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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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2 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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3 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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4 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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5 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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6 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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7 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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8 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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9 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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10 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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11 Random walk on a torus of M sites T = inter-contact time 9:00 9:30 10:00 10:30 11:0011:3012:0012:30 13:00 13:30 T = 4 h 30 min Mean inter-contact time, E(T) = M

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12 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, n 10 4 No power law ! Example:

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13 Random walk on a torus … (3) For infinitely many sites M, P(T > n) ~ const / n 1/2 P(T > n) Inter-contact time, n M = 500 Power law ! Example:

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14 Random walk on Manhattan street network P(T > n) Inter-contact time, n M = 500

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15 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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16 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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17 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: P 0 (S n > s) = (b/s) a, b > 0, 1 < a < 2 Ex 2: Random waypoint –Take f V 0 (v) = K v 1/2 1(0 v vmax) – E 0 (S n ) <, E 0 (S n 2 ) =

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18 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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