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MASSIMO FRANCESCHETTI University of California at Berkeley

Ad-hoc wireless networks with noisy links. MASSIMO FRANCESCHETTI University of California at Berkeley. Lorna Booth, Matt Cook, Shuki Bruck, Ronald Meester. Phase transition effect.

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MASSIMO FRANCESCHETTI University of California at Berkeley

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  1. Ad-hoc wireless networks with noisy links MASSIMO FRANCESCHETTI University of California at Berkeley Lorna Booth, Matt Cook, Shuki Bruck, Ronald Meester

  2. Phase transition effect when small changes in certain parameters of the network result in dramatic shifts in some globally observed behavior, i.e., connectivity.

  3. Percolation theory Broadbent and Hammersley (1957)

  4. P 1 0 pc p Percolation theory Broadbent and Hammersley (1957) H. Kesten (1980)

  5. Random graphs Erdös and Rényi (1959) if graphs with p(n) edges are selected uniformly at random from the set of n-vertex graphs, there is a threshold function, f(n) such that if p(n) < f(n)a randomly chosen graph almost surely has property Q; and if p(n)>f(n), such a graph is very unlikely to have property Q.

  6. Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component B A Continuum Percolation Gilbert (1961)

  7. B A Continuum Percolation Gilbert (1961) The first paper in ad hoc wireless networks !

  8. P 1 0 λc λ Continuum Percolation Gilbert (1961) P = Prob(exists unbounded connected component)

  9. Continuum Percolation Gilbert (1961) l=0.4 l=0.3 lc~0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

  10. Phase transitions in graphs Erdös and Rényi (1959) Broadbent and Hammersley (1957) Gilbert(1961) Physics Mathematics Models of the internet Impurity Conduction Ferromagnetism… Universality, Ken Wilson Nobel prize Percolation theory Random graphs Random Coverage Processes Continuum Percolation Grimmett (1989) Bollobas (1985) Hall (1985) Meester and Roy (1996) wireless networks (more recently) Gupta and Kumar (1998) Dousse, Thiran, Baccelli (2003) Booth, Bruck, Franceschetti, Meester (2003)

  11. An extension of the model Sensor networks with noisy links

  12. Experiment • 168 rene 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

  13. Prob(correct reception) Experimental results

  14. Connection probability Connection probability 1 1 d 2r d Random connection model Continuum percolation Connectivity with noisy links

  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. Conjecture Forall

  22. Theorem Consider annuli shapes A(r) of inner radius r, unit area, and critical density For all , there exists a finite , such that A(r*)percolates, for all It is possible to decrease the percolation threshold by taking a sufficiently large shift !

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

  24. 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.

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

  26. For papers, send me email: massimo@paradise.caltech.edu Percolation in wireless multi-hop networks, Submitted to IEEE Trans. Info Theory Covering algorithm continuum percolation and the geometry of wireless networks (Previous work) Annals of Applied Probability, 13(2), May 2003.

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