Percolation of Clustered Wireless Networks

1. Percolation of Clustered Wireless Networks MASSIMO FRANCESCHETTI University of California at Berkeley

2. What I cannot create, I cannot understand (Richard Feynman) Can I understand what I can create?

3. Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component Continuum percolation theory Meester and Roy, Cambridge University Press (1996)

4. B A Model of wireless networks Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component

5. Example l=0.4 l=0.3 lc=0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

6. Introduced by… Ed Gilbert (1961) (following Erdös and Rényi) To model wireless multi-hop networks Maybethe first paperon Wireless Ad Hoc Networks !

7. P 1 0 λ1 λc λ2 λ Ed Gilbert (1961) P = Prob(exists unbounded connected component)

8. A nice story Gilbert (1961) Physics Mathematics Phase Transition Impurity Conduction Ferromagnetism Universality (…Ken Wilson) Started the fields of Random Coverage Processes and Continuum Percolation Hall (1985) Meester and Roy (1996) Engineering (only recently) Gupta and Kumar (1998,2000)

9. our contribution Clustered Wireless Networks

10. the lazy Gardener Generalization of Continuum Percolation

11. Clustered wireless networks Client nodes Base station nodes

12. Application Commercial networks Sensor networks

13. Contribution Algorithm Random point process Connectivity Algorithmic Extension Algorithm: each point is covered by at least a disc and each disc covers at least a point.

14. P 0 λ1 λ2 λ New Question What is the Result of a deterministic algorithm on a randomprocess ? 1 P = Prob(exists unbounded connected component)

15. , then for high λ, percolation occurs if for any covering algorithm, with probability one. P 1 0 λ1 λ2 λ A Basic Theorem P = Prob(exists unbounded connected component)

16. some covering algorithm may avoid if percolation for any value of λ P 0 λ A Basic Theorem 1 P = Prob(exists unbounded connected component)

17. Note: Percolation any algorithm Interpretation One disc per point Percolation Gilbert (1961) Need Only

18. Counter-intuitive For any covering of the points covering discs will be close to each other and will form bonds

19. The first principle is that you must not fool yourself and you are the easiest person to fool

20. A counter-example many finite annuli 2r obtain } Draw circles of radii {3kr, k no Poisson point falls on the boundaries of the annuli cover the points without touching the boundaries

21. A counter-example 2r Cluster, whatever Each cluster resides into a single annulus

22. A counter-example counterexample can be made shift invariant (with a lot more work)

23. cannot cover the points with red discs without blue discs touching the boundaries of the annuli Counter-example does not work

24. Proof by lack of counter-example?

25. Define disc small enough, such that red disc intersects the disc blue disc fully covers it e r R/2 Coupling proof Let R > 2r

26. Define disc small enough, such that red disc intersects the disc blue disc fully covers it Coupling proof Let R > 2r choose l > lc(e), then cover points with red discs

27. Coupling proof every e disc is intersected by a red disc therefore all e discs are covered by blue discs

28. Coupling proof every e disc is intersected by a red disc therefore all e discs are covered by blue discs blue discs percolate!

29. any algorithm percolates, for high l some algorithms may avoid percolation even algorithms placing discs on a grid may avoid percolation Bottom line Be careful in the design!

30. Which classes of algorithms, for , form an unbounded connected component, a.s., when is high?

31. Classes of Algorithms • Grid • Flat • Shift invariant • Finite horizon • Optimal “Covering Algorithms, continuum percolation, and the geometry of wireless networks” Annals of Applied Probability, to appear (Coll. L. Booth, J. Bruck, R. Meester)

32. P 1 0 λ1 λc λ2 λ Open Problems 1. Classes of Algorithms with a Critical Density

33. Open Problems 1. Classes of Algorithms with a Critical Density 2. Uniqueness of the infinite cluster

34. Open Problems 1. Classes of Algorithms with a Critical Density 2. Uniqueness of the infinite cluster 3. Existence of optimal algorithms (partially solved)

36. Welcome to the real world http://webs.cs.berkeley.edu

37. Experiment • 168 nodes on a 12x14 grid • grid spacing 2 feet • open space • one node transmits “I’m Alive” • surrounding nodes try to receive message http://localization.millennium.berkeley.edu

38. Prob(correct reception) Connectivity with noisy links

39. Connection probability Connection probability 1 1 d 2r d Random connection model Continuum percolation Unreliable connectivity

40. Rotationally asymmetric ranges Start with simple modifications to the connection function

41. Squishing and Squashing Connection probability ||x1-x2||

42. Connection probability 1 ||x|| Example

43. Theorem Forall “it is easier to reach connectivity in an unreliable network” “longer links are trading off for the unreliability of the connection”

44. Shifting and Squeezing Connection probability ||x||

45. Connection probability 1 ||x|| Example

46. Do long edges help percolation? Mixture of short and long edges Edges are made all longer

47. Squishing and squashing Shifting and squeezing for the standard connection model (disc) CNP

48. lc=0.359 How to find the CNP of a given connection function Run 7000 experiments with 100000 randomly placed points in each experiment look at largest and second largest cluster of points (average sliding window 100 experiments) Assume lc for discs from the literature and compute the expansion factor to match curves

49. How to find the CNP of a given connection function

50. Prob(Correct reception) Rotationally asymmetric ranges