massimo franceschetti university of california at berkeley phase transitions an engineering...

MASSIMO FRANCESCHETTIUniversity of California at Berkeley

Phase transitions an engineering perspective

when small changes in certain parameters of a system result in dramatic shifts in some globally observed behavior of the system.

Phase transition effect

Can we mathematically explain these naturally observed effects?

Phase transition effect

Example 1percolation theory, Broadbent and Hammersley (1957)

Example 1

cp Broadbent and Hammersley (1957)

2

1cp H. Kesten (1980)

pc0 p

P1

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) < f(n) a

randomly chosen graph almost surely has property Q; and if p(n)>f(n), such a graph is very unlikely

to have property Q.

Example 2Random graphs, Erdös and Rényi (1959)

Uniform random distribution of points of density λ

One disc per pointStudies the formation of an unbounded connected component

Example 3Continuum percolation, Gilbert (1961)

Example 3Continuum percolation, Gilbert (1961)

The first paper in ad hoc wireless networks !

A

B

0.3 0.4

c0.35910…[Quintanilla, Torquato, Ziff, J. Physics A, 2000]

Example 3

Gilbert (1961)

Mathematics Physics

Percolation theoryRandom graphs

Random Coverage ProcessesContinuum Percolation

Wireless Networks (more recently)Gupta and Kumar (1998)Dousse, Thiran, Baccelli (2003)Booth, Bruck, Franceschetti, Meester (2003)

Models of the internetImpurity ConductionFerromagnetism…

Universality, Ken Wilson Nobel prize

Grimmett (1989)Bollobas (1985)

Hall (1985)Meester and Roy (1996)

Broadbent and Hammersley (1957) Erdös and Rényi (1959)

Phase transitions in graphs

Not only graphs…

Example 4Shannon channel coding theorem (1948)

C H(x)

H(x|y)

Attainable region

H(x|y)=H(x)-C

noise

source coding decoding destination

Example 5“The uncertainty threshold principle, some fundamental limitations

of optimal decision making under dynamic uncertainty”Athans, Ku, and Gershwin (1977)

1 T

...2,1,''min0

tuRuxQx

TEJ ttt

tt

ut

1

xy

BuAxx

tt

tttttt

An optimal solution for exists

T

/1|)(|max Aii

Our work

Kalman filtering over a lossy networkJoint work withB. SinopoliL. SchenatoK. Poolla M. JordanS. Sastry

Two new percolation modelsJoint work withL. Booth J. BruckM. CookR. Meester

Clustered wireless networks

Extending Gilbert’s continuum percolation model

Contribution

Random point

process

Algorithm Connectivity

Algorithm: each point is covered by at least a disc and each disc covers at least a point.

Algorithmic Extension

A Basic Theorem

0

λ

P

λ2λ1

1

r

R

2iffor any covering algorithm, with probability one.

, then for high λ, percolation occurs

P = Prob(exists unbounded connected component)

A Basic Theorem

0

λ

P

1

r

R

P = Prob(exists unbounded connected component)

2if some covering algorithm may avoidpercolation for any value of λ

2r

R Percolation any algorithm

One disc per point 0rNote:

PercolationGilbert (1961)

Need Only

Interpretation

Counter-intuitive

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

A counter-example

Draw circles of radii {3kr, k }

many finite annuliobtain

no Poisson point falls on the boundaries of the annuli

cover the points without touching the boundaries

2r

R

2r

Each cluster resides into a single annulus

Cluster, whatever

A counter-example 2r

R

2r

counterexample can be made shift invariant(with a lot more work)

A counter-example

2r

R

cannot cover the points with red discs without blue discs touching the boundaries of the annuli

Counter-example does not work

Proof by lack of counter-example?

Coupling proofLet R > 2r

R/2

r

disc small enough, such thatDefinered disc intersects the disc blue disc fully covers it

Coupling proofLet R > 2r

choose c(),then cover points with red discs

disc small enough, such thatDefinered disc intersects the disc blue disc fully covers it

Coupling proof

every disc is intersected by a red disctherefore all discs are covered by blue discs

Coupling proof

every disc is intersected by a red disctherefore all discs are covered by blue discsblue discs percolate!

2r

Rsome algorithms may avoid percolation

Bottom line

2r

Reven algorithms placing discs on a gridmay avoid percolation

Be careful in the design!

2r

Rany algorithm percolates, for high

Which classes of algorithms, for form an unbounded

connected component, a.s. ,when is high?

2

Classes of Algorithms

•Grid•Flat•Shift invariant•Finite horizon•Optimal

Recall Ronald’s lecture(… or see paper)

Another extension of percolationSensor networks with noisy links

Prob(correct reception)

Experiment

1

Connectionprobability

d

Continuum percolationContinuum percolation

2r

Our modelOur model

d

1

Connectionprobability

Connectivity model

Connectionprobability

1

x

A first order question

How does the percolation threshold cchange?

Squishing and Squashing

Connectionprobability

x

) ()( xpgpxgs

)(xg

2

)())((x

xgxgENC

))(())(( xgsENCxgENC

2

)(0x

xg

Theorem

))(())(( xgsxg cc

For all

“longer links are trading off for the unreliability of the connection”

“it is easier to reach connectivity in an unreliable network”

Shifting and Squeezing

Connectionprobability

x

)(

0

1

)()(

))(()(yhs

s

y

dxxxgxdxxgss

xhgxgss

)(xg

2

)())((x

xgxgENC

))(())(( xgssENCxgENC

)(xgss

Example

Connectionprobability

x

1

Mixture of short and long edges

Edges are made all longer

Do long edges help percolation?

2

)(0x

xg

Conjecture

))(())(( xgssxg cc

For all

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 )(0 * rc rr *

)(rc*

It is possible to decrease the percolation threshold by taking a sufficiently large shift !

CNP

Squishing and squashing Shifting and squeezing

What have we proven?

CNP

Among all convex shapes the hardest to percolate is centrally symmetricJonasson (2001), Annals of Probability.

Is the disc the hardest shape to percolate overall?

What about non-circular shapes?

CNP

To the engineer: above 4.51 we are fine!To the theoretician: can we prove “disc is hardest” conjecture?

can we exploit long links for routing?

Bottom line

Not only graphs…

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

A pursuit evasion game

• Goal: given observations find the best estimate (minimum variance) for the state

• But may not arrive at each time step when traveling over a sensor network

Intermittent observations

Problem formulation

System

Kalman Filter

M

z-1

ut

et

xt

M

z-1

K+

+

+

-

xt+1

yt+1

• Discrete time LTI system

• and are Gaussian random variables with zero mean and covariance matrices Q and R positive definite.

Loss of observation

• Discrete time LTI system

Let it have a “huge variance” when the observation does not arrive

Loss of observation

• The arrival of the observation at time t is a binary random variable

• Redefine the noise as:

Kalman Filter with losses

Derive Kalman equations using a “dummy” observation when

then take the limit for

t=0

Results on mean error covariance Pt

ci

cPt

ctt

PtMPE

PPE

||max

11

0condition initialany and 1for ][

0condition initial some and 0for ][lim

0

0

0

Special cases

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100S,

V

c

C is invertible, or A has a single unstable eigenvalue

Conclusion• Phase transitions are a fundamental effect in

engineering systems with randomness• There is plenty of work for mathematicians