MASSIMO FRANCESCHETTI

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Ad-hoc wireless networks with noisy links
MASSIMO FRANCESCHETTI University of California at
Berkeley
Lorna Booth, Matt Cook, Shuki Bruck, Ronald
Meester
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Phase transition effect

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

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Percolation theory
Broadbent and Hammersley (1957)
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Percolation theory
Broadbent and Hammersley (1957)
H. Kesten (1980)
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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) lt
  f(n) a randomly chosen graph almost surely has
  property Q and if p(n)gtf(n), such a graph is
  very unlikely to have property Q.

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Continuum Percolation
Gilbert (1961)
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Continuum Percolation
Gilbert (1961)
The first paper in ad hoc wireless networks !
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Continuum Percolation
Gilbert (1961)
P Prob(exists unbounded connected component)
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Continuum Percolation
Gilbert (1961)
l0.4
l0.3
lc0.35910Quintanilla, Torquato, Ziff, J.
Physics A, 2000
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Phase transitions in graphs
Erdös and Rényi (1959)
Broadbent and Hammersley (1957)
Gilbert (1961)
Physics
Mathematics
Models of the internet Impurity
Conduction Ferromagnetism Universality, Ken
Wilson Nobel prize
Percolation theory Random graphs Random Coverage
Processes Continuum Percolation
Grimmett (1989) Bollobas (1985) Hall
(1985) Meester and Roy (1996)
wireless networks (more recently)
Gupta and Kumar (1998) Dousse, Thiran, Baccelli
(2003) Booth, Bruck, Franceschetti, Meester (2003)
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An extension of the model
Sensor networks with noisy links
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Experiment

• 168 rene nodes on a 12x14 grid
• grid spacing 2 feet
• open space
• one node transmits Im Alive
• surrounding nodes try to receive message

http//localization.millennium.berkeley.edu
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Experimental results
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Connectivity with noisy links
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Squishing and Squashing
Connection probability
x1-x2
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Example
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Theorem
For all
it is easier to reach connectivity in an
unreliable network
longer links are trading off for the
unreliability of the connection
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Shifting and Squeezing
Connection probability
x
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Example
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Do long edges help percolation?
Mixture of short and long edges
Edges are made all longer
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Conjecture
For all
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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 !
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for the standard connection model (disc)
CNP
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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.
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Bottom line
To the engineer as long as ENCgt4.51 we are
fine! To the theoretician can we prove more
theorems?
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For papers, send me email
massimo_at_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.
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How to find the CNP of a given connection function
Run 7000 experiments
with 100000 randomly placed points in each
experiment
look at largest and second largest cluster of
points (average sliding window 100 experiments)
Assume lc 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