milan vojnović joint work with: jean-yves le boudec workshop on clean slate network design,...
TRANSCRIPT
Milan Vojnović
Joint work with: Jean-Yves Le Boudec
Workshop on Clean Slate Network Design, Cambridge, UK, Sept 18, 2006
On the Origins of Power Laws in Mobility Systems
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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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• Over a large range of values
• Power law exponent is time dependent
• Confirmed by several experiments (iMots/PDA)
– Ex Lindgren et al CHANTS ’06
Human inter-contact times follow a power law [Chainterau et al, Infocom ’06]
P(T
> 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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Mobility models need to be redesigned ! Exponential
decay of inter contact is wrong !
Need new mobility models (?)
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
9:009:3010:0010:3011:0011:3012:0012:3013:0013:30
T = 4 h 30 min
• Mean inter-contact time, E(T) = M
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Random walk on a torus … (2)
• For fixed number of sites M, P(T > n) decays exponentially with n, for large n
M = 500
P(T
> n
)
Inter-contact time, n104
• No power law !
Example:
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Random walk on a torus … (3)
• For infinitely many sites M, P(T > n) ~ const / n1/2
P(T
> n
)
Inter-contact time, n
M = 500
• Power law !
Example:
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Random walk on Manhattan street network
0 1 2 3 4 5 6 7
x 105
10-3
10-2
10-1
100
Inter-contact time
ECDF
100
101
102
103
104
105
106
10-3
10-2
10-1
100
Inter-contact time
ECDF
P(T
> n
)P
(T >
n)
Inter-contact time, n
Inter-contact time, n
M = 500
M = 500
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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 > s) = (b/s)a, b > 0, 1 < a < 2
• Ex 2: Random waypoint
– Take fV0(v) = K v1/2 1(0 v vmax)
– E0(Sn) < , 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
