Adhoc wireless networks with noisy links MASSIMO FRANCESCHETTI

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

Percolation theory Broadbent and Hammersley (1957)

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

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.

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

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

Continuum Percolation Gilbert (1961) l=0. 3 l=0. 4 lc~0. 35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

Phase transitions in graphs ent and b d a o r B Erdös a 957) 1 ( y e l s er Hamm Gilbert (1961) yi (1959 ) Physics Mathematics Percolation theory Random graphs Random Coverage Processes Continuum Percolation Grimmett (1989) Bollobas (1985) Hall (1985) Meester and Roy (1996) nd Rén Models of the internet Impurity Conduction Ferromagnetism… Universality, Ken Wilson Nobel prize wireless networks (more recently) Gupta and Kumar (1998) Dousse, Thiran, Baccelli (2003) Booth, Bruck, Franceschetti, Meester (2003)

An extension of the model Sensor networks with noisy links

Experiment • 168 rene nodes on a 12 x 14 grid • grid spacing 2 feet • open space • one node transmits “I’m Alive” • surrounding nodes try to receive message http: //localization. millennium. berkeley. edu

Experimental results Prob(correct reception)

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

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

Connection probability Example 1 ||x||

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

Shifting and Squeezing Connection probability ||x||

Example Connection probability 1 ||x||

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

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 !

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

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. Is the disc the hardest shape to percolate overall? CNP

Bottom line To the engineer: as long as ENC>4. 51 we are fine! To theoretician: can we prove more theorems? CNP

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.