critical density thresholds and complexity in wireless networks bhaskar krishnamachari school of...

Critical Density Thresholds and Complexity in Wireless Networks

Bhaskar Krishnamachari

School of Electrical and Computer Engineering Cornell University

Department of Electrical EngineeringUniversity of Southern California (Fall 2002)

Large Scale Wireless Networks: The VisionLarge Scale Wireless Networks: The Vision The “many - tiny” principle: wireless networks of thousands of inexpensive

miniature devices capable of computation, communication and sensing For smart spaces, environmental monitoring ...

Berkeley MoteBerkeley MoteFrom Pister et al., Berkeley Smart Dust Project

Tomorrow’s Embedded Wireless SystemsTomorrow’s Embedded Wireless Systems

2From Manges et al., Oak Ridge National Laboratory, Instrumentation and Controls Division

ORNL Telesensor ChipORNL Telesensor Chip22

Berkeley Dust MoteBerkeley Dust Mote11

1From Pister et al., Berkeley Smart Dust Project

ChallengesChallenges Unattended Operation: adaptive, self-configuration mechanisms Severe energy constraints : no battery replacement Large Scale : possibly thousands of nodes Are there qualitatively new phenomena at this scale?

Berkeley MoteBerkeley MoteFrom Pister et al., Berkeley Smart Dust Project

Complex SystemsComplex Systems Ordered global behavior and structure “emerging” from

multiple local interactions.

From the Oxford Cryodetector Group

Phase TransitionsPhase Transitions

Emergent structure - abrupt change in a global system property at a critical level of local interactions.

Example: SuperconductivityExample: Superconductivity

Connectivity in a Multi-hop NetworkConnectivity in a Multi-hop Network

© 2002 Bhaskar Krishnamachari

All nodes increasing transmission range R simultaneously


Phase Transition for ConnectivityPhase Transition for Connectivity

Communication radius RProb

abilit

y (N

etwo

rk is

Con

nect

ed)


abilit

y (N

etwo

rk is

Con

nect

ed)

undesirableregime

desirableregime

Energy-efficient operating point

n = 20



Communication radius R

Prob

abilit

y (N

etwo

rk is

Con

nect

ed)



xx11 xx22 xx33 xx44

RR

2D model of Gupta & Kumar (1998) shows log n density threshold; involves continuum percolation theory. Simpler 1D Model. Consider Poisson arrivals with rate . Connectivity between nodes within range R.

Probability that first n nodes form a connected network:

This model also shows an analogous phase transition with an O(log n) density threshold function.


Phase Transition for Bi-ConnectivityPhase Transition for Bi-Connectivity

2 node-disjoint paths between all pairs of nodes.

n = 100


Prob

abilit

y (N

etwo

rk is

Con

nect

ed)


Bernoulli Random Graphs G(n,p)Bernoulli Random Graphs G(n,p)

Studied by mathematicians since 50’s (Erdos, Renyi, Bollobas, Spencer, others)


Phase Transitions in Random GraphsPhase Transitions in Random Graphs

A number of zero-one laws developed over the years. E.g. (Fagin 1976): for all first order graph properties A,

(Friedgut 1996): ALL monotone graph properties undergo sharp phase transitions.

Examples: k-Connectivity k-Colorability Hamiltonian cycle


Fixed Radius Random Graphs G(n,R)Fixed Radius Random Graphs G(n,R)

A model for multi-hop wireless networks Density parameter: communication radius R


Phase Transition for Hamiltonian Cycle FormationPhase Transition for Hamiltonian Cycle Formation

Does there exist a cycle in the network graph that visits each node exactly once?


abilit

y (H

amilt

onia

n cy

cle e

xist

s)

n = 100 n = 100


A Wireless Sensor Tracking ProblemA Wireless Sensor Tracking Problem

Given: Multiple sensors and targets Sensors can only communicate locally Sensors can only “see” targets locally Need 3 communicating sensors tracking each target

sensosensorrtargettarget


Communication between sensors within range RCommunication between sensors within range R

sensorsensor

targettarget

RR



Communication graphCommunication graph

sensorsensor

targettarget



sensorsensor

targettarget

SS


Targets visible within range STargets visible within range S


Visibility graphVisibility graph

sensorsensor

targettarget



Decision problem: can all targets can be tracked Decision problem: can all targets can be tracked by three communicating sensors ?by three communicating sensors ?

sensorsensor

targettarget



Phase Transition for Sensor Tracking Problem Phase Transition for Sensor Tracking Problem

Probability of Tracking all Targets

Sensing range Communication range

s = 17s = 17t = 5t = 5

%%

1

23

2


Broadcast Scheduling/Channel AllocationBroadcast Scheduling/Channel Allocation

No nodes within 2 hops of each other can be allocated the same channel.

Are k channels enough ? Less likely with more edges in network graph, i.e. with higher transmission range.

NP-Hard in general, polynomial-time for 1D model.

1

23

1



Communication/Interference range R

Pr

obab

ility

(k c

hann

els

suffi

ce)

n = 1001D model



If k channels are used, and W is the aggregate available bandwidth/data-rate, the maximum network-wide throughput is T = n(W/k)

Let 1 = max(# of 1-hop neighbors), 2 = max(# of 2-hop neighbors), then:



n = 1001D model

99% connectivity threshold

Operating point w/maximum throughput



Communication/Interference range R

Pr

obab

ility

(k c

hann

els

suffi

ce)

n = 302D model


29

Communication/Interference radius R© 2002 Bhaskar Krishnamachari


n = 302D model



Expe

cted

nor

mal

ized

netw

ork

thro

ughp

ut T

/W

Communication/Interference radius R© 2002 Bhaskar Krishnamachari


n = 1002D model



Expe

cted

nor

mal

ized

netw

ork

thro

ughp

ut T

/W

Min-k-neighbor property: all nodes have at least k neighbors.

Let Ai = event that node i has at least k neighbors, then

where (for R 0.5, ignoring edge effects)


Min-k-NeighborMin-k-Neighbor

The min-k-neighbor property undergoes a zero-one phase transition asymptotically at transmission radius

For finite n, the probability that all neighbors have at least k neighbors is > 1 - , if


Critical Threshold for Min-k-NeighborCritical Threshold for Min-k-Neighbor


Critical Threshold for Min-k-neighborCritical Threshold for Min-k-neighbor

n = 100 n = 100

Probability that all nodes have at least 2 neighbors



Probabilistic FloodingProbabilistic Flooding

Prob

abilit

y( a

ll no

des r

ecei

ve p

acke

t)

Query forwarding probability q

Each node forwards packet with probability q

n = 100 n = 100

Resource-efficientoperating point


Phase Transitions and Computational Complexity: Phase Transitions and Computational Complexity: Constraint SatisfactionConstraint Satisfaction

In the early 90’s, AI researchers found phase transitions in NP-complete constraint satisfaction problems like SAT (Cheeseman et al. 1991, Selman et al. 1992, Hogg et al. 1994)

SAT: given a binary logic formula in CNF (conjunction of OR clauses), is there a truth assignment which makes the formula true?

((a a b b c c) ) ( ( d d e e f f ) ) ( ( a a c c ee))


Phase Transition in 3-SATPhase Transition in 3-SAT

0 2 3 4 5Ratio of Constraints to Variables6 7 8

1000

3000

Cost

of C

ompu

tati

on

2000

400050 var 40 var 20 var

0.02 3 4 5

Ratio of Constraints to Variables6 7 8

0.2

0.6

Prob

abili

ty o

f Sol

utio

n

0.4

50% sat0.8

1.0

Selman et al. (1994):


Phase Transition in 3-SATPhase Transition in 3-SAT

(Friedgut 1999): constant ck s.t. , formulas with at most (1-)ckn clauses are satisfiable w.h.p. and formulas with at least (1+)ckn are unsatisfiable w.h.p.

Critical ratio of clauses to variables for 3-SAT: empirically ~ 4.24, theoretically between 3.125 and 4.601


Worst-case vs. Average ComplexityWorst-case vs. Average Complexity

3-SAT remains NP-complete even for random instances with ratios between 1/3 and 7(n2-3n+2).

(Frieze et al. 1996): Polynomial heuristic finds satisfying solutions w.h.p. if ratio less than 3.006.

(Bearne et al. 1998): Can prove unsatisfiability in polynomial time w.h.p. if ratio more than n/log(n).

(Williams and Hogg, 1994): First-order, algorithm-independent analysis of CSPs showing that average computational effort peaks at phase transition point.


Distributed Constraint Satisfaction Distributed Constraint Satisfaction in Wireless Networksin Wireless Networks

Many NP-complete problems in wireless networks can be formulated as distributed constraint satisfaction problems (DCSP).

A DCSP consists of agents, each with a set of variables they need to set values to. There are intra-agent constraints and inter-agent constraints on these values.


Hamiltonian Cycle FormationHamiltonian Cycle Formation

Does there exist a cycle in the network graph that visits each node exactly once?


Hamiltonian Cycle FormationHamiltonian Cycle Formation


n = 100


Conflict Free Resource AllocationConflict Free Resource Allocation

Allocating channels with 1-hop constraint.

3 channels



3 channels



Interference Range

More bandwidth


Decision problem: can all targets can be tracked Decision problem: can all targets can be tracked by three distinct communicating sensors ?by three distinct communicating sensors ?

sensorsensor

targettarget

Sensor Tracking ProblemSensor Tracking Problem


Sensor Tracking ProblemSensor Tracking Problem Decision Problem: Can each target be tracked by a

set of three distinct communicating sensors?

NP-complete for arbitrary communication and visibility graphs, polynomial solvable when the communication graph is complete.

Can be formulated as a DCSP:



Mean computational cost

Probability of Tracking

Sensing range Communication rangeSensing range Communication range



Mean communication cost

Probability of Tracking

Sensing range Communication range Communication rangeSensing range


ConclusionsConclusions

Phase transition analysis appears to be a promising approach for determining resource-efficient operating points for various global network properties.

Also helpful in reducing the average computational cost of distributed algorithms for constraint satisfaction in wireless networks.

It may be possible to incorporate this approach into online, self-configuring mechanisms.

AcknowledgementsAcknowledgements

Professor Stephen Wicker (Cornell ECE) Professor Bart Selman (Cornell CS) Dr. Ramon Bejar (Cornell CS) Dr. Carla Gomes (Cornell CS) Marc Pearlman (Cornell ECE)


critical density thresholds and complexity in wireless networks bhaskar krishnamachari school of...

Documents