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

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

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

4.  0.3  0.4 Example  0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000] c r2r2

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

7. Ed Gilbert (1961) λcλc λ2λ2 1 0 λ P λ1λ1 P = Prob(exists unbounded connected component)

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

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

10. Welcome to the real world “Don’t think a wireless network is like a bunch of discs on the plane” (David Culler)

11. 168 nodes on a 12x14 grid grid spacing 2 feet open space one node transmits “I’m Alive” surrounding nodes try to receive message Experiment http://localization.millennium.berkeley.edu

12. Prob(correct reception) Connectivity with noisy links

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

14. Rotationally asymmetric ranges Start with simplest extensions

15. Random connection model Connection probability ||x 1 -x 2 || define Let such that

16. Squishing and Squashing Connection probability ||x 1 -x 2 ||

17. Connection probability 1 ||x|| Example

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

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

20. Example Connection probability ||x|| 1

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

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

23. c =  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 c for discs from the literature and compute the expansion factor to match curves

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

25. Prob(Correct reception) Rotationally asymmetric ranges

26. CNP Among all convex shapes the triangle is the easiest to percolate Among all convex shapes the hardest to percolate is centrally symmetric Jonasson (2001), Annals of Probability. Is the disc the hardest shape to percolate overall? Non-circular shapes

27. CNP To the engineer: as long as ENC>4.51 we are fine! To the theoretician: can we prove more theorems ? Conclusion

28. .edu WWW.. Download from: Or send email to: massimof@EECS.berkeley.edu Paper Ad hoc wireless networks with noisy links. Submitted to ISIT ’03. With L. Booth, J. Bruck, M. Cook.