conceptual issues in scaling sensor networks massimo franceschetti, uc berkeley
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Conceptual issues in scaling sensor networks
Massimo Franceschetti,UC Berkeley
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UCB Sensor Networks day, Jan 28, 2004 2
State of the art on scaling:
PEG ~100 sensor nodes, 1 evader, 2 pursuers
Cory Sharp & Shawn Schaffert
Shankar Sastry group
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Can we dramatically scale this?
• Practical problems• Conceptual problems
design for complexity
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UCB Sensor Networks day, Jan 28, 2004 4
• Percolation theory• Random graphs• Distributed computing• Distributed control• Channel physics• Distributed sampling• Network information theory• Network coding
• Connectivity• Routing• Storage• Failures• Packet loss• Malicious behavior• Remote operation
Theory Practice
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Some conceptual issues
• LARGE SCALE CONNECTIVITY• ROUTING• CAPACITY• CONTROL
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1
Connectionprobability
|x|
Continuum percolationContinuum percolation
2r
Random connection modelRandom connection model
|x|
1
Connectionprobability
Single hop model
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Multi-hop connectivity model
There is a phase transition at a critical node density value
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1
g1(x)
|x|2r |x|
1
g2(x)
How does the critical density change with the shape of the connection function?
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General Tendency
When the selection mechanism with which
nodes are connected to each other is sufficiently “spread out’’, then few links (in the limit one on average) will suffice to obtain global connectivity.
Balister, Bollobas, Walters (2003)
Franceschetti, Booth, Cook, Bruck, Meester (2003)
D. Dubhashi, O. Haggstrom, A. Panconesi (2003)
R. Meester, M. Penrose, A. Sarkar (1997)
M. Penrose (1993)
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General Tendency
In contrast, when connections do not spread
out, few links are not enough for connectivity.
Xue and P. R. Kumar (2003)
O. Haggstrom and R. Meester (1996)
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Connectionprobability
1
|x|
Spread out connections (1)
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UCB Sensor Networks day, Jan 28, 2004 12
2
)(0x
xg
Theorem
))(())(( xpgpxg cc
For all connection functions
“longer links are trading off for the unreliability of the connection”
“it is easier to reach connectivity in this model of unreliable network”
Franceschetti, Booth, Cook, Bruck, Meester (2003)
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Spread out connections (2)
Connectionprobability
|x|
1
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Mixture of short and long links
Two different spreading strategies
Links are made all longer
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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 connectivity threshold by taking a sufficiently large shift !
Balister, Bollobas, Walters (2003)
Franceschetti, Booth, Cook, Bruck, Meester (2003)
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CNL
squashing Shifting
What have we learned
CNL=average number of connections per node needed for connectivity
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• Navigation in the small world• Need links at ALL scale
lengths !
What about routing?
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Z
Intuition: scale invariance
r1
r2
Model of neighbors density:
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Z
Intuition: scale invariance
r1
r2
Model of neighbors density:
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UCB Sensor Networks day, Jan 28, 2004 20
Z
Intuition: scale invariance
r1
r2
Model of neighbors density:
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UCB Sensor Networks day, Jan 28, 2004 21
Z
Intuition: scale invariance
r1
r2
Slow far from destination
Slow close to destination
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UCB Sensor Networks day, Jan 28, 2004 22
Theorem
S
T
d
Franceschetti & Meester (2003)
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Bottom line
S
T
d
Build routing trees that are scale invariant
to route with few hops at all distance scales
Want to balance the number of short and long links
Need to exploit the “hairy edge”(D. Culler)
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UCB Sensor Networks day, Jan 28, 2004 24
Summary
• Towards a system theory of large scale networks
• Conceptual issues at different levels• Design for complexity strategy• Close the gap between theory and practice