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

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Abstract 2 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.

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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 , Mar 07 Project website: 3

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4 Opportunistic communications

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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) 5

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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 6

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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 7 CCDF = Complementary Cumulative Distribution Function Chaintreau et al 06 assume a Pareto CCDF of inter-contact time (sampled at contact instant): If < 1, expected packet forwarding delay infinite for any forwarding scheme If > 1, CCDF of inter-contact time observed from an arbitrary time instant:

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Why does it matter? (contd) Suggested to revisit current mobility models – Claim: current mobility models do not feature power-law but exponential tail 8

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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) 9

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

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Datasets 11 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)

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Power law 12

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Power law (contd) 13

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Exponential decay 14

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

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

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Simple random walk on a circuit 0 1 m

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Return time to a site 0 1 m R = 8

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

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

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Proof sketch (contd) Z-transform 21

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Proof sketch (contd) For infinite circuit 22 (Binomial Theorem) (Stirling)

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Return time for a finite state space Markov chain Let X n 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 that where > 0 and (n) is a trigonometric polynomial. 23 Proof: spectral analysis (see technical report)

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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 24 (Spitzer, 64)

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Inter-contact time 0 1 m T = 5

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

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

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

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

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Proof sketch (contd) Reduction to simple random walk on a circuit 30 m/20 Inter-contact time Number of verticals transitions between two successive horizontal transitions Number of horizontal transitions until hitting = z-transform of return time to site 0 from site 1 on a circuit of m/2 sites 1/4

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Random waypoint on a chain 01m next waypoint

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Random waypoint on a chain (contd) 32

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Random waypoint on a chain (contd) 33 Device 1 location Device 2 location Long inter-contact time

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Random waypoint on a chain (contd) 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 34

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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 35 1/4 (Spitzer, 64)

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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 36

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

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Return time Power-law exponential dichotomy 38

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Devices in contact at a few sites 39

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Aggregate inter-contact times 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 40 Device pair 1 in contact 0 1 Device pair 2 in contact 0 1 Device pair K in contact 0 1 T Inter-contact time 0 T T T 0 …

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Aggregate viewpoint stationary ergodic case Contact instance viewpoint: 41 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 Arbitrary time viewpoint: CCDF of inter-contact time for device pair p Contact and arbitrary time viewpoints related by residual time formula:

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Aggregate viewpoint (contd) stationary ergodic case 42 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? …

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Aggregate viewpoint (contd) 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 43 Averaging over time and over device pairs

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

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Averaging over time and over device pairs (contd) 45 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

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

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

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Time of day viewpoints (contd) 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 48

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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 49

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

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Conclusion The dichotomy hypothesis for distribution of inter-contact time: power law up to a point, exponential decay beyond – In sharp contrast to proposed power-law tail hypothesis – More optimistic view on delay of packet forwarding schemes Simple mobility models exhibit the same dichotomy – In sharp contrast to the claim that current mobility models are inadequate Empirical evidence that return time of a device to its frequently visited site feature the same dichotomy – More elementary metric – Suggests explanation of power-law inter-contact time Diversity of viewpoints – Aggregate vs specific device pair – Time-average vs specific time 51

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References F. Spitzer, Principles of Random Walk, Springer, 2 nd edt, 1964 M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad- hoc Wireless Networks, IEEE Infocom 2001 N. Bansal and Z. Liu, Capacity, Delay and Mobility in Wireless Ad- hoc Networks, IEEE Infocom 2003 A. El Gamal, J. Mammen, B. Prabhakar, D. Shah, Throughput-delay Trade-off Wireless Networks, IEEE Infocom 2004 G. Sharma and R. Mazumdar, Delay and Capacity Trade-off in Wireless Ad Hoc Networks with Random Waypoint Mobility, preprint, https://engineering.purdue.edu/people/gaurav.sharma.3, 2005 A. Chaintreau, P. Hui, J. Crowcroft, C. Diot, R. Gass, and J. Scott, Impact of Human Mobility on the Design of Opportunistic Forwarding Algorithms, IEEE Infocom

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