Problem Assumptions The survivability metric is measured as the connectivity of the given critical OD-pairs. The attacker and the defender have complete information about the targeted network topology. The defender’s budget allocation strategy is a given parameter.

6. Problem Assumptions (Cont’d) The objective of the attacker is to minimize the total attack cost of destroying all paths between one of the critical OD-pairs. We consider node attacks only. (No link attacks are considered). If a node is attacked, its outgoing links are not functional. We consider malicious attacks only. (No random failures are considered.)

7. Model

8. Model Description Given  Network topology  A set of critical OD-pairs  Total defense budget for the defender

9. Model Description (Cont’d) Objective:  To minimize the total cost of an attack Subject to:  There is no available path for one of the critical OD-pairs to communicate. To determine:  Which nodes will be attacked

10. Given Parameters

11. Decision Variables

12. Formulation Objective Function subject to (IP 1.1) Link cost representation (IP 1.2) (IP 1.3) (IP 1.4)

13. Formulation (Cont.) subject to (cont.) (IP 1.6) (IP 1.7) (IP 1.8) (IP 1.5)

14. Reformulation We reformulate the problem with one assumption and one argument. Assumption Argument  the optimality condition for the defender holds if and only if the total budget B is fully used. The threshold attack cost to compromise a node equals to the allocated budget on it.

15. Reformulation (Cont.) Objective Function subject to (IP 2.1) Link cost representation (IP 2.2) (IP 2.3) (IP 2.4)

16. Reformulation (Cont.) subject to (cont.) (IP 2.5) (IP 2.6) (IP 2.7) (IP 2.8) (IP 2.9)

17. Solution Approach

18. Max-Flow Min-Cut Theorem The maximum value of the flow from a source node to a sink node t in a capacitated network equals the minimum capacity among all s-t cuts. Therefore, we gain a byproduct of the minimum cut from the maximum flow algorithm.

19. Genetic Augmenting Path Algorithm

20. Example

21. Example (cont.)

22. Questions How to identify an augmenting path or show that the network contains no such path? Whether the algorithm terminates in finite number of iterations? Labeling algorithm is a specific implementation.

23. Exists if the residual capacity of the arc is not zero The Labeling Algorithm

25. S-T Cut A cut is a partition of the node N into two subsets S and =N – S. We refer to a cut as an s-t cut if.

26. Example of an S-T Cut

27. Theorem The maximum value of the flow from a source node s to a sink node t in a capacitated network equals the minimum capacity among all s-t cuts. Proof.  When the labeling algorithm terminates, it also discovered a minimum cut.

28. Theorem (Cont’d) A flow x* is a maximum flow if and only of the residual network G(x*) contains no augmenting path. Proof.  If the residual network G(x*) contains an augmenting path, clearly the flow x* is not a maximum flow.

29. Node Splitting 300

30. Solution Approach Combine max-flow min-cut theorem and node splitting method.

31. Example 300 200 50 400 70

32. Example (Cont’d) 300 50 200 70 400 Infinite Capacity -200 -50 Max Flow and Min Cut ： 250

33. Time Complexity Analysis Labeling Algorithm :O((|N|+|L|)xn)  n: number of augmentations Consider w OD-pairs  O(|W|x(|N|+|L|)xn)

34. Thanks for Your Listening