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

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

Percolation theory Broadbent and Hammersley (1957)

H. Kesten (1980) pcpc 0 p P 1 Percolation theory

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), such a graph is very unlikely to have property Q. Random graphs Erdös and Rényi (1959)

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

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

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

 0.3  0.4 c  …[Quintanilla, Torquato, Ziff, J. Physics A, 2000] Continuum Percolation Gilbert (1961)

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

An extension of the model Sensor networks with noisy links

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 Experiment

Prob(correct reception) Experimental results

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

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

Connection probability 1 ||x|| Example

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

Shifting and Squeezing Connection probability ||x||

Example Connection probability ||x|| 1

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

Conjecture For all

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 !

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

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

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

For papers, send me 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.

c =  How to find the CNP of a given connection function Run 7000 experiments with 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

How to find the CNP of a given connection function