massimo franceschetti university of california at berkeley
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Phase transitions an engineering perspective. MASSIMO FRANCESCHETTI University of California at Berkeley. Phase transition effect. when small changes in certain parameters of a system result in dramatic shifts in some globally observed behavior of the system. Example. - PowerPoint PPT PresentationTRANSCRIPT
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
Examplepercolation theory, Broadbent and Hammersley (1957)
Example
cp Broadbent and Hammersley (1957)
2
1cp H. Kesten (1980)
pc0 p
P1
Gilbert (1961)
Mathematics Physics
Percolation theoryRandom graphs
Random Coverage ProcessesContinuum Percolation
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 random graphs
Large scale networks of embedded devices
• Ad-Hoc networks• Sensor networks
Where are the phase transitions?
Uniform random distribution of points of density λ
One disc per pointStudies the formation of an unbounded connected component
First modelContinuum percolation, Gilbert (1961)
First modelContinuum 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
N Unit area
Penrose (1997) Penrose Pisztora (1996)
Xue Kumar (2003)
Gilbert (1961) [Meester, Roy book (1996)]
Haggstrom Meester (1996)
Dousse Baccelli Thiran (2003)
Booth, Bruck, Franceschetti, Meester (2003)
2Density of points in
N
nNr
)(log~2
Nr
1~2
Nneighborsnearest log~
Continuum Percolation
Nearest neighbors percolation
Percolation with interference
Clustered networks
More wireless models
Towards a less idealistic modelFranceschetti, Booth, Cook, Meester, Bruck (2003)
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
For all , there exists a finite , such that gss r*(x) percolates, for all )(1 * gc rr * *
It is possible to decrease the percolation threshold by taking a sufficiently large shift !
For all 2
)(0x
xg
CNP
Squishing and squashing Shifting and squeezing
What have we proven?
(sporadic) long links help the percolation process
CNP
Among all convex shapes the hardest to percolate is centrally symmetricJonasson (2001)
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
Small World Networks
Regular RandomSmall World
Watts Strogatz (1998)
Kleinberg (2000) Franceschetti & Meester (2003)
Routing in a small world
• Nodes on the grid• Fixed number of contacts• Probability scales with distance
• Nodes on the plane• Random number contacts in a given region • Density scales with distance
each node has only local information of the network connectivity
Routing in a small world
Connections of z are PPP of density
-delivery occurs when msg is delivered within to target
S
T
Routing in a small world
S
T
d
Routing in a small world
S
T
d
Scale the number of neighbors as 1/x2 to obtain efficient routing
Not only graphs…
One application
A pursuit evasion gameSinopoli, Schenato, Franceschetti, Poolla, Sastry (2003)
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 formal work to be done
For papers:
“Navigation in small world networks, a continuum scale-free model”Franceschetti and Meester
Preprint ‘03
“Percolation in wireless multi-hop networks”, Franceschetti, Booth, Cook, Meester, Bruck
ISIT ’03 and Submitted to IEEE Trans. Info Theory
“Covering algorithm continuum percolation and the geometry of wireless networks”Booth, Bruck, Franceschetti, Meester
Annals of Applied Probability, 13(2), May 2003.
“Kalman Filtering with intermittent observations”Sinopoli, Schenato, Franceschetti, Poolla, Sastry
CDC ’03 and Submitted to IEEE Trans. Automatic Control