Conceptual issues in scaling sensor networks

Massimo Franceschetti,UC Berkeley

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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• 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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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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Z

Intuition: scale invariance

r1

r2

Model of neighbors density:

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Z

Intuition: scale invariance

r1

r2

Slow far from destination

Slow close to destination

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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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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