1. Tools for the Analysis and Design of Complex Multi-Scale Networks: Dynamics; Security; Uncertainty
MURI Annual Review Columbus OH, October 14, 2010
J. Walrand, PI

2. Contents: Dynamics of Local Protocols Security of Graphs
Games with Uncertainty

3. Dynamics of Local Protocols:
Scientific Objectives
Approach: An Example
General Approach
Example 1
Example 2

4. Scientific Objectives
Researchers propose new protocols for networks
The protocols are based on local interactions
These protocols are difficult to evaluate
New methodology for
Delays
Transients
Impact of mobility

5. Approach: An Example
Link 1
Link 2
Link 3

6. Approach: An Example (continued)

7. Approach: An Example (continued)

8. Approach: An Example (continued)
How to evaluate delays, mobility, transients?
Service rate of queue 1 = P({1}) + P({1, 3})
=> Service rate of queues = f(R) = f(q)

9. Approach: An Example (continued)
How to evaluate delays, mobility, transients?

10. General Approach: Local interactions, randomized protocols
Use fluid limit approximation to get ODEs
Evaluate ODEs for transient analysis, delays, mobility

11. Example 1: ODEs vs. Simulations

12. Example 2: ODEs vs. Simulation

13. Example 2: ODEs + Simulations with Virtual Arrivals

14. Security of Graphs:
Model
Theorem
Examples
Application
Algorithm

15. Model: Graph Goal of Network Manager:
Choose a Spanning Tree T to minimize Probability of Attack
Goal of Attacker:
Attack a Link L to maximize Probability that L is in T
Attacker: + 1
Manager: - 1
One-Shot Zero-Sum Game (Simultaneous Actions)

16. Theorem: There is a Nash Equilibrium where
Attacker attacks only the links of a critical set C, with equal probabilities
Manager chooses only trees that have a minimal intersection with C, and have equal likelihood of using each link of C, not larger than that of using any link not in C. [Such a choice is possible.]
There is a polynomial algorithm to find C [Cunningham]
Critical Set: C if it has a maximal vulnerability ν(C) where

17. Example 1:
Critical Set: C if it has a maximal vulnerability ν(C) where
ν(C) = 4/7
Every spanning tree must use at least 4 links of C, i.e., 4 out of 7 links of C.
This fraction is maximal for C.
Note that the fraction is only ½ for the minimal cut set.

18. Example 1: ν(C) = 4/7 Nash equilibrium:
Attacker attacks each link of C with probability 1/7;
Manager chooses spanning trees that use only 4 links of C, in such a way that the probability that a chosen tree uses a particular link of C is 4/7, and
this probability is not larger than for any link not in C.]

19. Example 2:
Critical Set: C if it has a maximal vulnerability ν(C) where
ν(C) = 1/2
Every spanning tree must use at least 1 link of C, i.e., 1 out of 2 links of C.
This fraction is maximal for C.
In this case, C is the minimal cut set.

20. Example 2: ν(C) = 1/2 Nash equilibrium:
Attacker attacks each link of C with probability 1/2;
Manager chooses spanning trees that use only 1 link of C, in such a way that the probability that a chosen tree uses a particular link of C is ½, which is not larger than the probability of using any link not in C.

21. Application: Network Design:
How to choose a topology with minimal vulnerability?
V(G) = 3/4
V(G) = 3/5
V(G) = 2/3
2/3 > 3/5

22. Games with Uncertainty:
Excess of Caution can Hurt
Should one be Optimistic?

23. Excess of Caution can Hurt
Relay Network:

24. Excess of Caution can Hurt
Relay Network:

25. Optimism
Best Response:
Consistency:
Attitude:

26. Optimism
Attitude Game:
Example:
Consistent Sets:
Attitude Game: