massimo franceschetti university of california at berkeley ad-hoc wireless networks with noisy links...
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MASSIMO FRANCESCHETTIUniversity 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 theoryBroadbent and Hammersley (1957)
cp Broadbent and Hammersley (1957)
2
1cp H. Kesten (1980)
pc0 p
P1
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) 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.
Random graphsErdös and Rényi (1959)
Continuum PercolationGilbert (1961)
Uniform random distribution of points of density λ
One disc per pointStudies the formation of an unbounded connected component
A
B
The first paper in ad hoc wireless networks !
A
B
Continuum PercolationGilbert (1961)
1
0
λ
P
P = Prob(exists unbounded connected component)
Continuum PercolationGilbert (1961)
λc
0.3 0.4
c0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]
Continuum PercolationGilbert (1961)
Gilbert (1961)
Mathematics Physics
Percolation theoryRandom graphs
Random Coverage ProcessesContinuum Percolation
wireless networks (more recently)Gupta and Kumar (1998)Dousse, Thiran, Baccelli (2003)Booth, Bruck, Franceschetti, Meester (2003)
Models of the internetImpurity ConductionFerromagnetism…
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 modelSensor 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
http://localization.millennium.berkeley.edu
Prob(correct reception)
Experimental results
1
Connectionprobability
d
Continuum percolationContinuum percolation
2r
Random connection modelRandom connection model
d
1
Connectionprobability
Connectivity with noisy links
Squishing and Squashing
Connectionprobability
||x1-x2||
))(()( 2121 xxpgpxxgs
)( 21 xxg
2
)())((x
xgxgENC
))(())(( xgsENCxgENC
Connectionprobability
1
||x||
Example
2
)(0x
xg
Theorem
))(())(( xgsxg cc
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
Connectionprobability
||x||
)(
0
1
)()(
))(()(yhs
s
y
dxxxgxdxxgss
xhgxgss
)(xg
2
)())((x
xgxgENC
))(())(( xgssENCxgENC
)(xgss
Example
Connectionprobability
||x||
1
Mixture of short and long edges
Edges are made all longer
Do long edges help percolation?
2
)(0x
xg
Conjecture
))(())(( xgssxg cc
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 )(0 * rc rr *
)(rc*
It is possible to decrease the percolation threshold by taking a sufficiently large shift !
2
51.44)(
...359.0
2
2
rdxxgCNP
r
cc
c
CNP
Squishing and squashing Shifting and squeezing
for the standard connection model (disc)
CNP
Among all convex shapes the triangle is the easiest to percolateAmong 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 email:
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.