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Power law and exponential decay

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Power law and exponential decay

of inter contact times between mobile devices

Milan Vojnović

Microsoft Research Cambridge

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

Hynet colloquium series, University of Maryland, Mar 07

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.

- 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

- 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)

- 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

- Implications on delay of opportunistic packet forwarding
- For sufficiently heavy tail, the expected packet delay infinite for any packet forwarding scheme

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

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

CCDF = Complementary

Cumulative Distribution Function

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

- Suggested to revisit current mobility models
- Claim: current mobility models do not feature power-law but exponential tail

- 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)

- Power-law exponential dichotomy
- Mobility models support the dichotomy
- Return time and diversity of viewpoints
- Conclusion

- 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)

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

- Power-law exponential dichotomy
- Mobility models support the dichotomy
- Return time and diversity of viewpoints
- Conclusion

0

1

2

4

3

0

1

m-1

2

0

R = 8

1

8

2

4

3

0

5

6

7

1

m-1

2

- Expected return time:
- Power-law for infinite circuit:
- Exponentially decaying tail:

Trigonometric polynomial

f(n) ~ g(n) means f(n)/g(n) goes to 1 as n goes to infty

- Expected return time where ri = expected return time to site 0 starting from site i. Standard analysis yields

- Z-transform

- For infinite circuit

(Binomial Theorem)

(Stirling)

- 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 thatwhere b > 0 and f(n) is a trigonometric polynomial.

Proof: spectral analysis (see technical report)

- Power law holds quite generally for 1-dim random walk
- For any irreducible aperiodic random walk in 1-dim with finite variance2

(Spitzer, 64)

0

T = 5

1

2

4

3

0

5

1

m-1

2

- Power-law exponential dichotomy

- Power-law exponential dichotomy

- Expected inter-contact time:
- Power-law for infinite circuit:
- Exponentially decaying tail:

Qualitatively same as return time to a site

X2 (= location of device 2)

m

0

X1 (= location of device 1)

1/4

1/4

1/4

1/4

(-m/2,m/2)

- m

Hitting set := highlighted sites

- Reduction to simple random walk on a circuit

Number of horizontal transitions until hitting

Number of verticals transitions

between two successive horizontal

transitions

Inter-contact time

1/4

1/4

1/4

0

m/2

1/4

= z-transform of return time to site 0 from

site 1 on a circuit of m/2 sites

1

2

3

4

0

5

0

1

2

m-1

next waypoint

Device 2 location

Long inter-contact time

Device 1 location

- 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

- 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

1/4

1/4

1/4

1/4

(Spitzer, 64)

- 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

- Power-law exponential dichotomy
- Mobility models support the dichotomy
- Return time and diversity of viewpoints
- Conclusion

- Power-law exponential dichotomy

Device pair 1 in contact

1

- 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

0

T

Device pair 2 in contact

1

0

T

…

Device pair K in contact

1

0

T

Inter-contact time

0

0

T

Arbitrary time viewpoint:

Contact instance viewpoint:

CCDF of inter-contact time for device pair p

CCDF of inter-contact time“aggregate samples”

CCDF of inter-contact time for device pair p

Expected number of contacts per unit time for device pair p

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

- 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? …

- 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

Averaging over time and over device pairs

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

Averaging over device pairs

Time until next inter-contact for

device pair p observed at time s

Averaging over time

Empirical analogue of

residual time formula

Relation to aggregate CCDF

“Error term” due to

boundaries of observation

interval

Number of contacts

over the observation

interval over all device pairs

Relation of aggregate and device-pair CCDF

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

nth inter-contact time of device pair p

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

- 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

- 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

- Power-law exponential dichotomy
- Mobility models support the dichotomy
- Return time and diversity of viewpoints
- 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

- F. Spitzer, Principles of Random Walk, Springer, 2ndedt, 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