Percolation Theory in Wireless Sensor Networks with Noisy Links

Wireless sensor networkswith noisy links 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 Meester and Roy, Cambridge University Press (1996)

B A Model of wireless networks Uniform random distribution of points of density λ One disc per point Studies the formation of an unbounded connected component

Example l=0.3 l=0.4

lc(1) 0.35910 lc(r) = = r2 r2 Threshold known (only) experimentally 2r lc(r) 4p r2 =4.5 = ENC ENC is independent of r [Quintanilla, Torquato, Ziff, J. Physics A, 2000]

Theory Ed Gilbert (1961) (following Erdös and Rényi) To model wireless multi-hop networks Maybethe first paperon Wireless Ad Hoc Networks !

P 1 0 λ1 λc λ2 λ Ed Gilbert (1961) P = Prob(exists unbounded connected component)

A nice story Gilbert (1961) Physics Mathematics Phase Transition Impurity Conduction Ferromagnetism Universality (…Ken Wilson) Started the fields of Random Coverage Processes and Continuum Percolation Hall (1985) Meester and Roy (1996) Engineering (only recently) Gupta and Kumar (1998)

Engineering “What have we learned from this theory? That adding more transmitters helps reaching connectivity… …so what?” (Jan Rabaey)

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

Experiment • 168 nodes on a 12x14 grid • grid spacing 2 feet • open space • one node transmits “I’m Alive” • surrounding nodes try to receive message http://localization.millennium.berkeley.edu

Prob(correct reception) Connectivity with noisy links

Connection probability Connection probability 1 1 d 2r d Random connection model Continuum percolation Unreliable connectivity

Rotationally asymmetric ranges How do percolation theory results change?

Random connection model Connection probability Let define such that ||x1-x2||

Squishing and Squashing Connection probability ||x1-x2||

Connection probability 1 ||x|| Example

Theorem Forall “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||

Connection probability 1 ||x|| Example

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

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

Prob(Correct reception) Rotationally asymmetric ranges

Is the disc the hardest shape to percolate overall? CNP Non-circular shapes

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

The network is connected, buthow do I get packets to destination? • Two extreme cases: • Re-transmissions are independent (channel is highly variant) • Re-transmissions have same outcome (channel is not variant) Flip a coin at every transmission Flip a coin only once to determine network connectivity

Compare three cases Connection probability Connection probability 1 1 d d Reliable network • Unreliable network • independent retransmissions • dependent retransmissions ENCunrel= ENCrel

Is shortest path always good? Not for independent transmissions! Sink 0.9 0.2 B 0.9 A Source

Shortest path Max chance of delivery without retransmission Min expected number of transmissions Unreliable-dependent Unreliable-independent Reliable

Bottom line Long links are helpful if you can consistently exploit them Connection probability 1 p ||x||

N hops vs. N hops (no retransmission) N hops vs. hops (with indep. retransmission) Bottom line Long links are helpful if you can consistently exploit them Connection probability 1 p ||x||

Acknowledgments Connectivity: L. Booth, J. Bruck, M. Cook. Routing: T. Roosta, A. Woo, D. Culler, S. Sastry

Percolation Theory in Wireless Sensor Networks with Noisy Links

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