Evaluation of Geographic Vulnerabilities in Networks

Todd Gardner

Dr. Cory Beard

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.

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.

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.

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.

7: t

1

2

3

4

5

7 Node Network with 2-Terminal Vulnerability (6-7).

6: s

6 71

2

3

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7 Node Network with All -Terminal Vulnerability.

Some Notation First

r Threat radius F List of feasible failure events Q FIFO Queue of network events to test

E2 Event of form ([x y ], where node x is active and node y is

failed (for 2-Terminal method). Ea Event of form ([1,0,1] where 1 is not failed and 0 is failed

(for all- terminal method). G(V,E) Graph consisting of Nodes N and Edges E GE(VE,EE) Graph formed from G(V,E) formed by Event E L(G) Laplacian Matrix of G formed by (2) 2…n Eigenvalues of L(G)

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)

2-Terminal Method – BackgroundDotson’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.

2-Terminal Method1. Use G(N,E) and r to calculate ISO 2. Set F to [] 3. Add E2 = [] network event to Q. 4. Loop till Q is empty

a. Pop event E2 off Q. Create GE2(VE2,EE2) b. Find shortest path in new GE2 using Breadth First Shortest

algorithm (BFS). success if path found. c. If (success), create new events reflecting the complement of

new path not specified in E2 (1). i. Test new events for feasibility of single threat with radius r

using ISO ii. If (feasible) then add event to Q else discard

d. If (not success) add new failure mode to F 5. For each failure mode i in F

a. Find first node associated with failure mode i. Add threat radius around node as Vi area.

b. For remaining nodes n associated with failure mode i: Replace Vi with intersection of Vi and threat radius around node n. Repeat until no nodes remain in i.

6. Vulnerable area V = V1 U V2 U … U VF

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 (1, 2, 3, ... n) of L(G) where

n is the number of nodes in G

0 = 1 <= 2 <=3 <= ... <= n (3)

• Algebraic connectivity, (2) 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 1 is the number disconnected components [3].

• Since L is a semi-definite positive singular matrix 1 is always zero.

All-Terminal Method1. Use G(N,E) and r to calculate ISO. 2. Add Ea = [1,1,1,…,1] network event to Q 3. Loop till Q is empty

a. Pop Ea off Q. Create GEa(VEa,EEa) b. Calculate n of L(GE). If > 0 then success

(network connected) c. If (success), create new events by incrementally failing each

node remaining that is not part of Ea, add the new event to Q. i. Test new events for feasibility of single threat with radius r

using ISO ii. If (feasible) then add event to Q else discard.

d. If (failure) store new failure event in F. 4. For each failure mode i in F

a. Find first node associated with failure mode i. Add threat radius around node as Vi area.

b. For remaining nodes n associated with failure mode i: Replace Vi with intersection of Vi and threat radius around node n. Repeat until no nodes remain in i.

5. Vulnerable area V = V1 U V2 U … U VF

ResultsWireless 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.

2-Terminal All-Terminal

ResultsNorth 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

ResultsPerformance 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

0.01

0.1

1

10

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7Threat Ratio (Ratio of Threat Radius/Total Network Width)

Sec

on

ds

30 Nodes, Deg 5.7330 Nodes, Deg 9.67

30 Nodes, Deg 15.6040 Nodes, Deg 3.85

40 Nodes, Deg 9.1040 Nodes, Deg 12.75

0.1

1

10

100

1000

10000

0.1 0.2 0.3 0.4 0.5Threat Ratio (Ratio of Threat Radius/Total Network Width)

Cal

cula

tio

n T

ime

(Sec

)

30 Nodes, Deg 5.73

30 Nodes, Deg 9.67

30 Nodes, Deg 15.60

40 Nodes, Deg 3.85

40 Nodes, Deg 9.10

40 Nodes, Deg 12.75

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.

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).

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

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

7: t

1

2

3

4

5

7 Node Network with 2-Terminal Vulnerability (6-7).

6: s

1 3

1 3

1 3

Venn Diagram

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.

7: t

1

2

3

4

5

7 Node Network with 2-Terminal Vulnerability (6-7).

6: s

1 2 4

1 2 4

1 2 4

Venn Diagram

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

7: t

1

2

3

4

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7 Node Network with 2-Terminal Vulnerability (6-7).

6: s

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.....