Power law and exponential decay

Milan VojnovićMicrosoft Research Cambridge

Collaborators: T. Karagiannis and J.-Y. Le Boudec

Hynet colloquium series, University of Maryland, Mar 07

of inter contact times between mobile devices

Abstract

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We examine the fundamental properties that determine the basic performance metrics for opportunistic communications. We first consider the distribution of inter-contact times between mobile devices. Using a diverse set of measured mobility traces, we find as an invariant property that there is a characteristic time, order of half a day, beyond which the distribution decays exponentially. Up to this value, the distribution in many cases follows a power law, as shown in recent work. This power law finding was previously used to support the hypothesis that inter-contact time has a power law tail, and that common mobility models are not adequate. However, we observe that the time scale of interest for opportunistic forwarding may be of the same order as the characteristic time, and thus the exponential tail is important. We further show that already simple models such as random walk and random waypoint can exhibit the same dichotomy in the distribution of inter-contact times as in empirical traces. Finally, we perform an extensive analysis of several properties of human mobility patterns across several dimensions, and we present empirical evidence that the return time of a mobile device to its favorite location site may already explain the observed dichotomy. Our findings suggest that existing results on the performance of forwarding schemes based on power-law tails might be overly pessimistic.

Resources

• MSR technical report:

Power law and exponential decay of inter contact times between mobile devices, T. Karagiannis, J.-Y. Le Boudec, M. Vojnović, MSR-TR-2007-24, Mar 07

• Project website:

http://research.microsoft.com/~milanv/albatross.html

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Opportunistic communications

Until 2006

• Various studies of mobile systems under hypothesis: – Distribution of inter-contact time between mobile devices

decays exponentially

• Examples:– Grossglauser and Tse (Infocom 01)– Bansal and Liu (Infocom 03)– El Gamal et al (Infocom 04)– Sharma et al (Infocom 06)

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But in 2006…• Empirical evidence (Chaintreau et al, Infocom 06):

Distribution of inter-contact time between human carried devices exhibits power-law over a range from minute to half a day

• Suggested hypothesis:

Inter-contact time distribution has power-law tail

In sharp contrast to exponential decay

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Why does it matter?• Implications on delay of opportunistic packet forwarding

– For sufficiently heavy tail, the expected packet delay infinite for any packet forwarding scheme

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t

ttF 00 )( 0 ,0 0 tt

1

0)(

t

ttF 00 tt

CCDF = Complementary Cumulative Distribution Function

Chaintreau et al 06 assume a Pareto CCDF of inter-contact time (sampled at contact instant):

If < 1, expected packet forwarding delay infinite for any forwarding scheme

If > 1, CCDF of inter-contact time observed from an arbitrary time instant:

Why does it matter? (cont’d)

• Suggested to revisit current mobility models

– Claim: current mobility models do not feature power-law but exponential tail

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This slide deck• Empirical evidence of dichotomy in distribution of

inter-contact time– Power-law up to a point (order half a day), exponential decay beyond– In sharp contrast to the power-law tail hypothesis

• Dichotomy supported by (simple) mobility models

• Return time and diversity of viewpoints– Empirical evidence that the dichotomy characterizes return time of a

device to a home location– Diversity of viewpoints

(aggregate vs device pair, time average vs time of day)

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Outline

• Power-law exponential dichotomy

• Mobility models support the dichotomy

• Return time and diversity of viewpoints

• Conclusion

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Datasets

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• All but vehicular dataset are public and were used in earlier studies (see references in technical report)

• Vehicular is a private trace (thanks to Eric Hurwitz and John Krumm, Microsoft Research MSMLS project)

Power law

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Power law (cont’d)

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Exponential decay

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Summary

• Empirical evidence suggest dichotomy in distribution of inter-contact time– Power-law up to a point, exponential decay beyond

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Outline

• Power-law exponential dichotomy

• Mobility models support the dichotomy

• Return time and diversity of viewpoints

• Conclusion

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Simple random walk on a circuit0

1m-1

2

01234

Return time to a site0

1m-1

2

012345678

R = 8

Return time to a site of a circuit• Expected return time:

• Power-law for infinite circuit:

• Exponentially decaying tail:

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nn

nR large ,12

~)(P2/1

0, large ,)(~)(P nennR n

mR )(E

K

kkkkk nbnan

1

)]sin()cos([)( Trigonometric polynomialf(n) ~ g(n) means f(n)/g(n)

goes to 1 as n goes to infty

Proof sketch

• Expected return time

where ri = expected return time to site 0 starting from site i.

Standard analysis yields

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11)(E rR

1,,1,0 ),( miimiri

Proof sketch (cont’d)• Z-transform

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)(E)( 0 iXzzf Ri

1)()(0 zfzf m

11 )),()((2

1)( 11 ,m-,izfzfzzf iii

11 ,)11))(,(1()11)(,(

)(22

,m-,iz

zzmazzmazf

i

ii

i

mm

mm

zz

zzzma

)11()11(

)11(),(

22

2

Proof sketch (cont’d)

• For infinite circuit

22

1z0for ,0),( mzma

z

zzf

2

1

11)(

odd

12

1

)1(1)/2(n

1/2)(

n

nzzfn

nn

nR even 2/

2/1)(P

nnπ /

even large 12

~23

(Binomial Theorem)

(Stirling)

Return time for a finite state space Markov chain

• Let Xn be an irreducible Markov chain on a finite state space S.

• Let R be the return time to a strict subset of S.• The stationary distribution of R is such that

where > 0 and (n) is a trigonometric polynomial.

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nennR n large ,)(~)(P

Proof: spectral analysis (see technical report)

Power law for 1-dim random walk

• Power law holds quite generally for 1-dim random walk

• For any irreducible aperiodic random walk in 1-dim with finite variance2

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nn

σnR/

large ,12

~)(P21

(Spitzer, 64)

Inter-contact time0

1m-1

2

012345

T = 5

Inter-contact time on a circuit of 20 sites

• Power-law exponential dichotomy

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Inter-contact time on a circuit of 100 sites

• Power-law exponential dichotomy

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Inter-contact time on a circuit• Expected inter-contact time:

• Power-law for infinite circuit:

• Exponentially decaying tail:

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nn

nT large ,12

~)(P2/1

0, large ,)(~)(P nennT n

1)(E mT

Qualitatively same as return time to a site

Proof sketch

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m0

(-m/2,m/2)

- m

X2 (= location of device 2)

1/4

1/4 1/4

1/4 X1 (= location of device 1)

Hitting set := highlighted sites

Proof sketch (cont’d)• Reduction to simple random walk on a circuit

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m/20

H

nnVT

1

1

Inter-contact timeNumber of verticals transitions between two successive horizontal transitions

Number of horizontal transitions until hitting

)2/1(Geo~ i.i.d., nn VV

))(E(E)(E 1 HVT zzz

)2/()(E 1 zzzV

)(E Hz = z-transform of return time to site 0 from site 1 on a circuit of m/2 sites

1/4

1/41/41/4

Random waypoint on a chain

0 1 m-12

012345

next waypoint

Random waypoint on a chain (cont’d)

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Random waypoint on a chain (cont’d)

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0 200 400 600 800 10000

200

400

600

800

1000

X1

X2

Device 1 location

Dev

ice

2 lo

catio

n

Long inter-contact time

Random waypoint on a chain (cont’d)

• Numerical results suggest distribution of inter-contact time exhibit power-law over a range

• Previous claim on exponential decay limited to special case RWP (Sharma and Mazumdar, 05)– Unit sphere– Fixed trip duration between waypoints

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Manhattan street network• Does power-law characterize CCDF of

inter-contact time for simple random walk in 2-dim ?

– No

– Return time to a site R of an infinite lattice such that

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nn

nR large ,)log(

~)(P

1/41/4

1/4

1/4

(Spitzer, 64)

Summary

• Simple random walk on a circuit– Return time of a device to a site and inter-contact

time between two devices feature the same power-law exponential dichotomy

• Random waypoint on a chain– Numerical results suggest power-law over a range

• Simple models can support power law distribution of inter-contact time over a range

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Outline

• Power-law exponential dichotomy

• Mobility models support the dichotomy

• Return time and diversity of viewpoints

• Conclusion

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Return time

• Power-law exponential dichotomy38

Devices in contact at a few sites

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Aggregate inter-contact times• Inter-contact time

CCDF estimated by taking samples of inter-contact times – over an observation

time interval – over all device pairs

• Used in many studied

• Unbiased estimate if inter contacts for distinct device pairs statistically identical

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Device pair 1 in contact

0

1

Device pair 2 in contact

0

1

Device pair K in contact

0

1

T

Inter-contact time

0

T

T

T

0

…

Aggregate viewpointstationary ergodic case

Contact instance viewpoint:

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Pp

pp tFtF )()( 00

CCDF of inter-contact time“aggregate samples”

CCDF of inter-contact time for device pair p

Pp

p Expected number of contacts per unit time for device pair p

Arbitrary time viewpoint:

Pp

p tFP

tF )(||

1)(

CCDF of inter-contact time for device pair p

• Contact and arbitrary time viewpoints related by residual time formula:

t

dssFPtF )(|)|/()( 0

Aggregate viewpoint (cont’d)stationary ergodic case

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• Aggregate and specific device pair viewpoints, in general, not the same • Same if device inter contacts statistically identical• Contact time viewpoint weighs device pairs

proportional to their rate of contacts• Arbitrary time viewpoint weighs device pairs equally

• What does CCDF of inter-contact times collected over an observation interval and over all device pairs tell me? …

Aggregate viewpoint (cont’d)• Using the CCDF of all pair inter-contact times

sampled at contact instances with residual time formula interpreted as:

– Pick a time t uniformly at random over the observation interval

– Pick a device pair p uniformly at random

– Observe the inter-contact time for pair p from time t

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Averaging over time and over device pairs

Averaging over time and over device pairs

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Pp

p tssTP

stF ))((1||

1),(ˆAveraging over device pairs

Averaging over time dsstFT

TtFT

),(ˆ1),(ˆ

0

Relation to aggregate CCDF )(),(ˆ||

)(),(ˆ 0 TedsTsF

TP

TNTtF

t

Time until next inter-contact for device pair p observed at time s

“Error term” due to boundaries of observation interval

Number of contacts over the observation interval over all device pairs

Empirical analogue of residual time formula

Fraction of device pair with residual inter-contact time > t at time s

Averaging over time and over device pairs (cont’d)

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Relation of aggregate and device-pair CCDF ),(ˆ

)(

1)(),(ˆ 00 TtF

TN

TNTtF p

Pp

p

Pp

p TNTN )()(

1)(

1

0 )(11)(

1),(ˆ

TN

n

pn

pp

p

tTTN

TtF

Number of contacts of device pair p in [0,T]

nth inter-contact time of device pair p

Inter-contact time CCDF (sampled per contact)Aggregate vs per device pair

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Time of day viewpoints

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• Strong time-of-day dependence• Time-average viewpoint may deviate significantly from specific time-of-day viewpoint

Time of day viewpoints (cont’d)

• Dichotomy of contact durations (pass-by vs park-by) • Strong time-of-day dependence• Time-average viewpoint may deviate significantly from specific time-of-day viewpoint

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Summary

• Empirical evidence suggest dichotomy in distribution of return time of a device to its favourite site

• Diversity of viewpoints– Aggregate vs specific device pair– Time average vs specific time of day– Relevant for packet forwarding delay

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Outline

• Power-law exponential dichotomy

• Mobility models support the dichotomy

• Return time and diversity of viewpoints

• Conclusion

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Conclusion• The dichotomy hypothesis for distribution of inter-contact time:

power law up to a point, exponential decay beyond– In sharp contrast to proposed power-law tail hypothesis– More optimistic view on delay of packet forwarding schemes

• Simple mobility models exhibit the same dichotomy– In sharp contrast to the claim that current mobility models are

inadequate

• Empirical evidence that return time of a device to its frequently visited site feature the same dichotomy– More elementary metric– Suggests explanation of power-law inter-contact time

• Diversity of viewpoints– Aggregate vs specific device pair– Time-average vs specific time

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References• F. Spitzer, Principles of Random Walk, Springer, 2nd edt, 1964• M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad-

hoc Wireless Networks, IEEE Infocom 2001• N. Bansal and Z. Liu, Capacity, Delay and Mobility in Wireless Ad-

hoc Networks, IEEE Infocom 2003• A. El Gamal, J. Mammen, B. Prabhakar, D. Shah, Throughput-delay

Trade-off Wireless Networks, IEEE Infocom 2004• G. Sharma and R. Mazumdar, Delay and Capacity Trade-off in

Wireless Ad Hoc Networks with Random Waypoint Mobility, preprint, https://engineering.purdue.edu/people/gaurav.sharma.3, 2005

• A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, and J. Scott, Impact of Human Mobility on the Design of Opportunistic Forwarding Algorithms, IEEE Infocom 2006

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