massimo franceschetti university of california at berkeley phase transitions an engineering...
Post on 21-Dec-2015
214 views
TRANSCRIPT
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