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MASSIMO FRANCESCHETTI University of California at Berkeley

Percolation of Wireless Networks. MASSIMO FRANCESCHETTI University of California at Berkeley. Uniform random distribution of points of density λ. One disc per point. Studies the formation of an unbounded connected component. Continuum percolation theory.

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MASSIMO FRANCESCHETTI University of California at Berkeley

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

  2. 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)

  3. 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

  4. lc =0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000] r2 Example l=0.3 l=0.4

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

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

  7. 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)

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

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

  10. 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

  11. Prob(correct reception) Connectivity with noisy links

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

  13. Rotationally asymmetric ranges Start with simplest extensions

  14. Random connection model Connection probability Let define such that ||x1-x2||

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

  16. Connection probability 1 ||x|| Example

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

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

  19. Connection probability 1 ||x|| Example

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

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

  22. 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

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

  24. Prob(Correct reception) Rotationally asymmetric ranges

  25. Is the disc the hardest shape to percolate overall? CNP Non-circular shapes 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.

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

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

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