MASSIMO FRANCESCHETTI

1
Percolation of Wireless Networks
MASSIMO FRANCESCHETTI University of California at
Berkeley
2
(No Transcript)
3
Model of wireless networks
Uniform random distribution of points of density ?
One disc per point
Studies the formation of an unbounded connected
component
4
Example
l0.3
l0.4
5
(No Transcript)
6
Maybe the first paper on Wireless Ad Hoc
Networks !
7
Ed Gilbert (1961)
P Prob(exists unbounded connected component)
8
A nice story
Gilbert (1961)
Physics
Mathematics
Phase Transition Impurity Conduction Ferromagnetis
m 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 (1998,2000)
9
Welcome to the real world
http//webs.cs.berkeley.edu
10
Welcome to the real world
Dont think a wireless network is like a bunch
of discs on the plane (David Culler)
11
Experiment

• 168 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
12
Connectivity with noisy links
13
Unreliable connectivity
14
Rotationally asymmetric ranges
Start with simplest extensions
15
Random connection model
Connection probability
Let
define
such that
x1-x2
16
Squishing and Squashing
Connection probability
x1-x2
17
Example
18
Theorem
For all
it is easier to reach connectivity in an
unreliable network
longer links are trading off for the
unreliability of the connection
19
Shifting and Squeezing
Connection probability
x
20
Example
21
Do long edges help percolation?
Mixture of short and long edges
Edges are made all longer
22
for the standard connection model (disc)
CNP
23
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
24
How to find the CNP of a given connection function
25
Rotationally asymmetric ranges
26
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.
27
Conclusion
To the engineer as long as ENCgt4.51 we are
fine! To the theoretician can we prove more
theorems?
28
Paper
Ad hoc wireless networks with noisy
links. Submitted to ISIT 03. With L. Booth, J.
Bruck, M. Cook.
Download from
Or send email to
massimof_at_EECS.berkeley.edu