Milan Vojnovic

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On the Origins of Power Laws in Mobility Systems

• Milan Vojnovic

Joint work with Jean-Yves Le Boudec
Workshop on Clean Slate Network Design,
Cambridge, UK, Sept 18, 2006
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Abstract
Recent measurements suggest that inter-contact
times of human-carried devices are well
characterized by a power-low complementary
cumulative distribution function over a large
range of values and this is shown to have
important implications on the design of packet
forwarding algorithms (Chainterau et al, 2006).
It is claimed that the observed power-law is at
odds with currently used mobility models, some of
which feature exponentially bounded inter-contact
time distribution. In contrast, we will argue
that the observed power-laws are rather
commonplace in mobility models and mobility
patterns found in nature.
See also ACM Mobicom 2006 tutorial
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Networks with intermittent connectivity

• Context
• Pocket switched networks (ex Haggle)
• Ad-hoc networks
• Delay-tolerant networks
• Apps
• Asynchronous local messaging
• Ad-hoc search
• Ad-hoc recommendation
• Alert dissemination
• Challenges
• Mobility intermittent connectivity to other
  nodes
• Design of effective packet forwarding algorithms
• Critical node inter-contact time

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Human inter-contact times follow a power law
Chainterau et al, Infocom 06

• Over a large range of values
• Power law exponent is time dependent
• Confirmed by several experiments (iMots/PDA)
• Ex Lindgren et al CHANTS 06

P(T gt n)
Inter-contact time n
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The finding matters !

• The power-law exponent is critical for
  performance of packet forwarding algorithms
• Determines finiteness of packet delay Chainterau
  et al, 06
• Some mobility models do not feature power-law
  inter-contacts
• Ex classical random waypoint

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A brief history of mobility models(partial
sample)

• Manhattan street network (87)
• Random waypoint (96)
• Random direction (05)
• With wrap-around or billiards reflections
• Random trip model (05)
• Encompasses many models in one
• Stability conditions, perfect simulation

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Need new mobility models (?)
Mobility models need to be redesigned !
Exponential decay of inter contact is wrong !
Current mobility models are at odds with the
power-low inter-contacts !

• Do we need new mobility models ?

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Why power law ?

• Conjecture Heavy tail is sum of lots of cyclic
  journeys of
• a small set of frequency and phase difference
• Crowcroft et al 06 (talk slides)

Why power law ?
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This talk two claims

• Power-law inter-contacts are not at odds with
  mobility models
• Already simple models exhibit power-law
  inter-contacts
• Power laws are rather common in the mobility
  patterns observed in nature

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Outline

• Power-law inter-contacts are not at odds with
  mobility models
• Power laws are rather common in the mobility
  patterns observed in nature
• Conclusion

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Random walk on a torus of M sites

• T inter-contact time

930
900
1000
1300
1330
1030
1100
1130
1200
1230
T 4 h 30 min

• Mean inter-contact time, E(T) M

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Random walk on a torus (2)
Example
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Random walk on a torus (3)
Example
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Random walk on Manhattan street network
P(T gt n)
M 500
Inter-contact time, n
P(T gt n)
M 500
Inter-contact time, n
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Outline

• Power-law inter-contacts are not at odds with
  mobility models
• Power laws are rather common in the mobility
  patterns observed in nature
• Conclusion

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Power laws found in nature mobility

• Albatross search
• Spider monkeys
• Jackals
• See Klafter et al, Physics World 05, Atkinson et
  al, Journal of Ecology 02

• Model Levy flights
• random walk with heavy-tailed trip distance
• anomalous diffusion

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Random trip model permits heavy-tailed trip
durations

• But make sure that mean trip duration is finite
• Ex 1 random walk on torus or billiards
• Simple take a heavy-tailed distribution for trip
  duration (with finite mean)
• Ex. Pareto P0(Sn gt s) (b/s)a, b gt 0, 1 lt a lt
  2
• Ex 2 Random waypoint
• Take fV0(v) K ?v1/2 1(0 ? v?? vmax)
• E0(Sn) lt ?, E0(Sn2) ?

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Conclusion

• Power-law inter-contacts are not at odds with
  mobility models
• Already simple models exhibit power-law
  inter-contacts
• Power laws are rather common in the mobility
  patterns observed in nature
• Future work
• Algorithmic implications
• Ex delay-effective packet forwarding (?)
• Ex broadcast (?)
• Ex geo-scoped dissemination (?)
• Realistic, reproducible simulations (?)
• Determined by (a few) main mobility invariants

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?

• milanv_at_microsoft.com