Coping with Physical Attacks on Random Network Structures

1. Coping with Physical Attacks on Random Network Structures Omer Gold omergolden@gmail.com Joint work with Reuven Cohen (BIU) To appear in the Proceedings of IEEE International Conference on Communications (ICC), June, 2014.

2. Content • Problem and motivation • Previous work Overview • Random Network model • Algorithms and Analyses • Summary

3. Problems and motivation • Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an Electromagnetic Pulse (EMP) attack. • Such real-world events happen in specific geographical locations and disrupt specific parts of the network. • Therefore, the geographical layout of the network determines the impact of such events on the network’s connectivity.

4. Large Scale Physical Attacks/Disasters Source: Report of the Commission to Assess the threat to the United States from Electromagnetic Pulse (EMP) Attack, 2008 • EMP (Electromagnetic Pulse) attack, Solar Flares, and other Natural Disasters • will destroy backbone nodes and links • Physical attacks or disasters affect a specific geographical area • Fibers, routers, generators, and power lines have a physical location

5. Problems and motivation • An interesting question is to identify the most vulnerable parts of the network. • That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. • We consider graph models in which nodes and links are geographically located on a plane, and model the disaster event as a line segment or a circular cut.

6. Problems and motivation • Definition 1 (Performance Measures): The performance measures of a cut are (the last 3 are defined as the values after the removal of the intersected links): • TEC - The total expected capacity of the intersected links. • ATTR - The average two terminal reliability of the network. (Connectivity measure) • MFST - The maximum flow between a given pair of nodes s and t. • AMF - The average value of maximum flow between all pairs of nodes.

7. Example – Circular cut 7 2 3 7 9 Q. What is the TEC measure for this cut? A. 2+3+7+9+7 = 28*The black links are not affecetd by the cut.

8. Problems and motivation • Definition 2 (Worst-Case Cut): Under a specific performance measure, a worst-case cutis a cut which maximizes/minimizes the value of the performance measure.

9. Problems and motivation • Geographical Network Inhibition by Circles (GNIC) Problem (2009): Given a graph, cut radius, link probabilities, and capacities, find a worst-case circular cut under performance measure TEC. • After solving TECmeasure problem in polynomial time, it can be shown that the other performance measures(ATTR, MFST, AMF) are also polynomial. (Zussman, Neumayer, Cohen, Modiano. 2009). • We will focus from now on the TEC measure.

10. Example – Real Network The fiber backbone operated by a major U.S. network provider We want to find a cut that maximizes TEC,denote as “worst-cut”.

11. Previous Work Overview • S. Neumayer, G. Zussman, R. Cohen, E. Modiano (IEEE INFOCOM 2009) Showed polynomial time algorithms that finds Worst-Case Circular Cut in time • Improvements have been made for some performance measures using tools from Computational Geometry (Arrangements)P. K. Agarwal, A. Efrat, S. K. Ganjugunte, D. Hay, S. Sankararamany and G. Zussman. (IEEE MILCOM 2010)

12. Previous Work Overview • Network Reliability Under Random Line-Segment Cut: Calculate some network performance metrics to such a disaster in polynomial time. S. Neumayer, E. Modiano (IEEE Infocom 2010) • Network Reliability Under Random Circular-Cut disasters that take the form of a `randomly' located disk in a plane. Approximate some network performance metrics in case of such a disaster in polynomial time.S. Neumayer, E. Modiano (IEEE Globecom 2011)

13. Previous Work: Probabilistic Failures Major work has been made recently about generalizing previous failure model to probabilistic failure model and simultaneous attack failures. Work by: Pankaj K. Agarwal, AlonEfrat, Shashidhara K. Ganjugunte, David Hay, SwaminathanSankararaman, Gil Zussman: The resilience of WDM networks to probabilistic geographical failures. INFOCOM 2011: 1521-1529 Later improved version in: IEEE/ACM Trans. Netw. 21(5): 1525-1538 (2013) Another improvement and variation has been recently made by: Pankaj K. Agarwal, SarielHar-Peled, Haim Kaplan, MichaSharir: Union of Random Minkowski Sums and Network Vulnerability Analysis.

14. Previous Work OverviewFailure Models • Deterministic: • Fails definitely if within range • Probabilistic: • Simple: fails with a probability q if within range • Spatial Probability Functions • Linear, Gaussian, Arbitrary* P. Agrawal, A. Efrat, S. Ganjugunte, D. Hay, S. Sankararaman,G. Zussman IEEE Infocom 2011

15. Random Network • As we saw, models with deterministic, random and probabilistic failures have been recently studied extensively . • What about Random Networks? This is what we are going to talk about here. Our work is about developing algorithms for finding “worst-cuts” in Random Networks, as well as developing methods to model a random network from a given data, such as: demographic map, terrain conditions, economic considerations, etc.

16. Random Network- Motivation • The attacker (adversary) has partial or no knowledge about the network topology. • Adversary has a “noisy” network topology map. • Assessing the reliability of hidden networks. • Real-life networks topology sometimes presents similar characteristics to a random network topology.

17. Random Network- Our Model • We model a random network in a rectangle which bounds the country’s extreme points +cut’s radius .

18. Random Network- Our Model • The main idea of our stochastic modeling is in considering the configuration of the stations as realizations of non-homogeanus Poisson point process. The main advantage of a Poisson process is it's simplicity. • The distribution of a Poisson point processis completely defined through the intensity measure representing the mean density of points.

19. Random Network- Our Model • Network Model Formulation: • Nodes are distributed in the rectangle Rec through a Spatial Non-Homogenous Poisson Point Process where is the intensity function of the . • Let be the probability for the existence of a link between two nodes located at and in Rec. • is the cumulative distribution function of the link capacity between two connected nodes.i.e. where and are the locations of nodes and , respectively.

20. Random Network- Model • It is reasonable to assume that and can be computed easily as a function of the distance from to and that the possible capacity between them is bounded (denote by ).

21. Random Network- Model • We assume the following: the intensity function and (the derivative of ) are functions of constant description complexity, they are continuously differentiable and Riemann-integrable over Rec. Implies that our probability functions are of bounded variation over Rec as their derivatives receives maximum over the compact set .

22. Algorithm – Evaluate Damage • We present Polynomial Time Approximation Algorithms for finding the ExpectedWorst Case cutslocation and damage in the Random Network model, i.e. cut that maximizes the total expected capacity of the intersected links (TEC). • For this goal wefirst develop an algorithm to evaluate the TEC (Damage) for a given cut.

23. Evaluate Damage of a Circular Attack (i.e. “cut”) • For a cut we divide the intersections of with the graph's edges into 3 independent types: • , is the case where the entire edge is inside . which means both endpoints of the edge are inside . • , is the case where one endpoint is insideand the other is outside. • is the case where both endpoints are outside from and the link which connects the endpoints intersects.

24. Example: demonstration demonstration demonstration

25. Example – Circular cut 7 2 3 7 9 A red link is an A blue link is a An Orange link is a

26. Evaluate Damage of Circular Attack • For, let be the total capacity of the intersected type edges with cut .Namely, the damage determined by . • Thus, it holds that the expected capacity of the intersected links of types , and is determined by:

27. Evaluate Damage of Circular Attack where is the expected capacity between two nodes at points and .

28. Evaluate Damage of Circular Attack is the indicator function, giving if the segment intersects the circle and otherwise.

29. Evaluate Damage of Circular Attack • Let be the total damagedetermined by all the intersected links with cut . • Due to the linearity of expectation, we get that the total expected capacity of the intersected links (TEC) is

30. Algorithm to Evaluate the Expected Damage for a given Cut. • Algorithm to evaluate the TEC (Damage) for a given cut : Input: • Output: TEC (Damage) caused by

31. Algorithm to Evaluate the Expected Damage for a given Cut. • Computing the Damage: • Computing the Damage:

32. Algorithm to Evaluate the Expected Damage for a given Cut. • How to compute the damage? • Requires more sophisticated approach. • First, let’s ask the following: For a node , what is the set of points in that satisfies: For every node , the edge intersects , and for every node the edge does not intersect ?

33. Example:

34. Evaluate the Expected Damage for a given Cut. Computing the Damage: • Following the idea we just saw, we want to integrate for all such possible nodes and • We use numerical integration: We divide into a Grid of squares.

35. Algorithm to Evaluate the Expected Damage for a given Cut. • Evaluate_ • Create two tangents to going out from . • Denote the line segments of the tangents which their endpoints are the intersection point of the tangent with boundary by and . • Denote by is the set of points which is bounded by , , and the boundary of . • Return

36. Algorithm to Evaluate the Expected Damage for a given Cut. Computing the Damage: • This integral is computed numerically over the Grid. • The denser the Grid, the more accurate result we obtain. • Finally, Return

37. Graphic Example to -link damage computation How? • We run over the squares center-point. • Size of the Grid (squares) is determined by the requested accuracy parameters.

38. Algorithm to Evaluate the Expected Damage for a given Cut. • We define an additiveto the cut capacity as a quantity satisfying: Where is the actual capacity intersecting the cut.

40. Numerical Accuracy • We refer the side length of the grid squares as the “grid-constant” . • We denote the set of squares center-points as “grid points”. • We restrict our results to the case where , as otherwise the approximation is too crude to consider.

41. Numerical Accuracy • The leading term in the error when integrating numerically over the grid points, as is obtained by the following geometric results

42. Numerical Accuracy • and are tangents to a circle of radius centered at and to the circular cut. • The colored areas depicts the extremum of difference for possible endpoints emanating from a point within the circle centered at at one side of the cut.

43. Numerical Accuracy For simplicity, let’s look at this • We want to bound the grey area using .

44. Numerical Accuracy • Using trigonometry and technique this area can be bounded by where and is the diagonal length of . • Basic idea is calculation of the head angle using and of the grey triangle where bounding its sides with .

45. Numerical Accuracy • First, let’s look at the extreme case: Using We can obtain that . Thus

46. Numerical Accuracy • First, let’s look at the extreme case: Using We can obtain that . Thus This area is bounded by . Where is the diagonal length of . Bounded by , thus bounded also by

47. Numerical Accuracy • Now, back to the general case: is the same as before (barely visible in this setting) • Using geometry we obtain: . • If the point is located within a distance of . Thus, the grey area is bounded as in the previous case. • If we have . Thus, the grey area is bounded by.

48. Additive Approximation • Thus, the leading term (we will see why next) of the accumulated error from the numerical integration, as is Where

49. Additive Approximation • Other error terms are linear in , thus when are negligible by the leading term which is with factor of . • Using our analysis the function can be implemented by selecting small enough such that the total error will be at most .

50. Additive Approximation – Running Time • Denote the area of Rec by • The algorithm is based on performing numerical integration over pairs of grid points (square-center points). • The number of grid points in the rectangle is . Thus the running time is at most proportional to the number of pairs of grid points, which is .