Milan Vojnovi ć

1. On the Origins of Power Laws in Mobility Systems Milan Vojnović Joint work with: Jean-Yves Le Boudec Workshop on Clean Slate Network Design, Cambridge, UK, Sept 18, 2006

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. 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 > 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. 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 ?

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 9:30 13:30 10:00 9:00 13:00 10:30 12:00 12:30 11:30 11:00 T = 4 h 30 min • Mean inter-contact time, E(T) = M

12. M = 500 • For fixed number of sites M, P(T > n) decays exponentially with n, for large n P(T > n) • No power law ! 104 Inter-contact time, n Random walk on a torus … (2) Example:

13. M = 500 P(T > n) • For infinitely many sites M, P(T > n) ~ const / n1/2 Inter-contact time, n • Power law ! Random walk on a torus … (3) Example:

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

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: 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) = 

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

19. ? milanv@microsoft.com