1. Percolation of Clustered Wireless Networks
MASSIMO FRANCESCHETTI
University of California at Berkeley

2. What I cannot create, I cannot understand
(Richard Feynman)
Can I understand what I can create?

3. Continuum percolation theory
Meester and Roy, Cambridge University Press (1996)
Uniform random distribution of points of density λ
One disc per point
Studies the formation of an unbounded connected component

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

5. Example
l=0.4
l=0.3
lc= …[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

6. Introduced by… Maybe the first paper on Wireless Ad Hoc Networks !
To model wireless multi-hop networks
Ed Gilbert (1961)
(following Erdös and Rényi)
Maybe the first paper on
Wireless Ad Hoc Networks !

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

8. A nice story Gilbert (1961) Physics Mathematics
Started the fields of
Random Coverage Processes
and Continuum Percolation
Phase Transition
Impurity Conduction
Ferromagnetism
Universality (…Ken Wilson)
Hall (1985)
Meester and Roy (1996)
Engineering (only recently)
Gupta and Kumar (1998,2000)

9. our contribution
Clustered Wireless Networks

10. Generalization of Continuum Percolation
the lazy Gardener
Generalization of Continuum Percolation

11. Clustered wireless networks
Client nodes
Base station nodes

12. Application
Commercial networks
Sensor networks

13. Contribution Algorithmic Extension Random Algorithm point process
Connectivity
Algorithmic Extension
Algorithm: each point is covered by at least a disc and each disc
covers at least a point.

14. New Question What is the Result of a deterministic
algorithm on a random process ? λ P λ2 λ1 1
P = Prob(exists unbounded connected component)

15. P = Prob(exists unbounded connected component)
A Basic Theorem if
for any covering algorithm, with probability one.
, then for high λ, percolation occurs λ P λ2 λ1 1
P = Prob(exists unbounded connected component)

16. P = Prob(exists unbounded connected component)
A Basic Theorem if
some covering algorithm may avoid
percolation for any value of λ λ P 1
P = Prob(exists unbounded connected component)

17. Percolation any algorithm
Interpretation
Note:
One disc per point
Percolation
Gilbert (1961)
Percolation any algorithm
Need Only

18. Counter-intuitive For any covering of the points covering discs
will be close to each other and will form bonds

19. The first principle is that you must not fool yourself
and you are the easiest person to fool

20. A counter-example Draw circles of radii {3kr, k } many finite annuli
obtain
no Poisson point falls on the boundaries of the annuli
cover the points without touching the boundaries

21. Each cluster resides into a single annulus
A counter-example 2r
Each cluster resides into a single annulus
Cluster, whatever

22. counterexample can be made shift invariant
A counter-example
counterexample can be made shift invariant
(with a lot more work)

23. Counter-example does not work
cannot cover the points with red discs without
blue discs touching the boundaries of the annuli

24. Proof by lack
of counter-example?

25. Coupling proof Let R > 2r Define red disc intersects the e
disc small enough, such that
Define
red disc intersects the
disc
blue disc fully covers it
R/2 r e

26. Coupling proof Let R > 2r Define red disc intersects the
disc small enough, such that
Define
red disc intersects the
disc
blue disc fully covers it
choose l > lc(e),
then cover points with red discs

27. Coupling proof every e disc is intersected by a red disc
therefore all e discs are covered by blue discs

28. Coupling proof every e disc is intersected by a red disc
therefore all e discs are covered by blue discs
blue discs percolate!

29. Bottom line Be careful in the design!
any algorithm percolates, for high l
some algorithms may avoid percolation
even algorithms placing discs on a grid
may avoid percolation

30. Which classes of algorithms, for , form an unbounded
connected component, a.s. ,
when is high?

31. Classes of Algorithms and the geometry of wireless networks”
Grid
Flat
Shift invariant
Finite horizon
Optimal
“Covering Algorithms, continuum percolation,
and the geometry of wireless networks”
Annals of Applied Probability, to appear
(Coll. L. Booth, J. Bruck, R. Meester)

32. 1. Classes of Algorithms with a Critical Density
Open Problems
1. Classes of Algorithms with a Critical Density λc λ2 1 λ P λ1

33. Open Problems 1. Classes of Algorithms with a Critical Density
2. Uniqueness of the infinite cluster

34. Open Problems 1. Classes of Algorithms with a Critical Density
2. Uniqueness of the infinite cluster
3. Existence of optimal algorithms (partially solved)

36. Welcome to the real world

37. 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

38. Connectivity with noisy links
Prob(correct reception)
Connectivity with noisy links

39. Unreliable connectivity1
Connection
probability d
Continuum percolation 2r
Random connection model d 1
Connection
probability

40. Rotationally asymmetric ranges
Start with simple modifications to the connection function

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

42. Example
Connection
probability 1
||x||

43. Theorem “it is easier to reach connectivity in an unreliable network”
For all
“it is easier to reach connectivity in an unreliable network”
“longer links are trading off for the unreliability of the connection”

44. Shifting and Squeezing
Connection
probability
||x||

45. Example
Connection
probability
||x|| 1

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

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

48. How to find the CNP of a given connection function
lc= 0.359
How to find the CNP of a given connection function
Run 7000 experiments
with 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

49. How to find the CNP of a given connection function

50. Rotationally asymmetric ranges
Prob(Correct reception)
Rotationally asymmetric ranges

51. 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.
Is the disc the hardest shape to percolate overall?
CNP

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

53. .edu/~massimo WWW. . Papers Download from: Or send email to:
Covering algorithms continuum percolation and the geometry of wireless networks
ISIT ’02, and to appear in the Annals of Applied Probability.
With L. Booth, J. Bruck, R. Meester.
Ad hoc wireless networks with noisy links.
Submitted to ISIT ’03.
With L. Booth, J. Bruck, M. Cook.
Download from:
.edu/~massimo
WWW. .
Or send to: