Power law and exponential decay

1. Power law and exponential decay of inter contact times between mobile devices Milan Vojnović Microsoft Research Cambridge Collaborators: T. Karagiannis and J.-Y. Le Boudec Hynet colloquium series, University of Maryland, Mar 07

2. Abstract We examine the fundamental properties that determine the basic performance metrics for opportunistic communications. We first consider the distribution of inter-contact times between mobile devices. Using a diverse set of measured mobility traces, we find as an invariant property that there is a characteristic time, order of half a day, beyond which the distribution decays exponentially. Up to this value, the distribution in many cases follows a power law, as shown in recent work. This power law finding was previously used to support the hypothesis that inter-contact time has a power law tail, and that common mobility models are not adequate. However, we observe that the time scale of interest for opportunistic forwarding may be of the same order as the characteristic time, and thus the exponential tail is important. We further show that already simple models such as random walk and random waypoint can exhibit the same dichotomy in the distribution of inter-contact times as in empirical traces. Finally, we perform an extensive analysis of several properties of human mobility patterns across several dimensions, and we present empirical evidence that the return time of a mobile device to its favorite location site may already explain the observed dichotomy. Our findings suggest that existing results on the performance of forwarding schemes based on power-law tails might be overly pessimistic.

3. Resources • MSR technical report:Power law and exponential decay of inter contact times between mobile devices, T. Karagiannis, J.-Y. Le Boudec, M. Vojnović, MSR-TR-2007-24, Mar 07 • Project website:http://research.microsoft.com/~milanv/albatross.html

4. Opportunistic communications

5. Until 2006 • Various studies of mobile systems under hypothesis: • Distribution of inter-contact time between mobile devices decays exponentially • Examples: • Grossglauser and Tse (Infocom 01) • Bansal and Liu (Infocom 03) • El Gamal et al (Infocom 04) • Sharma et al (Infocom 06)

6. But in 2006… • Empirical evidence (Chaintreau et al, Infocom 06): Distribution of inter-contact time between human carried devices exhibits power-law over a range from minute to half a day • Suggested hypothesis: Inter-contact time distribution has power-law tail In sharp contrast to exponential decay

7. Why does it matter? • Implications on delay of opportunistic packet forwarding • For sufficiently heavy tail, the expected packet delay infinite for any packet forwarding scheme If a < 1, expected packet forwarding delay infinite for any forwarding scheme If a > 1, CCDF of inter-contact time observed from an arbitrary time instant: CCDF = Complementary Cumulative Distribution Function Chaintreau et al 06 assume a Pareto CCDF of inter-contact time (sampled at contact instant):

8. Why does it matter? (cont’d) • Suggested to revisit current mobility models • Claim: current mobility models do not feature power-law but exponential tail

9. This slide deck • Empirical evidence of dichotomy in distribution of inter-contact time • Power-law up to a point (order half a day), exponential decay beyond • In sharp contrast to the power-law tail hypothesis • Dichotomy supported by (simple) mobility models • Return time and diversity of viewpoints • Empirical evidence that the dichotomy characterizes return time of a device to a home location • Diversity of viewpoints (aggregate vs device pair, time average vs time of day)

10. Outline • Power-law exponential dichotomy • Mobility models support the dichotomy • Return time and diversity of viewpoints • Conclusion

11. Datasets • All but vehicular dataset are public and were used in earlier studies (see references in technical report) • Vehicular is a private trace (thanks to Eric Hurwitz and John Krumm, Microsoft Research MSMLS project)

12. Power law

13. Power law (cont’d)

14. Exponential decay

15. Summary • Empirical evidence suggest dichotomy in distribution of inter-contact time • Power-law up to a point, exponential decay beyond

16. Outline • Power-law exponential dichotomy • Mobility models support the dichotomy • Return time and diversity of viewpoints • Conclusion

17. Simple random walk on a circuit 0 1 2 4 3 0 1 m-1 2

18. Return time to a site 0 R = 8 1 8 2 4 3 0 5 6 7 1 m-1 2

19. Return time to a site of a circuit • Expected return time: • Power-law for infinite circuit: • Exponentially decaying tail: Trigonometric polynomial f(n) ~ g(n) means f(n)/g(n) goes to 1 as n goes to infty

20. Proof sketch • Expected return time where ri = expected return time to site 0 starting from site i. Standard analysis yields

21. Proof sketch (cont’d) • Z-transform

22. Proof sketch (cont’d) • For infinite circuit (Binomial Theorem) (Stirling)

23. Return time for a finite state space Markov chain • Let Xn be an irreducible Markov chain on a finite state space S. • Let R be the return time to a strict subset of S. • The stationary distribution of R is such thatwhere b > 0 and f(n) is a trigonometric polynomial. Proof: spectral analysis (see technical report)

24. Power law for 1-dim random walk • Power law holds quite generally for 1-dim random walk • For any irreducible aperiodic random walk in 1-dim with finite variance2 (Spitzer, 64)

25. Inter-contact time 0 T = 5 1 2 4 3 0 5 1 m-1 2

26. Inter-contact time on a circuit of 20 sites • Power-law exponential dichotomy

27. Inter-contact time on a circuit of 100 sites • Power-law exponential dichotomy

28. Inter-contact time on a circuit • Expected inter-contact time: • Power-law for infinite circuit: • Exponentially decaying tail: Qualitatively same as return time to a site

29. Proof sketch X2 (= location of device 2) m 0 X1 (= location of device 1) 1/4 1/4 1/4 1/4 (-m/2,m/2) - m Hitting set := highlighted sites

30. Proof sketch (cont’d) • Reduction to simple random walk on a circuit Number of horizontal transitions until hitting Number of verticals transitions between two successive horizontal transitions Inter-contact time 1/4 1/4 1/4 0 m/2 1/4 = z-transform of return time to site 0 from site 1 on a circuit of m/2 sites

31. Random waypoint on a chain 1 2 3 4 0 5 0 1 2 m-1 next waypoint

32. Random waypoint on a chain (cont’d)

33. Random waypoint on a chain (cont’d) Device 2 location Long inter-contact time Device 1 location

34. Random waypoint on a chain (cont’d) • Numerical results suggest distribution of inter-contact time exhibit power-law over a range • Previous claim on exponential decay limited to special case RWP (Sharma and Mazumdar, 05) • Unit sphere • Fixed trip duration between waypoints

35. Manhattan street network • Does power-law characterize CCDF of inter-contact time for simple random walk in 2-dim ? • No • Return time to a site R of an infinite lattice such that 1/4 1/4 1/4 1/4 (Spitzer, 64)

36. Summary • Simple random walk on a circuit • Return time of a device to a site and inter-contact time between two devices feature the same power-law exponential dichotomy • Random waypoint on a chain • Numerical results suggest power-law over a range • Simple models can support power law distribution of inter-contact time over a range

37. Outline • Power-law exponential dichotomy • Mobility models support the dichotomy • Return time and diversity of viewpoints • Conclusion

38. Return time • Power-law exponential dichotomy

39. Devices in contact at a few sites

40. Aggregate inter-contact times Device pair 1 in contact 1 • Inter-contact time CCDF estimated by taking samples of inter-contact times • over an observation time interval • over all device pairs • Used in many studied • Unbiased estimate if inter contacts for distinct device pairs statistically identical 0 T Device pair 2 in contact 1 0 T … Device pair K in contact 1 0 T Inter-contact time 0 0 T

41. Aggregate viewpointstationary ergodic case Arbitrary time viewpoint: Contact instance viewpoint: CCDF of inter-contact time for device pair p CCDF of inter-contact time“aggregate samples” CCDF of inter-contact time for device pair p Expected number of contacts per unit time for device pair p • Contact and arbitrary time viewpoints related by residual time formula:

42. Aggregate viewpoint (cont’d)stationary ergodic case • Aggregate and specific device pair viewpoints, in general, not the same • Same if device inter contacts statistically identical • Contact time viewpoint weighs device pairs proportional to their rate of contacts • Arbitrary time viewpoint weighs device pairs equally • What does CCDF of inter-contact times collected over an observation interval and over all device pairs tell me? …

43. Aggregate viewpoint (cont’d) • Using the CCDF of all pair inter-contact times sampled at contact instances with residual time formula interpreted as: • Pick a time t uniformly at random over the observation interval • Pick a device pair p uniformly at random • Observe the inter-contact time for pair p from time t Averaging over time and over device pairs

44. Averaging over time and over device pairs Fraction of device pair with residual inter-contact time > t at time s Averaging over device pairs Time until next inter-contact for device pair p observed at time s Averaging over time Empirical analogue of residual time formula Relation to aggregate CCDF “Error term” due to boundaries of observation interval Number of contacts over the observation interval over all device pairs

45. Averaging over time and over device pairs (cont’d) Relation of aggregate and device-pair CCDF Number of contacts of device pair p in [0,T] nth inter-contact time of device pair p

46. Inter-contact time CCDF (sampled per contact)Aggregate vs per device pair

47. Time of day viewpoints • Strong time-of-day dependence • Time-average viewpoint may deviate significantly from specific time-of-day viewpoint

48. Time of day viewpoints (cont’d) • Dichotomy of contact durations (pass-by vs park-by) • Strong time-of-day dependence • Time-average viewpoint may deviate significantly from specific time-of-day viewpoint

49. Summary • Empirical evidence suggest dichotomy in distribution of return time of a device to its favourite site • Diversity of viewpoints • Aggregate vs specific device pair • Time average vs specific time of day • Relevant for packet forwarding delay

50. Outline • Power-law exponential dichotomy • Mobility models support the dichotomy • Return time and diversity of viewpoints • Conclusion