Post on 30-Jan-2016
description
MASSIMO FRANCESCHETTIUniversity of California at Berkeley
Percolation of Wireless Networks
Continuum percolation theoryMeester and Roy, Cambridge University Press (1996)
Uniform random distribution of points of density λ
One disc per pointStudies the formation of an unbounded connected component
Model of wireless networks
Uniform random distribution of points of density λ
One disc per pointStudies the formation of an unbounded connected component
A
B
0.3 0.4
Example
0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]cr2
Maybe the first paper on Wireless Ad Hoc Networks !
Introduced by…
To model wireless multi-hop networks
Ed Gilbert (1961)(following Erdös and Rényi)
Ed Gilbert (1961)
λc λ2
1
0
λ
P
λ1
P = Prob(exists unbounded connected component)
A nice story
Gilbert (1961)
Mathematics Physics
Started the fields ofRandom Coverage Processesand Continuum Percolation
Engineering (only recently)Gupta and Kumar (1998,2000)
Phase TransitionImpurity Conduction
FerromagnetismUniversality (…Ken Wilson)
Hall (1985)Meester and Roy (1996)
Welcome to the real world
http://webs.cs.berkeley.edu
Welcome to the real world
“Don’t think a wireless network is like a bunch of discs on the plane” (David Culler)
•168 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)
Connectivity with noisy links
Unreliable connectivity
1
Connectionprobability
d
Continuum percolationContinuum percolation
2r
Random connection modelRandom connection model
d
1
Connectionprobability
Rotationally asymmetric ranges
Start with simplest extensions
Random connection model
Connectionprobability
||x1-x2||
)( 21 xxg
2
)())((0x
xgxgENC
]1,0[:)( 221 xxgdefine
Let 221, xx
such that
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
51.44)(
...359.0
2
2
rdxxgCNP
r
cc
c
CNP
Squishing and squashing Shifting and squeezing
for the standard connection model (disc)
c=0.359
How to find the CNP of a given connection functionRun 7000 experimentswith 100000 randomly placed points in each experimentlook 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
Prob(Correct reception)
Rotationally asymmetric ranges
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?
Conclusion
.eduWWW. .
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massimof@EECS.berkeley.edu
PaperAd hoc wireless networks with noisy links.Submitted to ISIT ’03.With L. Booth, J. Bruck, M. Cook.