1. Game-Theoretic Analysis of Mobile Network Coverage David K.Y. Yau

2. Outline  Introduction  Mobility models  Cats’ strategies  Mouse’s strategies  Experimental results  Conclusion  Introduction  Mobility models  Cats’ strategies  Mouse’s strategies  Experimental results  Conclusion

3. Motivation – Why Mobile?  The mouse  Evade detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies  The mouse  Evade detection  Nature of “mission”  The cat  Improved coverage with fewer sensors  Robustness against contingencies

4. Problem Formulation  Two player game—cats and mouse  Closed rectangular area  Cats try to shorten detection time  Mouse tries to lengthen detection time  Both move at constant speed  Both have finite sensing range  Ends when mouse is within cat’s sensing range  Two player game—cats and mouse  Closed rectangular area  Cats try to shorten detection time  Mouse tries to lengthen detection time  Both move at constant speed  Both have finite sensing range  Ends when mouse is within cat’s sensing range

5. Mobility Model  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection  Four-tuple  N: network area  M: accessibility constraints -- the “map”  T: trip selection  R: route selection  Random waypoint model is a special case  Null accessibility constraints  Uniform random trip selection  Cartesian straight line route selection

6. The Sensing Range  Cats’ sensing range  R c  Mouse’s sensing range  R m  Blind mouse  R m < R c  Caught before evasion  Seeing mouse  R m > R c  Active evasion possible  Cats’ sensing range  R c  Mouse’s sensing range  R m  Blind mouse  R m < R c  Caught before evasion  Seeing mouse  R m > R c  Active evasion possible

7. Cats’ Strategies  Uniform scan  Bouncing  Random waypoint model  Uniform scan  Bouncing  Random waypoint model

8. Mouse Strategy  Blind  Hide at safe haven  Assume cats’ presence statistics is known  Seeing  Cats presence  Run  Maximize the minimum distance to all cats  Cats absence  Bouncing  Random waypoint model  Static  Don’t move  Blind  Hide at safe haven  Assume cats’ presence statistics is known  Seeing  Cats presence  Run  Maximize the minimum distance to all cats  Cats absence  Bouncing  Random waypoint model  Static  Don’t move

9. The Presence Matrix – ∏  Probability of a cat presence at an area  Divide network area into m × n cells  ∏ i,j = Probability of one or more cats present in cell (i, j)  Probability of a cat presence at an area  Divide network area into m × n cells  ∏ i,j = Probability of one or more cats present in cell (i, j)

10. Best Blind Mouse Play  Find an optimal path to move to a safest cell  Detection time is maximized along the path  ∏ i,j is lowest at the safest cell (usually)  Dynamic programming  Greedy does not always work  Find an optimal path to move to a safest cell  Detection time is maximized along the path  ∏ i,j is lowest at the safest cell (usually)  Dynamic programming  Greedy does not always work

11. Comparison with Local Greedy Strategy Greedy Dynamic Programming 0.00030.03000.00030.0300

12. Optimal Escape Path Formulation  Using ∏, computes  E j [T stay ] = Expected detection time if staying at cell i  E j [T move(k) ] = Expected detection time if moving to cell k  Cell k is a neighboring cell of i  Make decision—stay or move  Maximize expected detection time  Optimal escape path = sequence of movement until stay is chosen  How to compute the expected detection time?  Using ∏, computes  E j [T stay ] = Expected detection time if staying at cell i  E j [T move(k) ] = Expected detection time if moving to cell k  Cell k is a neighboring cell of i  Make decision—stay or move  Maximize expected detection time  Optimal escape path = sequence of movement until stay is chosen  How to compute the expected detection time?

13. Compute Expected Detection Time  Initialize E j [T detect ] as E j [T stay ]  Insert all the cells into max-heap  i := Extract-max  Update E j [T detect ] for each neighbor cell k of i  E k [T detect ] := max(E k [T detect ], E i [T move(k) ])  Heapify  Repeat until heap becomes empty  Initialize E j [T detect ] as E j [T stay ]  Insert all the cells into max-heap  i := Extract-max  Update E j [T detect ] for each neighbor cell k of i  E k [T detect ] := max(E k [T detect ], E i [T move(k) ])  Heapify  Repeat until heap becomes empty

14. Example Optimal Paths 0.00000.03740.00000.0374 V m = 10 m/s V m = 15 m/s

15. Blind Mouse Strategies Compared Expected Detection Time Cat Strategies ScanBouncingRWP Mouse Strategies DP1083.26628.662823.26 RWP415.31442.23271.73 Stay511.50305.03226.13 V c = 10 m/s, V m = 10 m/s, R c = 25 m, R m = 0 m

16. Other Options for Cats  Increase sensing range  Increase speed  Increase quantity  Increase sensing range  Increase speed  Increase quantity R c (m) 15152550 T detect (s) 151363056948585271 V c (m/s) 10204080160 T detect (s) 26181253640245130 NcNcNcNc121050100 5732784562

17. Best Seeing Mouse Play  Find and move at the optimum direction  Minimum distance to all cats is maximized  Distance between cat and mouse  d(β, t) = ║C(t) – M(β, t)║  Minimum distance moving at direction β  d * (β) = min t ≥ 0 { d(β, t) }  Optimal escape direction  β * = argmax d * (β)  Find and move at the optimum direction  Minimum distance to all cats is maximized  Distance between cat and mouse  d(β, t) = ║C(t) – M(β, t)║  Minimum distance moving at direction β  d * (β) = min t ≥ 0 { d(β, t) }  Optimal escape direction  β * = argmax d * (β)

18. Strategies When Cat Absence  Bouncing  Centric  Random waypoint model  Static  Don’t move  Bouncing  Centric  Random waypoint model  Static  Don’t move

19. Seeing Mouse Strategies Compared Expected Detection Time Cat Strategies BouncingRWP Mouse Strategies Bouncing149.531455.28 Centric340.851092.29 Static92.39899.07 Stay10.2321.99 V c = 10 m/s, V m = 10 m/s, R c = 5 m, R m = 10 m

20. Result Explained  Why Bouncing is better for cats?  All area are equally likely to be visited (approx.)  Uniform presence matrix (approx.)  Safe haven eliminated  Why Centric is better for mouse?  More choices of direction  Why Bouncing is better for cats?  All area are equally likely to be visited (approx.)  Uniform presence matrix (approx.)  Safe haven eliminated  Why Centric is better for mouse?  More choices of direction

21. Presence Matrices Random Waypoint Model Bouncing

22. Where The Mouse Were Caught? 0.01543.2 Detection Time (s)

23. Other Options—Sensing Range

24. Other Options—Speed

25. Other Options—Quantity

26. Conclusions  Detect intelligent mobile target using mobile sensor  Mobile sensors increase robustness  Strategies to evade detection without full knowledge of cat movement  Movement model is important  Presence matrix determine the coverage performance  Bouncing movement is better than random waypoint model  Stochastic movement prevent movement prediction  Optimal escape direction helps seeing mouse  Dynamic programming algorithm helps blind mouse  Effects of sensing range, speed and number of cats are quantified  Detect intelligent mobile target using mobile sensor  Mobile sensors increase robustness  Strategies to evade detection without full knowledge of cat movement  Movement model is important  Presence matrix determine the coverage performance  Bouncing movement is better than random waypoint model  Stochastic movement prevent movement prediction  Optimal escape direction helps seeing mouse  Dynamic programming algorithm helps blind mouse  Effects of sensing range, speed and number of cats are quantified

27. Future Work  Radioactive, chemical plume detection  Explosion  Dispersion  Mobile target detection with presence of obstacle  Model for sensor reliability, interference, etc.  Quantification of sensing uncertainty  Radioactive, chemical plume detection  Explosion  Dispersion  Mobile target detection with presence of obstacle  Model for sensor reliability, interference, etc.  Quantification of sensing uncertainty

28. Question…