Towards a theory of large scale networks

Massimo Franceschetti

Where are we heading

year

Lo

g (

peop

le p

er

com

pu

ter)

Number CrunchingData Storage

NetworkingCommunication

Interaction with the physical world

Building Comfort,Smart Alarms

Great Duck Island

Elder Care

Fire Response

Factories

Wind ResponseOf Golden Gate Bridge

Vineyards

Redwoods

Instrumenting the world

Soil monitoring

• Connected ?

• Information throughput ?

• Transmission power ?

• routing ?

• delay ?

• reliability ?

Extreme scaling

Large scale networks theory

• Spatial stochastic networks

• Connectivity

• Sensor placement algorithms

• Throughput capacity

• Closing the loop

• Physics of propagation

• Distributed Algorithms

Talk Outline

Poisson distribution of points of density λ

Points are connected if their distance is less than 2rStudies the formation of connected components

Continuum percolation

S

D

Gilbert J.SIAM (1961)

Book: Meester & Roy (1995)

There is a phase transition at a critical node density value

Phase transition

Simulation of phase transition

Gradually increase the node density on the plane

λc λ2

1

0

λ

P

λ1

P = Prob(exists unbounded connected component)

Phase transition Theorem Gilbert J.SIAM (1961)

Networks with interference

Communication range is different from connectivity

Nodes in close range might not be connected

Dependent percolation model

What happens to the phase transition?

A percolation model for wireless networks

Node j can receive data from node i if the signal to noise plus interference ratio is above a threshold

All nodes transmit at power P

A percolation model for wireless networks

Gupta Kumar, IEEE Trans. IT, 2000]

Gilbert, J. SIAM, 1961]

[Dousse Baccelli Thiran, IEEE Trans. Net, 2004]

[Dousse Franceschetti Meester Thiran, preprint, 2004]

10

All nodes transmit at power PNode j can receive data from node i if the signal to noise plus interference ratio is above a threshold

Phase transition theorem (Dousse Baccelli Thiran)

Can we prove a stronger result ?

c

super-c

ritica

l

sub-cr

itical

No interference model (Gilbert)

Interference model

super-c

ritica

l

Theorem

c

super-c

ritica

l

sub-cr

itical

(Dousse Franceschetti Meester Thiran)

No interference model (Gilbert)

Interference model

super-c

ritica

l

Bottom line

An ideal network with perfect interference cancellation (independent percolation model) exhibits a phase transition for c

A network where nodes cause interference (dependent percolation model) exhibits a phase transition for c,

More work on connectivity

• Gilbert’s model

• Interference model

• Spread out, unreliable connections

Prob(correct reception)

Let’s look at some real data

•168 nodes on a 12x14 grid• grid spacing 2 feet• only one node transmits “I’m Alive” (no interference)• surrounding nodes try to receive message

http://localization.millennium.berkeley.edu

Absence of sharp threshold

1

Connectionprobability

||xi-xj ||||xi-xj||

1

Connectionprobability

How does the critical density change with the shape of the connection

probability?

c

Connectionprobability

||xi-xj||

Basic transformation: spreading probability

Example

Connectionprobability

1

||x||

longer links are trading off for the unreliability of the connection

TheoremFranceschetti Booth Cook Bruck Meester (2003)

It is easier to reach connectivity with unreliable spread out connections

Conjecture

More complex spreading of g also helps percolation

1

Disc is hardest shape to percolate

More work on connectivity

• Gilbert’s model

• Extension with interference

• Spread out, unreliable connections

• Sensor placement algorithms

Clustered wireless networks

Random point

processAlgorithm Connectivity

each point is covered by at least a red disc and each red disc covers at least a point

Franceschetti PhD Thesis, CIT 2003

r

R

Theorem

iffor any covering algorithm, with probability one

then for c percolation occurs1

if then some covering algorithms may avoidpercolation for any value of λ, with probability one

1

R radius blue discr radius red disc r

Covering algorithms

“Covering Algorithms, continuum percolation, and the geometry of wireless networks”

Annals of Applied Probability, 2003Booth Bruck Franceschetti Meester

PhD Thesis, CIT, 2003Franceschetti

When which classes of algorithms form an unbounded connected component, a.s., When is high?

From connectivity to network capacity

• Gilbert’s model

• Extension with interference

• Spread out, unreliable connections

• Sensor placement algorithms

• Throughput capacity

Throughput capacity

How much information can flow through the network?

Throughput capacity without interference

How many disjoint paths there are that traverse the network?

Nodes closer than a given range are connected

High density = more disjoint paths

Interference

!

Main claim

Operate the network near percolation threshold not in the high density regime

Previous results, routing in high density regime

Gupta Kumar (2000)

Kulkarni Viswanath (2002)

El Gamal, Mammen, Prabhakar, Shah (2004)

High density regime routing

• Divide area into small boxes • Scale down power to allow only

transmission to adjacent boxes• Route along almost straight lines

Previous results, high density regime

Our strategy, percolation regime

Franceschetti Dousse Tse Thiran (2004)

• Adopt thermodynamic scaling• Some boxes are empty

Our strategy

• Adopt thermodynamic scaling• Take c large to have many crossing paths of adjacent full

boxes (by percolation)• Use these paths as the “wireless backbone” to relay traffic

crossing path

Our strategy

Theorem

crossing path

This strategy achieves

maximum throughput, minumum delay

Throughput per node vs Range

Interference limited networkNo interference ideal network

0

c

Percolation

=log n =n

High density regime

Interference

Bottom line

Scaling power at a slow rate, order, can use straight line routing

Scaling power at a fast rate, disorder, no backbone forms

Phase transition, backbone forms, rich in crossing paths, not straight lines, carry most traffic over short hops

• Gilbert’s model

• Extension with interference

• Spread out, unreliable connections

• Sensor placement algorithms

• Throughput capacity

• Closing the loop

Let’s look at the Application level

Pursuit evasion games at Berkeley

Random losses in the feedback loop

Sinopoli Schenato Franceschetti

Poolla Sastry Jordan IEEE Trans-AC (2004)

SystemSensor

web

ControllerState

estimator

WirelessMulti-hop

• What happens to the Kalman filter when some sensor readings are lost?

• Can we bound the error covariance

Theorem Sinopoli Schenato Franceschetti

Poolla Sastry Jordan IEEE Trans-AC (2004)

c10

Theorem Sinopoli Schenato Franceschetti

Poolla Sastry Jordan IEEE Trans-AC (2004)

The road ahead

• Towards a system theory of large scale networks• Spatial stochastic networks as a core discipline• Intellectual unification across disciplines

New branches

• Time varying stochastic networks• Impact of mobility• Games on graphs

Phase transitions

• Phase transitions are a fundamental effect in engineering systems with randomness

• Optimal operation regions are often at the boundary of these transitions

A random walk model of wave propagation IEEE Trans.-AP

Franceschetti Bruck Schulman

Stochastic rays propagationIEEE Trans.-AP

Franceschetti

Some more work…

Interaction with physical level

Small world networksSmall-world networks a continuum model

Franceschetti Meester

A geometric theorem for network designIEEE Trans.-Comp.

Franceschetti Cook Bruck

Lower bounds on data collection times in sensory networksIEEE-JSAC

Florens Franceschetti McEliece

A group membership algorithm with a practical specificationIEEE Trans.-PDS

Franceschetti Bruck

Algorithms and protocols