Evaluation of Geographic Vulnerabilities in Networks

1. Evaluation of Geographic Vulnerabilities in Networks Todd Gardner Dr. Cory Beard

2. Publications • M. Todd Gardner, Cory Beard, “Evaluating Geographic Vulnerabilities in Networks”, Accepted for publication at the 2011 IEEE International Communications Quality and Reliability (CQR) Workshop, May 10-12, 2011. • M. Todd Gardner, Cory Beard, Deep Medhi, “Resilient Network Design to Prevent Geographic Vulnerabilities ”, Submitted for publication at the 2011 IEEE International Workshop on the Design of Reliable Communication Networks (DRCN 2011), October 10-12, 2011.

3. Evaluating Geographic Vulnerabilities in Networks • Wireless ad-hoc networks and wireline networks • Applications include search and rescue, military operations, and emergency communications • Many failure modes are geographic in nature. • Jammers, explosions, enemy attacks, terrain issues, and natural causes like floods, storms, and fires.

4. Evaluating Geographic Vulnerabilities in Networks • We did not find evaluation methods in for geographic vulnerabilities existing research. • We found methods like avg. nodal degree, algebraic connectivity, random walk betweenness, etc... • Two Methods • 2-Terminal (point to point demand) • All-Terminal (point to all terminal demand) • Use to optimize network node selection, placement and design.

5. 5 1 3 6: s 7: t 5 4 2 1 3 7 6 7 Node Network with 2-Terminal Vulnerability (6-7). 4 2 7 Node Network with All -Terminal Vulnerability. Geographic Evaluation Methods • The 2-Terminal method finds areas that given an impact of a certain radius can disconnect the source and destination pair. • The All-Terminal method finds areas that given a threat of an impact of a certain radius can disconnect any component of the network. • An event/attack in the shaded area will have an impact radius that could disable the network. • Striped area shows possible impact radius.

6. Some Notation First

7. 2-Terminal Method - Background • Dotson’s method [1] to calculate 2-terminal reliability • Finds a list of disjoint success modes and failure modes to connect source (s) and destination (t). • Uses DeMorgan’s Law (1) to divide search space. • This approach significantly reduces the search space and thus computation time over an exhaustive search approach. • Sparse networks benefit computationally from this approach. • We modified Dotson’s Method to discard geographically non-feasible failure modes. If P = {1,2,3,…} then P = {} + {1} + {1 2} + … (1)

8. 2-Terminal Method – Background Dotson’s Method • Start by finding the shortest path from source s to destination t. • If the path P is found, it is added to a list of successful combinations. • The complement of P is added to a list of combinations to try. • If a combination is tried and no path can be found, that combination is added to a list of failure modes that would disconnect s and t • The next combination on the list is tried. • When the list to try is exhausted, a complete list of disjoint successes and failures exist. • Each success or failure can be analyzed for its likelihood of occurrence and the 2-terminal reliability of the network is found.

9. 2-Terminal Method

10. All-Terminal Method - Background • Algebraic Connectivity concept was developed by Fiedler in [3]. • The Algebraic Connectivity is the 2nd smallest eigenvalue of the Laplacian Matrix L(G). L(G) = D(G) – A(G) (2) • D(G) is a diagonal matrix with the nodal degree of each node as the diagonal. • A(G) is the adjacency matrix of G. • The spectrum of a graph is the eigenvalues (l1, l2, l3, ... ln) of L(G) where n is the number of nodes in G 0 = l1 <= l2 <=l3 <= ... <= ln (3) • Algebraic connectivity, (l2) gives a strong indication of the connectivity of the graph. • One unique quality of the algebraic connectivity is that if a component of the graph is disconnected, the algebraic connectivity of the graph is zero. • The number of zero eigenvalues other than l1 is the number disconnected components [3]. • Since L is a semi-definite positive singular matrix l1 is always zero.

11. All-Terminal Method

12. Results Wireless Ad-Hoc Networks (Disk Communication Links) • Average Node Degree of 4.6 and 6.4 (isotropic transmission range of 30 and 40) • The areas of vulnerability are shown as the threat radius is changed, lighter shading corresponding to larger threat radii. • Notice changes in vulnerability shapes and locations as link density is increased. All-Terminal 2-Terminal

13. Results North American Reference Wireline Network (39 Node JANUS-US-CA from SNDlib 1.0) • The average node degree for this network is 3.0. • It is interesting to note the vulnerabilities in the highly dense areas on the east coast. • Node 34 in the All-Terminal case can be isolated due to the serial type of connection in that portion of the network. • This was not intuitive, given the ring type configuration. The cause is the proximity of the nodes in the ring to each other. 2-Terminal All-Terminal

14. Results Performance Testing • Using both methods implemented with Matlab on a PC with a dual-core processor and 2 Gb of RAM. • Computation time versus threat radius for 30/40 nodes and various average node degrees. • Threat Radius and number of nodes have largest effect on performance. • Increasing the threat radius eliminates benefits caused by discarding geographically non-feasible combinations. 2-Terminal Method All-Terminal Method

15. Insights from Results • Modifying the link density can significantly change the geographic vulnerability of a network (and not always in expected ways). • Ring topologies may be geographically vulnerable if the nodes of the ring are physically close together. • Using long links can significantly improve the geographic resilience of a network without increasing the link density. • Even though some insights are somewhat intuitive, we can now quantify those insights.

16. Future Work • Created an Integer Linear Program (ILP) that selects locations to add nodes to a mobile communications network to greatly improve its robustness to geographic events • Could use Swarm Optimization to select new node locations. • Swarm optimization also may provide a way to tie the evaluation technique to the optimization problem since the evaluation technique is not linear and cannot be used in an ILP. • In addition, we would like to use both the evaluation methods with more realistic networks that have transmission impairments and obstructions. • Using non-circular impact zones (simulate an earthquake).

17. References [1] William P. Dotson, Jurgen O. Gobien, “A new Analysis Technique for Probabilistic Graphs”, IEEE Transactions on Circuits and Systems, Vol. CAS-26, No. 10, October 1997. [2] Y. B. Yoo, “A Comparison of Algorithms for Terminal-Pair Reliability”, IEEE Transactions on Reliability, Vol. 37, No. 2, 1988 JUNE [3] M. Fiedler. “Algebraic Connectivity of Graphs”, Czechoslovak Math.Journal, 23(98):298–305, 1973. [4] Jia-Yu Shao, Ji-Ming Guo, Hai-Ying Shan, “The Ordering of Trees and Connected Graphs by Algebraic Connectivity”, Linear Algebra and its Applications 428 (2008), pp1421-1438, Elsevier ScienceDirect

18. 5 1 3 6: s 7: t 4 2 7 Node Network with 2-Terminal Vulnerability (6-7). Supplemental Slides • 2-Terminal Example • Shortest path: 5-1-3-7 • exclude s and t • 1-3 is success • Add 1 and 1 3 to queue to test • 1 2 4 success • Add 1 2 to queue • 1 2 4 not feasible, discard 1 3 1 3 1 3 Venn Diagram

19. 5 1 3 6: s 7: t 4 2 7 Node Network with 2-Terminal Vulnerability (6-7). Supplemental Slides • 2-Terminal Example • 1 3 2 4 success • 1 3 2 not feasible, discard • 1 3 2 4 feasible, add to Q • 1 2 fail, add to Fail List • 1 3 2 4 fail add to Fail List • Queue empty done • Fail list is 1 2 and 1 3 2 4. 1 2 4 1 2 4 1 2 4 Venn Diagram

20. 5 1 3 6: s 7: t 4 2 7 Node Network with 2-Terminal Vulnerability (6-7). Supplemental Slides • 2-Terminal Example • Fail list is 1 2, 1 3 2 4. • Vuln area is intersection • of circles 1,2 and 3,4. • Given an impact (threat) radius as shown

21. Supplemental Slides • Thoughts on Time Complexity (this is not published) • BFS Method has complexity of O(bd) • b is branching factor, d is graph depth • Eigenvalue decomposition is ~O(n3) • Reliability calculation worst case is O(2n-1) • Dotson has been shown to be much better most of the time. No proofs or analytic results. • 2-Terminal worst case grows at 2nbd • All-Terminal worst case grows at 2nn3 • Reality • Sparse networks are much less. 2n is not accurate. • Threat Radius also causes not to be accurate • More study.....