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

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

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

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

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

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

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 ?

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 ?

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

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

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

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:

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:

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

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

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

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 ) =

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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