MASSIMO FRANCESCHETTI University of California at Berkeley

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MASSIMO FRANCESCHETTI University of California at Berkeley Percolation of Wireless Networks

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Percolation of Wireless Networks. MASSIMO FRANCESCHETTI University of California at Berkeley. Uniform random distribution of points of density λ. One disc per point. Studies the formation of an unbounded connected component. Continuum percolation theory. - PowerPoint PPT Presentation

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Page 1: MASSIMO FRANCESCHETTI University of California at Berkeley

MASSIMO FRANCESCHETTIUniversity of California at Berkeley

Percolation of Wireless Networks

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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

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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

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0.3 0.4

Example

0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]cr2

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Maybe the first paper on Wireless Ad Hoc Networks !

Introduced by…

To model wireless multi-hop networks

Ed Gilbert (1961)(following Erdös and Rényi)

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Ed Gilbert (1961)

λc λ2

1

0

λ

P

λ1

P = Prob(exists unbounded connected component)

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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)

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Welcome to the real world

http://webs.cs.berkeley.edu

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Welcome to the real world

“Don’t think a wireless network is like a bunch of discs on the plane” (David Culler)

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•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

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Prob(correct reception)

Connectivity with noisy links

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Unreliable connectivity

1

Connectionprobability

d

Continuum percolationContinuum percolation

2r

Random connection modelRandom connection model

d

1

Connectionprobability

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Rotationally asymmetric ranges

Start with simplest extensions

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Random connection model

Connectionprobability

||x1-x2||

)( 21 xxg

2

)())((0x

xgxgENC

]1,0[:)( 221 xxgdefine

Let 221, xx

such that

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Squishing and Squashing

Connectionprobability

||x1-x2||

))(()( 2121 xxpgpxxgs

)( 21 xxg

2

)())((x

xgxgENC

))(())(( xgsENCxgENC

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Connectionprobability

1

||x||

Example

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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”

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Shifting and Squeezing

Connectionprobability

||x||

)(

0

1

)()(

))(()(yhs

s

y

dxxxgxdxxgss

xhgxgss

)(xg

2

)())((x

xgxgENC

))(())(( xgssENCxgENC

)(xgss

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Example

Connectionprobability

||x||

1

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Mixture of short and long edges

Edges are made all longer

Do long edges help percolation?

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2

51.44)(

...359.0

2

2

rdxxgCNP

r

cc

c

CNP

Squishing and squashing Shifting and squeezing

for the standard connection model (disc)

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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

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How to find the CNP of a given connection function

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Prob(Correct reception)

Rotationally asymmetric ranges

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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

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CNP

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

Conclusion

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PaperAd hoc wireless networks with noisy links.Submitted to ISIT ’03.With L. Booth, J. Bruck, M. Cook.