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

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

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

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

Example l=0.4 l=0.3 lc=0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

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 !

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

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)

our contribution Clustered Wireless Networks

Generalization of Continuum Percolation the lazy Gardener Generalization of Continuum Percolation

Clustered wireless networks Client nodes Base station nodes

Application Commercial networks Sensor networks

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.

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)

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)

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)

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

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

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

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

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

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

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

Proof by lack of counter-example?

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

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

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

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

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

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

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)

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

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

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

Welcome to the real world http://webs.cs.berkeley.edu

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

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

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

Rotationally asymmetric ranges Start with simple modifications to the connection function

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

Example Connection probability 1 ||x||

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”

Shifting and Squeezing Connection probability ||x||

Example Connection probability ||x|| 1

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

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

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 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

How to find the CNP of a given connection function

Rotationally asymmetric ranges Prob(Correct reception) Rotationally asymmetric ranges

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

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

.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 email to: massimof@eecs.berkeley.edu