Demystifying Levy Walk Patterns

Demystifying Levy Walk Patterns in Human Walks

Kyunghan Lee†, Seongik Hong††, Seong Joon Kim††,Injong Rhee†† and Song Chong†

† School of EECS, KAIST, Daejeon, Korea, [email protected], [email protected]†† Dept. of Computer Science, NC State University, Raleigh, NC. USA, {shong, sjkim2, rhee}@ncsu.edu

ABSTRACTThis paper reports that bursty hot spot sizes are a key fac-tor in causing the heavy-tail distribution of flights in hu-man walks. A heavy-tailed distribution of flights is a sig-nature feature of Levy walks. The data analysis based onGPS traces of human walks reveals that the sizes of a fewextremely large hot spots are dominating the mean size ofhot spots and they cause the bursty (i.e., long-range depen-dent) dispersion of visit points where people make a stop.Bursty visit points cause the characteristic distance amongvisit points to have a heavy-tail distribution. On top of burstyvisit points, humans perform a distance-optimizing algorithmto plan their trips much like a heuristic to the traveling sales-man problem. These factors in combination make humanwalks to have a heavy-tailed flight distribution. The abovefindings enable the construction of a simple human mobilitymodel that taking as input the degree of burstiness in visitpoint dispersion, can naturally emulate hot spots as well as aheavy-tail flight distribution, both known to be important inmeasuring the realistic performance of mobile networks.

1. INTRODUCTIONRhee et al. [25][24] show that human walk patterns

contain similar statistical features found in Levy walkswhich biologists have observed from the mobility pat-terns of animals like spider monkeys [23] and seabirds[28]. Levy walks (LW) are characterized by a power dis-tribution of flights where a flight is the distance that awalker travels without making a pause or a directionalchange. [25] shows that human walks are statisticallyand fundamentally different from walks generated fromcommonly used mobility models such as random waypoint (RWP), random direction and Brownian motion(BM) whose flight distributions have a short tail. Itshows that the flight and pause time distributions ofhuman walks have a strong heavy tail tendency, andfurther, while the mobility of humans is super-diffusive,its diffusivity falls somewhere between the diffusivity ofRWP and that of BM. These features of human walkslead to performance characteristics of network protocolsthat form a middle ground between those seen from BMand RWP. For instance, the heavy-tail flight distribu-

tion of LW induces on average much longer inter-contacttimes (ICTs) than RWP, but shorter than BM. Thatis, Levy walkers meet much more often than Brownianwalkers, but much less often than RWP walkers. Thisresults in much longer DTN (delay-tolerant network)routing performance than RWP, but much shorter DTNrouting performance than BW. Thus, many existingDTN routing performance studies using RWP are signif-icantly over-estimating the DTN routing performance.

Unfortunately, human walks are not LW, though theyhave some similar statistical features, since unlike LWpeople hardly move randomly. They move with andwithin numerous contexts such as buildings, classrooms,market places, shopping malls, restaurants, tourist at-tractions, schedules and appointments. Random walkmodels are too simplistic to represent these man-madecontexts. Obviously not all contexts are relevant formobility modeling. Then, what contexts make humanwalks have such a heavy tail tendency? Finding fun-damental causes of these LW features in human walkshelps identify the contexts inducing such features. Fromthese “seed” contexts, human walks can be naturallygenerated without altering its distinctive statistical fea-tures. This way of generating mobility traces has thefollowing unique advantages over existing mobility mod-els.

1. Generating natural hot spots. Since it rep-resents important contexts where people walk inand out and also gather around, it naturally formsswarms or hot spots where many people “meet”.Representing hot spots naturally captures the re-alistic inter-contact properties of humans. In con-trast, random walks cannot represent hot spotsand thus are limited in capturing such properties.

2. Generating natural Heavy-tail flights. It nat-urally represents the similar statistical patterns asLW, but without forcing walkers to “jump” to ran-dom locations in order to make heavy-tail flights.As discussed above, heavy-tail flights induce a power-law ICT distribution, an important statistical char-acteristic for DTN routing. Artificially placing hot

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spots as done in many other mobility models [5][17] incorporating hot spots does not necessarilycreate such features.

3. Simplifying modeling. One way to faithfullyrepresent man-made contexts is to model detailedinformation about many man-made contexts suchas schedules, appointments, hot spot locations andsizes, human activities around hot spots, as donein detailed simulation (e.g., [16]). But such model-ing is too costly in terms of time and resource. Incontrast, modeling seed contexts that cause heavy-tail flights limits the amount of efforts required forconstructing such a mobility model without sacri-ficing important features of human mobility.

This paper provides reasonable answers to the causesof heavy-tail flights in human walk patterns. Using theGPS traces used in [25] which are taken from about 100participants in two university campuses, Disney World,New York City and a state fair involving about 200 dailytraces, we make the following observations.

First, daily walk traces contain highly bursty visitpoints which are defined as the places where the par-ticipant makes a stop. They form swarms of highlyvarying sizes and their swarm sizes show that their nor-malized variance over increasing ranges of aggregationdecays very slowly, with Hurst values ranging from 0.8to 0.6, and their distribution shows a heavy-tail ten-dency. A surprising finding is that the aggregation ofvisit points of all the participants within the same site isalso very bursty and long-range dependent. This is sur-prising because participants are chosen arbitrarily andhave very little in common except that their primarywalk-about areas are confined to the same area. Thisimplies that people tend to visit locations that otherpeople also visit, and popular places tend to be ex-tremely popular whose popularity dominates the mean.This finding is important because it reveals the statisti-cal features of hot-spots where people swarm, and it isknown that hot-spots strongly influence wireless routingperformance [2].

Second, from the daily traces, we find that partici-pants plan their trip to their visit points to optimizethe total sum of flights using a higher order heuristicfunction of distance. This way of planning is simi-lar to heuristics to a combinatorial NP-hard problemcalled traveling salesman (TS) [4] which minimizes thetotal travel distance while visiting all the input loca-tions from a starting location, and finally coming backto the original location. This observation coincides withthe least-action theory of Maupertuis [6]. We find thatthe heuristics using a function 1/da where d is the dis-tance from a current point to its next point and a isa positive constant, gives extremely good matching tothe original GPS traces with less than 1 to 11% error

margin. If a is infinite, then it uses the heuristic al-gorithm called nearest neighbor first (NNF)[26] [22] forTS. In most cases, we find a between 1 and 3 providesvery good matching.

From these observations, we identify the seed con-texts for mobility modeling to be bursty visit points.This leads to a construction of simple contexts fromwhich both hot-spots and heavy-tail flight distributionsof human walks can be naturally produced from the seedcontext. We generate bursty hot spots by emulatingthe burstiness of visit points measured from real traces.Then for a daily trip, each mobile node chooses its visitpoints from these points also in a bursty manner. Syn-thetic walk traces are generated by visiting these pointsusing a function of 1/da where we vary a from 1 to 3.We call our model bursty spot model (BSM).

We apply BSM to the evaluation of several DTNrouting protocols and verify that our model generatesheavy-tail flight. The emulation of hot spots reducesDTN routing performance substantially from popularrandom walk models– an order-of-magnitude less thanLW and even 2 to 3 times less than RWP. This resultindicates that the statistical features of hot spots andflights make significant impacts on the performance ofrouting in mobile networks and realistic representationsof them are important.

The remainder of this paper is organized as follows.Section 2 discusses related work, Section 3 describes thehuman walk traces from [25] used for our data analy-sis, Section 4 presents the results of our data analysis,Section 5 describes BSM, Section 6 presents the DTNrouting performance results using BSM and Section 7concludes this paper.

2. RELATED WORKA mobility model is placed somewhere between a pure

random model and a target real environment. Some areclose to a pure random model like as BM, RWP, randomdirection mobility model, and so on, others try to berealistic. Human mobility is affected by geographicalconstraints and his intention.

Jardosh et al. [13] incorporate obstacles in their mo-bility model called obstacle mobility (OM) to emulatemore realistic navigations of humans around obstacles.They approximate obstacles as polygons, and place themin a simulation terrain. Pathways are created usingVoronoi diagrams. The routing performance using OMis significantly different from that using RWP. The location-based preferences and hot spots have also been modeledusing a weighted way point model [14] and a Markovianwaypoint mobility model [12]. These models do not con-sider realistic statistical features of human mobility.

Hong et al. [11] model groups of users that move to-gether, but the model does not consider realistic repre-sentations of groups. Based on Albert and Barabassi’s

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preferential attachment theory, Herrmann [10] first in-corporate a power-law social inter-action model into hu-man mobility. Musolesi et al. [21] extend the work ofHerrmann by incorporating geographical movements ofgroups. But these models do not consider the geograph-ical positions of meetings and hot spots.

Borrel et al. [5] present a mobility model using pref-erential attachment. In the model, attractors and peo-ple dynamically arrive to the simulation terrain accord-ing to a Poisson process and stay for a random dura-tion. An attractor can be considered as a landmark or ahot spot. Each individual chooses a destination attrac-tor with a probability proportional to its attractivenesswhich is proportional to the number of attracted indi-viduals and inversely proportional to distance betweenthe individual and the attractor. Since the model doesnot consider bursty placements of hot spots, the gener-ated flights do not follow a heavy-tail distribution. Limet al. [17] also use preferential attachment. It first di-vides the simulation area into several subareas and usethem as hot spots. Initially, users are assigned to a sub-area using preferential attachment, then they choose anext subarea according to attractiveness proportionalto (k + 1)α, where k and α are the number of users inthe subarea and the clustering exponent, respectively.The strength of scale-free phenomena can be adjustedaccording to α. This model also has the same prob-lem as Borrel’s in that it does not create a heavy-taildistribution of flights because attractors are distributeduniformly and no distance consideration is given for tripplanning.

There have been a few studies to analyze mobilitypatterns of users using wireless devices [9] [20]. Thesestudies use periodic log data or event log data associ-ated with mobile devices at IEEE 802.11 access points(APs). Kim et al. [16] estimate the locations and move-ment paths of users from the sets of AP log data. Basedon the estimated information on locations, pause time,and velocity, hot spot regions and the transition prob-ability for moving from one hot spot to another areextracted. This information is used to construct a mo-bility model. This model can generate a fairly realisticand detailed representation of human mobility. But itrequires a considerable amount of effort to generate themobility model because hot spot locations and tran-sition probability between hot spots must be given asinput (instead of being generated).

3. HUMAN WALK TRACESFive sites are chosen for collecting human mobility

traces. These are two university campuses (NCSU andKAIST), New York City, Disney World, and North Car-olina State Fair. The total number of traces from thesesites is over 150 daily traces. Garmin GPS 60CSx hand-held receivers are used for data collection which are

Site (# of # of Duration # of X length Y lengthpartici- traces (avg) visit pts of site of sitepants) [hour] (avg) [meter] [meter]KAIST

761445.15 10598 10256.28 18650.72

(34) (19.02) (139.45)NCSU

31310.62 3316 2586.85 2347.01

(20) (10.02) (106.97)State fair

1847.01 691 1141.02 995.75

(18) (2.61) (38.39)Disney World

38378.08 4085 8214.56 9446.70

(18) (9.95) (107.50)New York

32293.21 1105 31432.72 18900.42

(10) (9.16) (34.53)

Table 1: Statistics of collected mobility tracesfrom five sites.

(a) NCSU (b) KAIST

(c) New York City (d) Disney World

(e) State fair

Figure 1: Visit points registered in our traces.Visit points are marked by ‘+’.

WAAS (Wide Area Augmentation System) capable witha position accuracy of better than three meters 95 per-cent of the time, in North America [1]. Occasionally,track information has discontinuity mainly when bear-

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Figure 2: Measuring aggregated variance of visitpoints aggregated from all walk traces. We di-vide the area by non-overlapping d by d squares,and count the number of visit points registeredin each square and then normalize the sampledcount by the size of the unit square. We com-pute the normalized variance as we increase d.

ers move indoor where GPS signals cannot be received.The GPS receivers take a reading of their current posi-tions at every 10 seconds and record them into a dailytrack log. The summary of daily traces is shown in Ta-ble 1. The radius of each trace is a half of the maximumdistance that a participant travels during a day.

20 participants in NCSU were randomly selected fromthe students who took a course in the computer sciencedepartment. Every week, 2 or 3 randomly chosen stu-dents carried the GPS receivers. The KAIST traces aretaken by 34 students who live in a campus dormitory.Since the participants in NCSU and KAIST occasion-ally moved outside their campuses, we use only thoselogs recorded within a radius of 10 km from the cen-ter of each campus. The New York City traces wereobtained from 10 volunteers living in Manhattan or itsvicinity. Most of the participants have offices in Man-hattan. Their track logs contain relatively long distancetravels because of their long commuting paths. Theirmeans of travel include subway trains, buses and mostlywalking. The State fair track logs were collected from18 volunteers who visited a local state fair that includesmany street arcades, small street food stands and show-cases. The site is completely outdoor and is smallestamong all the sites. Each participant in the State fairscenario spent less than three hours in the site. TheDisney World traces were obtained from 18 volunteerswho spent their Thanksgiving or Christmas holidays inDisney World, Florida, USA. For our study, we use onlythe track logs from the inside of the theme parks. Theparticipants mainly walked in the parks and occasion-ally rode trolleys.

4. MEASUREMENT STUDY

In this section, we examine the causes of heavy-taildistributions of flights in human walks. We conjecturethat they are highly related to the locations of destina-tions where people choose to walk to. To confirm this,we define a visit point to be the GPS location where aparticipant stays more than 30 seconds within a circleof 5 meter radius of that location. For each site, weplot the visit points registered by every walk trace ofthat site. We call these points aggregated visit points.Figure 1 shows the aggregated visit point of each site.

To measure the burstiness in the dispersion of visitpoints, we divide the site map into a grid of unit squares(initially of 5 by 5 meters). We count all the visit pointswithin each square and then normalize the count by thearea of the square. We measure the variance in thesenormalized count samples and call it aggregated vari-ance. Figure 2 illustrates the method. If there exists along-range dependency in the samples, the aggregatedvariance should not decay faster than -1 in a log-logscale as we increase the size of the square. To see this,we plot aggregated variance in a log-log scale as we in-crease the square size and measure its absolute slope β.The Hurst parameter of the samples is 1 − β/2. Thesample data are said to be bursty or long-range depen-dent (and therefore, self-similar) if the Hurst parameteris in between 0.5 and 1. Aggregated variance can also becomputed over one dimension by mapping visit pointsto X or Y axis of the map. In this case, we use a lineinstead of a square.

4.1 Bursty aggregated visit pointsThe bursty dispersion of visit points (or simply bursty

visit points) implies that people tend to swarm nearto a few popular locations and their popularity mea-sured by the number of visit points within the swarmsof visit points formed around the locations shows highburstiness: the popular locations tend to be very pop-ular while most other areas are not. Figure 3 plots theaggregated visit points of KAIST while we zoom in tosmaller areas (denoted by small boxes) in the map. Thepatterns of swarming distinctively appear similar inde-pendent of their zoom resolution (or scale). Figure 4shows the Hurst parameter measured from the aggre-gated visit points of KAIST. These values show a verystrong long-range dependency with a Hurst value largerthan 0.8. Figure 5 plots the Hurst values measured fromall the sites with their 95% confidence intervals. All thesites except NYC show a high degree of burstiness whilethe NYC traces show only slight burstiness. This outlieris, we conjecture, due to the very small number of par-ticipants relative to the size of the area and the numberof registered visit points are relatively small. ExceptNYC, the burstiness of the visit points is evident inde-pendent of their site locations although the degree ofburstiness may vary from one site to another.

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(a) 4800m×1200m

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Figure 3: It shows the self-similar nature of thedispersion of visit points in the KAIST trace.At different scales, the dispersion of visit pointslooks similarly bursty.

4.2 Bursty hot spotsFigure 1 shows that aggregated visit points are clus-

tered to form hot spots. This is because different peoplemay visit similar locations. A hot spot is defined to bea cluster of visit points that are connected to each otherby the transitive closure of the connected relation. Wesay that two visit points are connected if they are withina predefined radio range. The size of a hot spot is thenumber of visit points within that hot spot. Figure 6shows a clear pattern of heavy-tail distributions in hotspot sizes over several orders of magnitudes. It is a nat-ural consequence of the burstiness in aggregated visitpoints shown in Section 4.1. This phenomenon coin-cides with the preferential attachment theory of Albertand Barabassi [3] where popular places become morepopular to form a heavy tail distribution of popularity.

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Figure 5: Hurst parameter values of visit pointsin each site map. All show Hurst values higherthan 0.6 except NYC.

4.3 Bursty individual tracesWe also observe that the visit points registered in

each individual trace are bursty. For each trace, weperform the aggregated variance test on its visit points.Figure 7 shows the Hurst parameter values of individualtraces from the five sites. Their H values are slightlyless than those from the aggregated visit points shownin Figure 5, confirming that burstiness gets intensifiedas individually bursty traces are superimposed together.

Does the burstiness of individual traces come fromthe burstiness of hot spots or vice versa? Our data anal-ysis suggests the former. To see this, we randomly picka subset of visit points from the aggregated visit pointsof each site without any bias to locations. We find thatthese random subsets also show burstiness similar tothat in individual traces. Figure 8 shows the Hurst val-ues of visit points randomly taken from the aggregatedvisit points of KAIST. All 76 traces show a high degreeof burstiness. However, aggregating any random, butbursty visit points does not necessarily result in burstyhot spots. Our data shows that there appears cleargravity to spots that allow visit points to form clustersand that the burstiness in individual traces arises fromthat of hot spots, but not vice versa.

Table 2 shows the statistics from individual traces

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Figure 6: The hot spots in KAIST formed with100 meter range and CCDFs of their sizes fordifferent range values. The boxes in (a) repre-sent the tightest rectangles enclosing all the visitpoints in the same clusters.

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NCSU KAIST NYC DW SF

Site 43 98 117 52 6Trace 4.55 3.66 6.13 3.34 1.67

Table 2: The number of hot spots (clusters) ineach site with a radio range of 100 meters andthe average number of hot spots visited in eachtrace.

on the number of hot spots (when a radio range is 100meters) in each site and the average number of hot spotsthat each trace visits. On average, each trace visitsabout 5 to 10% of hot spots present in that site.

4.4 Characteristic distanceFlights are created on top of visit points. Depending

on the choices of next visit points, the traces may havedifferent flight distributions. What aspect of mobilitycauses a heavy-tail flight distribution? We find that thisis in part due to heavy-tail distributions of distance be-tween clusters of visit points at different scales. Theintuition is as follows. Long flights are created when aperson leaves one hot spot to get to another hot spot.Since the dispersion of visit points is long-range depen-dent, the probability that there exists a neighboring hotspot (swarm or cluster of visit points) at the increasingscales of range, decays very slowly even if clusters arecoalesced together as the range increases. If flights arecreated by moving from one cluster to another, flightswill also likely have a heavy tail distribution because alonger distance between clusters always exists.

Mendelbrot [19] made a related observation aboutfractal points (or the dispersion of points in a burstymanner)that when fractal points are dispersed over 1-D, their gaps (the distance between two neighboringpoints) have a power-law distribution. This observa-tion has been extended to multi-dimensional gaps [7]where fractal points are dispersed in multi-dimensionalspaces, their multi-dimensional gaps (or voids) also havea power-law distribution. To show this, Delaunay tri-angulation is used to measure gaps. Delaunay trianglesare defined to be triangles formed over a set of points ina plane in which no points are inside of the circumcircleof any triangles.

Figures 9 show Delaunay triangles drawn on top ofthe aggregated visit points of each site and the CCDFof the lengths of the lines forming the triangles. TheseCCDFs are showing the same heavy tail distributionsfound from real human walk traces. These CCDFs andthe CCDFs of flights from real traces are surprisinglysimilar. This strongly implies that humans tend tochoose as their flights the lines used in Delaunay tri-angulation. Since Delaunay triangles are formed be-tween only “natural” neighbors in a planar graph, Thismatch of CCDFs indicates that humans tend to visit all

neighboring points first before they make a long-jump toneighboring clusters of visit points. This phenomenonis further verified in the next section.

4.5 Least Action Trip PlanningGiven a set of visit points, how do humans plan their

trips around these visit points to form a heavy-tail dis-tribution of flights? Flights are highly influenced bythe ways that humans choose the next visit points fromtheir current visit points. In this decision, many fac-tors play a role. Since every person may have differentfactors, cost functions, personal situations and personaltendency, it is almost impossible to derive one algorithmthat can apply to all cases. The results from Delaunaytriangulation strongly indicate that people visit nearbyvisit points first before jumping to visit points in a dif-ferent cluster (or hot spot). In a way, this pattern mini-mizes the total amount of distance that a person travels,implying that distance is a stronger determinant in thisdecision.

Maupertuis’ principle of least action [6] provides someclues to this phenomenon: humans tend to make ac-tions that require the least amount of effort. Helbin etal. [?] also show that when people make trails in apark, they use a cost function minimizing discomfort inmoving from one place to another. This discomfort canbe translated into the distance as well as the conditionof roads (i.e., whether it is paved or not). To studythe influence of distance in choosing paths by humans,we run simple simulation mimicking a distance-basedpath choosing mechanism of humans when a set of visitpoints V is given. The simulation runs using the fol-lowing path selection algorithm.Path selection algorithm (least action trip plan-ning). At the current position i in V , the probabil-ity that a next position j is chosen is computed as

1/daij∑

for all k∈V−V ′ 1/daik

where dij is the Euclidean distancefrom i to j, a is a fixed floating number constant within0 to infinity, and V ′ is the set of positions in V thathave been visited so far. Based on this probability, anext position is randomly chosen from V ′.

From the algorithm, if a is set to zero, the next visitedpoint is completely independent of the distance to thatnext point, much like RWP, and if a is infinity, thenan unvisited point with the shortest distance to p ischosen as the next position and this algorithm is in factthe nearest neighbor first (NNF) heuristic used for thetraveling sales problem [26] [22]. As a increases, peopleplace more weights on distance in their path selectiondecisions.

We perform the simulation on top of visit points takenin each individual traces using the algorithm and the re-sults are given in Figure 10. We also compare the resultswith those from Levy Walks (LW) and RWP. With aset to 1.5 or 3, the difference in the sums of flights from

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Orig : 0 % 0 : 535.905 % 1 : 40.4744 % 1.5 : 7.57236 % 2 : 24.6979 % 2.5 : 33.0752 % 3 : 36.6555 % 4 : 41.477 % NN : 49.7957 % RWP : 1800.11 % LW : 82.7108 %

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Orig : 0 % 0 : 610.12 % 1 : 15.1723 % 1.5 : 11.3239 % 2 : 23.1826 % 2.5 : 28.5494 % 3 : 31.8748 % 4 : 35.5607 % NN : 42.3481 % RWP : 1388.03 % LW : 155.265 %

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Orig : 0 % 0 : 451.892 % 1 : 12.0707 % 1.5 : 6.92921 % 2 : 16.5669 % 2.5 : 20.421 % 3 : 23.6334 % 4 : 26.1072 % NN : 33.6126 % RWP : 510.363 % LW : 36.1963 %

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Orig : 0 % 0 : 731.943 % 1 : 155.099 % 1.5 : 63.4191 % 2 : 23.457 % 2.5 : 7.23247 % 3 : 2.93423 % 4 : 12.4258 % NN : 32.0131 % RWP : 1235.63 % LW : 119.157 %

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Orig : 0 % 0 : 275.834 % 1 : 88.7784 % 1.5 : 43.946 % 2 : 18.0205 % 2.5 : 6.73736 % 3 : 1.25554 % 4 : 10.5933 % NN : 24.4022 % RWP : 343.126 % LW : 64.897 %

(e) State Fair

Figure 10: The CCDF of aggregated flightsformed by the path selection algorithm with var-ious values of a. The legend inside each figuredenotes the value of a and its corresponding er-rors in flight sums.

the simulation and the real traces is always between 1%to 11%, and the CCDFs of flights also match extremelywell, indicating they have very similar means and vari-ances. Compared to LW and RWP, our model producesfar better matches.

We can interpret the above result as follows. a = 3produces a good match to flight distributions takenfrom theme-parks (Disney world and State Fair) as hu-mans plan their travel by the distance to each attractionin theme-parks and try to minimize the total travelingdistance as most try to visit as many attractions as pos-sible within a given time, and with a 1.5, other factorsplay slightly bigger roles (e.g., with time critical eventslike predetermined meetings and class schedules, peoplehave to travel to the meeting location irrespective of itsdistance). Thus this case fits better to the campus sce-narios. Even in these scenarios, distance has a stronginfluence.

8

Heavy Tail Hot Spot Sizes

Bursty Visit Points

Bursty Visit Points in

Individual Traces

Heavy Tail Distances of

Gaps

Least Action Tendency in

Human Mobility

Heavy Tail Flights

Figure 11: The summary of our finding from thedata analysis of human walk traces.

This result permits the logical link between the burstyvisit points in human traces and the heavy-tail flightdistributions. In Section 4.4, we show that the burstyvisit points induce the heavy tail distributions of thelengths of the lines formed by Delaulay triangles ofvisit points. As people give more importance to dis-tance when making path selections, the nearest unvis-ited neighboring point is more likely chosen and thatpath is likely part of a Delaunay triangle. Since thelengths of the lines forming the Delaunay triangles havea heavy-tail distribution, it is quite natural that flightsalso have the same tendency.

4.6 Summary of measurement studyIn summary, our measurement study provides a plau-

sible explanation for the causes of Levy walk patternsin human walks. (1) Section 4.2 shows that human visitpoints form clusters whose sizes are bursty (i.e., long-range dependent) and also has a heavy-tail distribution,(2) Section 4.3 shows that visit points found in individ-ual traces are also dispersed in a bursty manner, andthis tendency arises from the bursty hot spots, (3) Sec-tion 4.4 shows that visit points dispersed in a burstymanner in an area induces a heavy-tail distribution ofthe distance of “gaps” between visit points and (4) Sec-tion 4.5 shows that distance-based path selection on topof bursty visit points results in a heavy-tail distributionof flights observed in real walk traces. Figure 11 sum-marizes our data analysis results.

5. BURSTY SPOT MODELOur data analysis indicates that bursty visit points

and distance-based trip planning over those visit pointsare the keys in generating a heavy-tail flight distri-bution observed in human walk traces. This obser-vation greatly simplifies the construction of a humanwalk model that can emulate both hot spot and flightstatistics found in real traces. This section discusses the

construction of such a model called Bursty Spot Model(BSM).

We generate hot spots by dispersing visit points in abursty manner in a given area S. This dispersion cre-ates a synthetic map G containing the locations of afixed number of visit points given as input. The degreeof the burstiness must match closely that from a realmap T produced from human walk traces. To achievethis, we first divide an input area A into N square seg-ments of an equal size and then we distribute n visitpoints given also as input over to the N segments whileensuring that the normalized variance in the numbersof points assigned to each segment is matched to theinput variance R. The normalized variance is definedto be Var( X

E[X] ) where X is the number of visit pointsassigned to a segment. It can be proven that normal-ized variance is equivalent to aggregated variance in Sec-ton 4.1. Note that this is different from the aggregatedvariance in Section 4 where the sample is divided bythe area. We recursively apply the same technique: foreach segment i, A is set to i, the number of points as-signed to i as n, and v̄T

l /v̄Gl−1 − 1 as R. v̄T

l and v̄Gl−1

are respectively defined as follows. l is the resolutionlevel (scale) and initially set to one. For each recursivecall, l is incremented by one. The structure of segmentsdivided into l levels looks exactly like a full N -ary treewith height l. The number of segments at level l is N l.v̄T

l is the normalized variance at level l measured fromT where a sample X is the number of visit points in asegment at level l in T . v̄G

l−1 is the normalized variancemeasured from the synthetic trace G generated at theprevious level (i.e., level l − 1). Initially, A is set to S,n is the total number of visit points in T , and v̄G

0 is setto 1. The maximum number of levels is fixed to someconstant (typically we use 9).

The rationale for this scheme starts from the notionthat by emulating the same variance observed at eachscale from the original trace, we can emulate the self-similarity of the original trace. However, matching thevariances in the synthetic trace and the original trace isnot straightforward. At level 1, matching the varianceis simple since the same number of visit points as in Tmust be distributed. But at a higher level, the numberof visit points assigned to each segment may now bedifferent. Since we need to match only the variance butnot the mean, we normalize each sample (X) by themean. Our goal is to ensure v̄T

l = v̄Gl . The problem gets

a little complicated since we need to achieve this goal byadjusting the variance at each segment independently.This problem can be mapped into a line segmentingproblem where a line with length 1 is divided into N linesegments whose length distribution has a normalizedvariance v̄T

1 and each segment i is then again dividedinto N segments whose distribution R of lengths dividedby the length of segment i is set to produce the variance

9

of lengths of line segments measured from level 2, i.e.,v̄T2 . This process continues until we reach the final level.Now the problem is reduced to a problem of finding

such an R for each level. To recap the problem, atlevel l − 1, there are N l−1 line segments with a distri-bution A = [p1, p2, . . . , pN l−1 ] and the normalized vari-ance is v̄G

l−1 and each segment pi is divided into an-other N segments with the following ratio distributionB = [q1, q2, q3, . . . , qN ] whose variance is R and sum isalso one. Thus, the resulting line segments have a dis-tribution C which contains the line segments of lengthspiq1, piq2, . . . , piqN for all i’s. Let X, Y and Z be ran-dom variables that have the same distributions as A, Band C respectively. Note that E[Z] is 1/N l since thetotal length is one and there are N l segments.

v̄Gl = E[Z2]

E[Z]2 − 1

= (42l)E[Z2]− 1

= (4l)∑

Z2 − 1

= (42E[Y 2])(42(l−1)E[X2])− 1

= (R + 1)(v̄Gl−1 + 1)− 1

(1)

Therefore, if R = v̄Tl /v̄G

l−1−1, then we have v̄Tl = v̄G

l .This method takes as input the total number of visitpoints, v̄T

l for each level l, and the area S.Note that normalized variance can be plotted as a

straight line in a log-log variance vs. scale plot as shownin Figurer̃effig:KAIST-hurst. Thus, instead of providinga set of normalized variance as input, we can simplyprovide a starting variance (the variance at the firstlevel) and a power-law slope (between 0.5 and 1) asinput to the hot spot generator.

Figure 12 shows the sample synthetic trace generatedfrom the above method from the statistics measuredfrom each site.

After we generate a map of visit points, each nodeselects a subset of visit points in the map and plans atrip over the subset using the path selection algorithmdescribed in Section 4.5. From Table 2, we find thateach individual walk trace visits about 5% to 10% of hotspots in the map. Thus, after we form hot spots with afixed range using the technique described in Section 4.2,the subset V of visit points to visit in a single syntheticwalk trace being generated is selected as follows. (1) Werandomly (i.e., with uniform distribution) select 5 to 10% of hot spots to visit while assigning proportionallymore weights for the selection of a bigger hot spot, andthen (2) from the visit points belonging to the selectedhot spots, again randomly select V . The number of visitpoints to select for V is given from user input.

Figure 13 describes the scheme for generating BSM

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Figure 12: Maps containing visit points regis-tered in real traces are compared to syntheti-cally generated maps.

traces. All the components used in this scheme exceptthe generations of hot spots and per-trace visit pointsare discussed in Sections 4.3 and 4.5.

Figure 14 measures the CCDF of the flights in thesynthetic walk traces of BSM constructed using the in-put values extracted from real traces of each site (i.e.,the same number of visit points and the same normal-

10

Hot Spot Generator

Selection of Visit Points

for Individual Traces

Least Action Trip Planning

on the Selected Visit Points

Input Parameters

• The number of visit points

• Size of a map

• Starting normalized variance

• Power-law slope of normalized

variance of visit points

Figure 13: The schematic diagram of Burstyspot model.

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Figure 14: The CCDFs of flights from syn-thetic maps of visit points generated by BSMusing normalized variance values taken from realtraces. They match very well with those fromtheir corresponding real traces.

ized variance values). It is compared to those from realwalk traces in the same sites. They all show a very closematch, verifying that BSM reliably generates realisticflight distributions from the given input.

6. ROUTING PERFORMANCEIn this section, we will examine the impact of the

features of our mobility model on the performance ofhuman driven DTNs. We have the following setup forthe simulation.

6.1 Simulation Setup

We simulate 250 hours of human walks of 50 per-sons. We generate five different simulation sites similarto the five sites where the human walk traces are taken.Based on information in Table 1, we fix the size of thesimulation areas and the number of visit points (appliedonly to BSM). We also fix the slope of normalized vari-ances to the average β values of aggregated variancemeasured from each site. Their values can be deducedfrom Figure 5. For BSM, we vary a from 0 to 3. We de-note BSM(k) to indicate BSM simulation with a = k.Every mobility model has the same truncated power-law distribution of pause times (its slope is one and itsmaximum pause time is 28 hours) – the distribution isobtained from real walk traces [25]. For LW, we use thetruncated power-law distribution of flights with slope 1and the maximum flight length 2.5km. During the sim-ulation, the contact information is checked at every 1minute. The initial position of every person is selectedrandomly from visit points and all mobility models havethe same starting points. We discard the first 50 hoursof simulation results to avoid any transient effects. Thespeed of every user is set to 1 m/s for simplicity. Un-less specified otherwise, we set the transmission rangeof mobile devices to 250m which is the typical value ofWiFi.

We run four of the most widely studied DTN rout-ing protocols over the generated mobility traces. Thesimplest one is Direct Transmission where a message istransferred only when a source node finds its destinationnode in its radio range [8]. It is a very trivial algorithmbut is used as a baseline. Randomized Routing [27] al-lows the holder of a message to send the message toanother node in a radio range with probability p whichis between 0 and 1. In Utility-based Routing WithoutTransitivity [15] [18] [27], each node forwards its mes-sage to the neighboring node with the maximum utilityvalue among all of its neighbors. The utility function isset to be the age of the last encounter with a particularnode. Thus, a node forwards to a neighbor that hasmet the destination most recently among all its neigh-bors. Seek and Focus [27] is a hybrid approach rhatselectively uses randomized routing and utility-basedrouting. If the utility value (the time duration afterthe last encounter with the destination) is less than athreshold Th, then utility-based routing is used. Oth-erwise, randomized routing is used. We fix Th to 500seconds.

6.2 Performance resultsWe run the four DTN routing algorithms in the NCSU

site. Figure 15 shows the routing delay. In this sim-ulation, for each run, we generate 300 messages withrandom source and destination pairs and we wait until200 messages are delivered to their destinations before

11

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Figure 15: Average routing delays of four DTNrouting protocols under various mobility mod-els.

measuring the routing delays. We repeat the test for10 times for each setup. The figure shows the averagerouting delays for the delivery of 200 messages.

Several significant observations can be made aboutthe difference in the routing performance when differ-ent routing protocols and mobility models are used. (1)Overall, the average routing performance of various pro-tocols under BSM is much shorter than RWP and LWand among BSM, larger a values give longer routingdelays. Our measurement study indicates that humanwalks have a within a range of [1:3]. So BSM(0) tends tooverestimate the routing delays compared to BSM(3).(2) The staleness of utility information increases witha smaller transmission range. In BSM(3), utility-basedrouting performs much worse than randomized routingunder 150m Tx range. However this phenomenon is notshown with random walk models: BSM(0), RWP andLW. This is because the impact of wrong or stale util-ity information on routing performance is well compen-sated by random walk mobility patterns of these mod-els. Since human mobility is not completely random,it is natural that a “mistake” in routing using stale in-formation should be penalized more in real scenarios.This is consistent with the results from BSM(1.5) and

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Figure 16: The delay improvement ratio of vari-ous protocols over utility-based protocols undervarious transmission ranges.

BSM(3). (3) The performance improvement of a hy-brid routing protocol like Seek and Focus over utility-based and randomized protocols is much larger withBSM(3) than the other models. To illustrate this, wemeasure the delay improvement of different protocolsover utility-based protocol in Figure 16. It shows thatunder 150m Tx range, in BSM(1.5) and BSM(3), Seekand Focus shows about 50 to 100% improvement whilein the others, their improvement is much smaller. Theperformance gain gets reduced with a larger Tx rangebecause of increased connectivity. Overall, the emula-tion of hot spots and heavy-tail flights is important incharacterizing routing performance.

To see the effect of the degree of burstiness in the dis-persion of visit points, we measure routing performancealso on top of the synthetic maps constructed usingthe trace information of New York City and KAIST.The NYC traces show a very steep slope of the ag-gregated variance of visit points due to the relativelysmaller number of participants and visit points com-pared to its size of the site. On the other hand, theKAIST traces exhibit much more burstiness than theNYC traces for the comparable size of the site. In thissimulation, we use a = 1.5, the best matching a val-

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Figure 17: The routing performance under thesynthetic maps created from the KAIST andNYC data. NYC has a steeper aggregated vari-ance of visit points than KAIST for approxi-mately the same size of the area. The figureshows that the routing performance under theNYC map is much worse than under the KAISTmap.

ues taken from Figure 10. Figure 17 show the result.The performance under the KAIST traces is an-order-of-magnitude better than under the NYC. In fact, theNYC traces have a much fewer number of visit pointsthan KAIST, but in the simulation, we have the samenumber of walkers. Thus, they are more likely to visitthe same visit points. But their performance is muchworse. In contrast, KAIST has several big hot spots(due to its burstiness) where almost every walkers visitevery day. This has made a big impact on the perfor-mance. Thus, the burstiness of hot spots is critical inrealistic estimation of routing performance.

In summary, the performance of DTN routing is verysensitive to the burstiness in the distribution of hotspots and heavy-tail flight distributions. The BSM mod-els are shown to effectively capture this sensitivity.

7. CONCLUSIONHumans never walk randomly. Nonetheless, many

mobility models use random walks. Random walks com-pletely lack in man-made contexts, and thus signifi-cantly distort any performance evaluation results per-formed using these models. Especially, hot spots wherepeople meet and visit commonly are very important forcorrect estimation of mobile network performance. Ex-isting work lacks in realistically representing hot spotsand their statistical properties. Our paper addressesthis issue by emulating the statistical patterns of hotspots, namely highly bursty natures of hot spot dis-persion and sizes. We found this is one of the causesfor a heavy-tail distribution of flights. Furthermore, wefind that humans perform a distance optimizing heuris-tic when planning trips over a set of destinations. Thisis another cause of the heavy tail flight distribution.We propose a bursty spot model (BSM) based on theseobservations, which is a simple human mobility model

taking as input the degree of burstiness in visit pointdispersion. We generate bursty hot spots by emulat-ing the burstiness of visit points measured from realtraces. We apply BSM to the evaluation of several DTNrouting protocols and verify that our model generatesheavy-tail flights. The emulation of hot spots reducesDTN routing performance substantially from popularrandom walk models– an order-of-magnitude less thanLW and even 2 to 3 times less than RWP. This resultindicates that the statistical features of hot spots andflights make significant impacts on the performance ofrouting in mobile networks and realistic representationsof them are important.

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Demystifying Levy Walk Patterns