Monitoring physical distancing for crowd management: real-time trajectory and group analysis
Caspar A. S. Pouw, Federico Toschi, Frank van Schadewijk, Alessandro Corbetta
MMonitoring physical distancing for crowd management:real-time trajectory and group analysis
Caspar A.S. Pouw a,b , Federico Toschi a,c , Frank van Schadewijk b , Alessandro Corbetta a, ∗ a Department of Applied Physics, Eindhoven University of Technology 5600 MB Eindhoven, The Netherlands b ProRail Stations, 3511 EP Utrecht, The Netherlands c CNR-IAC I-00185, Rome, Italy
Abstract
Physical distancing, as a measure to contain the spreading of Covid-19, is defining a “new normal”. Unless belonging to a family,pedestrians in shared spaces are asked to observe a minimal (country-dependent) pairwise distance. Coherently, managers ofpublic spaces may be tasked with the enforcement or monitoring of this constraint. As privacy-respectful real-time tracking ofpedestrian dynamics in public spaces is a growing reality, it is natural to leverage on these tools to analyze the adherence to physicaldistancing and compare the e ff ectiveness of crowd management measurements. Typical questions are: “in which conditions non-family members infringed social distancing?”, “Are there repeated o ff enders?”, and “How are new crowd management measuresperforming?”. Notably, dealing with large crowds, e.g. in train stations, gets rapidly computationally challenging.In this work we have a two-fold aim: first, we propose an e ffi cient and scalable analysis framework to process, o ffl ine or in real-time, pedestrian tracking data via a sparse graph. The framework tackles e ffi ciently all the questions mentioned above, representingpedestrian-pedestrian interactions via vector-weighted graph connections. On this basis, we can disentangle distance o ff endersand family members in a privacy-compliant way. Second, we present a thorough analysis of mutual distances and exposure-timesin a Dutch train platform, comparing pre-Covid and current data via physics observables as Radial Distribution Functions. Theversatility and simplicity of this approach, developed to analyze crowd management measures in public transport facilities, enableto tackle issues beyond physical distancing, for instance the privacy-respectful detection of groups and the analysis of their motionpatterns. Keywords:
COVID-19 automated physical distancing analysis, high-statistics pedestrian dynamics, crowd management, statisticalmechanics of human crowds, privacy-respectful tracking
1. Introduction
Crowd management is a challenging scientific topic di-rectly impacting on the functioning of tra ffi cked urban infras-tructures such as, e.g., train or metro stations. Even more so, intime of Covid-19 pandemic, after an initial lock-down period,communities are still wondering how to resume a “new nor-mal” life, while the virus is still circulating among the popula-tion. One of the key control measures has been to maintain aminimal physical distance (often also called “social distance”)between any two individuals not belonging to the same fam-ily [1]. This distance is country-specific and it ranges from1 m (e.g. China and France), as recommended by WHO, up to2 m (e.g. UK and Canada), being 1 . ∗ Corresponding author
Email address: [email protected] (Alessandro Corbetta ) order to respect individual privacy one needs to employ sen-sors and techniques that ensure privacy by design while, at thesame time, providing accurate space-time information on indi-vidual positions with sub-meter accuracy.Secondarily, one needs to develop algorithms that, whilepreserving privacy, are capable to autonomously discern, witha good degree of accuracy, families and family members fromstrangers. This identification should be performed in real-time,raising a number of non-trivial technical challenges.Additionally, in recent months a number of countries havedeveloped contact tracing apps that allow to receive an alertwhen somebody has been in “close” contact with somebodythat, later on, will turn out to be positive to the Covid-19 [6].Countries are developing apps based on di ff erent alert thresh-olds, typically a combination of having been closer than a givendistance, for a time longer than an established reference. Thesethresholds, again, are country specific. In Italy and Germanythe national apps alert for contacts longer than 15 minutes ata distance below 2 meters. The proper balance of these twoaspects, distance and time, is key to avoid too many false posi-tives or false negatives due to e.g. to low or high risk exposuresas well as to the inaccuracy of distance estimation via the in-tensity of Bluetooth signals. It is therefore extremely interest- Preprint submitted to Arxiv July 15, 2020 a r X i v : . [ phy s i c s . s o c - ph ] J u l ng to be able to analyze, in a number of key urban settings,the combination of contact times for given distances betweentwo persons. This knowledge may provide key information forthe calibration of contact tracing apps in di ff erent context.In this paper we employ data from commercial pedestriantracking sensors placed overhead at Platform 3 in Utrecht cen-tral train station (The Netherlands), in order to develop an ef-ficient algorithm, capable of running in real-time, and able todistinguish infringements of the physical distancing rule fromthe behaviour of family members, that are allowed not to re-spect such a rule. We introduce the concept of “Corona event”,to indicate events when two people, not belonging to the samefamily, get closer than a threshold distance D .We focus on contact times and mutual distances consider-ing statistical observables as the radial distribution functions(RDFs), which can conveniently be employed to quantify av-erage exposure times. This enables a two-fold task: automatiz-ing the definition of families and groups (from now on namedfamily-groups) and characterizing the statistical distribution ofviolations, which we compare with analogous pre-Covid mea-surements. Based on the space-time dynamics of groups, wetry to identify family members as those individuals that con-sistently stay closer than a given threshold distance for su ffi -ciently long time. This, in turn, allows us to define physicaldistance violators as those individuals that only occasionally(i.e. inconsistently) yield Corona events infringing the mini-mal distance rule.This paper is structured as follows: in Section 2, we surveythe pedestrian dynamics literature and computer science meth-ods in connection with group-dynamics and mutual distances.We outline both fundamental outstanding questions and exis-tent analysis methods. In Section 3, we describe the locationand measurement setup at Utrecht Central train station used toacquire the analyzed pedestrian data. In Section 4 we reviewthe concept and basic properties of Radial Distribution Func-tions, extensively leveraged on by our method. In Section 5 wepresent our method and possible variations, that we employ toanalyze mutual distancing data in Section 6. A final discussionin Section 7 closes the paper.
2. Related works: (social) distance in pedestrian dynamics
The analysis of pairwise distances and the automated iden-tification of family-groups triggered by the Covid-19 pandemicsconnect with outstanding technological and fundamental is-sues in the broader field of crowd dynamics . Crowd dynam-ics is a multidisciplinary research area aiming at understand-ing and modeling the motion of pedestrians in crowds (see,e.g., [7, 8, 9], for introductory references). Outstanding ques-tions specifically connected to mutual distances and groupsare, e.g.: “what is the impact of the group on the individualdynamics observables such as position and velocity?” “Howdo people in social groups interact?”, “How does informa-tion propagates throughout groups?” (see e.g. [10, 11, 12, 13,14] and e.g. [15] for a group-psychology review). Althoughthese questions are longstanding, and have been investigated via models or in laboratory settings extensively, first quanti-tative studies in pedestrian dynamics driven by real-life bigexperimental datasets are relatively recent (see, e.g., [16, 17,18, 19]). Large-volumes of experimental data, in the order ofhundred of thousands real-life trajectories, are indeed essen-tial in order to analyze quantitatively and systematically thephysics of pedestrian motion, disentangling the high variationsin individual behaviors from average patterns, and characteriz-ing typical fluctuations and universal features [19, 20]. Thisrelative delay in performing high-statistics based analyses ofpedestrian motion (especially in comparison with other “ac-tive matter” physical systems [21]), is most likely due to thecomplex technical challenge of achieving accurate, privacy-preserving, individual tracking in real-life conditions (see, e.g.,[20, 22, 23], or [24] for approaches targeting even higher res-olution). Market solutions, as the one considered in this paper,are also becoming accessible, o ff ering various trade-o ff s be-tween accuracy and costs (see, e.g., [25]).On top of automated tracking, higher-level automated un-derstanding of individual behaviors – a concept also known incomputer science as trajectory pattern mining [26, 27] – re-mains also outstanding in many aspects. The automated iden-tification of pedestrian groups, or pedestrian “group mining”,is a notable example in this context. On one side, in cur-rent pedestrian dynamics research, the definition and classi-fication of groups and social structures in experimental datahas been manual, i.e. based on labor-intensive visual inspec-tion (e.g. [28]). While this ensures high-quality validated mea-surements, it limits the possibility to establish vast statisti-cal datasets towards data-driven characterizations of averagesand fluctuations in the dynamics. On the other side, auto-matic strategies to identify groups have been proposed by thedata mining community. These approaches primarily hinge onanalyzing (instantaneous) spatial clusters of pedestrians andthe consistency with which these adhere over time to somegroup semantics (flocking, convoying, aggregation / desegre-gation, see [27, 29, 30, 31] and references within).Here we pursue distance analyses and family-group identi-fication via discretized mutual pairwise distance distributions– represented in physics terms via Radial Distribution Func-tions among relevant pedestrian pairs (the concept of RDF isfurther reviewed in Section 4). We accumulate information ona “social” interaction graph with vector edge weights. Thisdata structure holds all the relevant contact times and distancestatistics; besides, family-groups emerge as incremental fea-tures queryable by a space-time distance semantics. Graphsare classic tools in discrete mathematics to represent networksof interactions, or connections between entities (e.g. [32]). For-mally speaking, a graph H is a set of nodes, H = { p i } , endowedwith edges, say e = ( p i , p j ), connecting node pairs. Providinga weight function, w ( e ), defined on the edges, makes the graph“weighted”. In our case, nodes are in 1:1 correspondence withobserved pedestrians, whereas edges underlie distance-basedinteractions, that are characterized by a weight function withvalues in a real vector space of pre-fixed dimension. Graphshave been often used for data-driven studies on social behav-ior both of humans, e.g. to analyze social networks [33], GPS-2 . 120 m3 m 5 m2020-05-10b. 2019-05-27c. Figure 1: (a) Floorplan of platform 3 at Utrecht Central Station (NL). The area monitored by the sensors is highlighted in grey. (b) Sample of 75 passengers waitingfor a train to arrive on the 10 th of May 2020, during the Covid-19 pandemic. Pedestrians which respect the 1 . ff enders are colored in red. This classification is performed via the method proposed inSection 5. In this situation only 3 out of the 75 people violate the physical distancing rules. (c) Same number of people distributed over the platform on the 27 th ofMay 2019, one year prior to the Covid-19 outbreak, here about one-third of the people stand closer than 1 . data [34], but also of social animals (e.g. [35, 36]). In [19, 37],graphs have also been used to address big-data analyses andrepresentation of pedestrian dynamics aiming at e ffi cient datasearches.
3. Pedestrian tracking setup at Utrecht Central Station
We benchmark our approach considering pedestrian track-ing data acquired on platform 3 at Utrecht Central station, TheNetherlands (cf. Figure 1). Utrecht Central, with roughly 57million annual users, is the nation-wide busiest railway station.Since 2017, platform 3 has been equipped with 19 commercialpedestrian tracking sensors, each of which captures 3D stereoimages at f =
10 frames per second and processes them todeliver individual tracking data in a privacy-friendly manner(cf. sketch in Figure 1). The sensor view-cones are in partialoverlap, which enables the sensor network to stitch togethertrajectory pieces acquired by the single devices. The total areacovered by the set of sensors consist of the full platform width(about 3 m) for 120 linear meters next to track 5, plus the areaunderneath escalators and staircases connecting the platformto the central hall. This yields a covered area of approximately450 m . Track 5 is among Utrecht’s busiest tracks and is pri-marily utilized by trains heading to Amsterdam Central Sta-tion and Schiphol Airport. The complex and multi-directionalcrowd flows on the platform are recorded with high space- andtime-resolution 24 / O (5 −
10) cm, similar technology to what employed in [25]). Innormal operation conditions, the system would capture about100 .
000 trajectories per day while, on average, only 16 .
4. Pedestrian radial distribution functions
In theoretical physics and molecular dynamics, the radialdistribution function, g ( r ) (RDF), and the radial cumulativedistribution function (RCDF), G ( r ), are established tools tocharacterize the distribution of pairwise distances between par-ticles (see e.g. [38]), i.e., in our case, pedestrians.By definition of RCDF, for a crowd with uniform spatialdensity ρ , on average, i.e. in the mean-field of many realiza-tions, the number of people, N ρ ( r ), at a distance up to r from ageneric individual satisfies N ρ ( r ) = ρ G ( r ) = ρ (cid:90) r g ( z ) dz , (1)therefore, g ( r ) = ∂ r G ( r ) holds. Thus, the functions g ( r ), G ( r )(and derived quantities) do not carry any space / time specificinformation, rather they relate to average properties, depend-ing only on mutual distances.For instance, in unconfined space, G ( r ) grows as the circlearea, i.e. N uncon f ρ ( r ) = π r ρ. (2)In our train platform, such a ∼ r growth ratio is possible onlywhen r is su ffi ciently smaller than the platform width. Else,we expect a ∼ r (linear) trend, as the area growth is bound tothe platform length only. This holds until platform finite sizee ff ects come into play. In Figure 3 we compare the RDF andRCDF for the two density levels highlighted in Figure 2. Wenotice a depletion in the radial distribution functions at shortdistances when comparing with the situation pre-outbreak. Asa partial anticipation of the results of this paper, in the figurewe report also the RDF discounted of short-distance family-group interactions (such interactions are allowed by present3 ρ p [ped / m ] c oun t ( ρ p ) Figure 2: Histogram of observed crowd density levels comparing a day before the Covid-19 outbreak (27 th of May 2019, blue dots) and for a day during theCovid-19 pandemic (27 th of May 2020, red triangles). Prior to the Covid-19 outbreak, densities in excess of 1 ped / m occurred daily. One year later, during theCovid-19 pandemic, the maximum crowd density recorded is only about 0 . / m . We compare measurements acquired at similar density levels, i.e. where theaverage available space per person is comparable. We focus on two density levels: 40-50 passengers (purple band, cf. Figures 3(a,c) and 6(a)) and 70-80 passengers(green band, cf. Figures 1, 3(b,d) and 6(b)). regulations). As expected, this yields a further depletion of theRDF in the region r (cid:47) . N pedestrians; individuals have a random spatial distribution sat-isfying a minimal mutual distance of 0 . ff ect. Measurements well-conform with simu-lations.Short mutual distances over extended time duration areknown to increase the contagion probability: RDF and RCDFcan be used to evaluate the average exposure time. A pedes-trian that was on the platform for a time interval ∆ T , was ex-posed, on average, for a time T r c = ∆ T ρ G ( r c ) , (3)with r c being a critical distance threshold (e.g. r c can be thenation-wide physical distancing requirement). Similarly, thefunction t ( r ) = ∆ T ρ g ( r ) , (4)quantifies the contribution to the total average exposure timegiven by peer pedestrians at distance r .
5. Distance-Interaction network
In this section, we describe our scalable framework to char-acterize pedestrians pairwise distances, identify family-groupsand distance o ff enders.Our measurements come in the form of time-stamped tra-jectories. As no further information is available, such as bodyorientation or gaze direction [39] or body size / approximate age, our identification of family-groups relies only on mutualproximity and its time-consistency. Whenever two or morepedestrians maintain a mutual short distance consistently through-out a su ffi cient fraction of their trajectory, they should auto-matically emerge as belonging to the same family-group. Ad-ditionally, we deem implementation simplicity, while main-taining e ffi ciency and su ffi cient accuracy in identifying family-groups relations, possibly in real-time, and without complex / costlydata-searches. Hence, our approach is “additive” (or “incre-mental”) and RDF-like information is increased, on the go, ina graph data structure (at minimal memory costs), and, with-out computationally-costly searches in stored records, family-groups and o ff enders remain identified immediately. In otherwords, by additivity, we stress that our data structure is builtonline and usable after only one time-forward pass of the tra-jectory data. In conceptual terms, we represent the pedestrian trajecto-ries as individual nodes of a graph H . Each node includesinformation specific to the trajectories, such as overall obser-vation time, τ , source and destination. These three quantitiesare incremental as, respectively, τ scales with the number offrames a pedestrian is observed, the source point is the ini-tial point of a trajectory while the destination gets constantlyupdated with the current position until a pedestrian leaves themeasurement area. Whenever two pedestrians, say p and p ,are observed simultaneously (i.e. in the same frame) and theirEuclidean distance, r = d ( p , p ), is below a critical thresh-old D (cid:48) > D , we memorize (properties of) this event within theweight, (cid:126) w ( e ), of the edge e = ( p , p ), that connects the twopedestrian-nodes p , p . Specifically, the weight (cid:126) w ( e ) aims ata discrete counterpart of the RDF ( g ( r ), cf. Eq. (1)) restrictedto pedestrians p , p and with support 0 ≤ r ≤ D (cid:48) . Similarly tothe RDF, also the graph H does not hold detailed microscopic4 -3 -2 -1 l og ρ G (r) log r [m]~r ~r a. Regulations (D)Platform width D’Monte Carlo model Covid-19 working dayPre Covid-19 Covid-19 working dayw/o families10 -3 -2 -1 l og ρ G (r) log r [m]~r ~r b. c • g (r) r [m] c. Regulations (D)Platform width D’Fit ~r Fit ~r Covid-19 working day Covid-19 working dayw/o families 0 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 5 6 c • g (r) r [m] d. Figure 3: (a, b) Radial cumulative distribution functions (RCDF), g ( r ), and (c, d) radial distribution functions (RDF), G ( r ), for density levels on a typical workingday. On the left (a, c) for density level 1, with 40-50 pedestrians on the platform, (green domain in Figure 2) and on the right (b, d) for density level 2, with 70-80pedestrians on the platform (purple domain in Figure 2). Vertical dashed lines at 1 . . . r < D ), the usable width of the platform (without danger zone) and the critical threshold D (cid:48) . The solid black line at small r values highlights the ∼ r growth ratioup to 2 . ∼ r trend at larger mutual distances. In (b,d) the normalization constant c is chosen such that (cid:82) ∞ cg ( z ) dz = N , where N is thenumber of people on the platform. Similar plots for a weekend day are reported in Figure 6. We compare the pre-Covid situation with the present, and with a MonteCarlo model of a random distribution of passengers across a region identical to the platform. We report the RDF and RCDF of the current situation including andexcluding family-groups contributions, as made possible by the method introduced in Section 5. [0:0.5] m [0.5:1] m [1:1.5] m [1.5:2] m [2:2.5] m020 C o n t a c tt i m e [ s ] Dutchregulations Outside danger zoneInside danger zone b. P e d e s t r i a n s Figure 4: (a) Conceptual sketch representing the accumulation of information on the graph H . Whenever two pedestrians, say p , p stand at a distance d smallerthan D (cid:48) , this gets recorded in the histogram weight of the edge between nodes p and p as an additive contribution to the bin approximating d . In the sketch wereport a section of the platform: edge appear between nodes according to the distance; the histogram weights are reported atop and beneath the sketch with the samecolor coding of the edges and scaled with the sampling time (thus they translate to the contact time conditioned by the distance). Nodes are reported in red if theyhave performed at least one Corona event (thus they have an edge with non-zero contributions at distances below D (cid:48) ), else they are in green. (b) Examples of graphsacquired in windows of about ten minutes around each train arrival (determining the peaks in the counts at the bottom). We report a magnified version of one amongthese graphs. Nodes are colored by the node degree, i.e. by the number of first neighbors, ranging from yellow to red. Edge thickness scaled by the contact time, T de , Eq. (7). The higher the degree of a node, the larger the number of distance o ff enses performed by the associated pedestrian. a. b. I n d i v . e x p o s . t i m e [ s ] Figure 5: (a) Detected clique consisting of two nodes representing two people traveling together. Both entering the platform through the stairs, waiting together forthe next train to arrive and finally boarding the train through the same door. The hue of the trajectories is proportional to the time spent on the platform. Lighter huewhen the people enter and a darker hue when they leave. Jump in hue indicating the place where the travelers were waiting. (b) Detected node with degree higherthan 10, i.e. a repeated o ff ender who violates physical distancing with more than 10 other people. The trajectory of the repeated o ff ender is reported in shades scaledto the exposure time, while the trajectories of other people that were met violating physical distancing are in gray. The considered o ff ender entered the platform viathe escalators and waited underneath the escalators for their train. information in space / time, such as instantaneous positions. Tofocus on the identification of Corona events and disentangle-ment of family-groups, we set D (cid:48) = . D = . < D (cid:48) . This aims at exploring the distance dynam-ics in the neighborhood of the current regulations and leavesflexibility should the regulations become stricter and requireadditional mutual separation.The vector weight (cid:126) w ( e ) keeps record of the number of oc-currences of distance events, r , after a given radial quantization(binning). In the following, we consider five evenly-sized binswith sides at r = { r , . . . , r d , . . . } = { , . , . , . , . , . } m; (5)for the sake of readability, we also indicate, the individual components of (cid:126) w ( e ), (cid:126) w d ( e ), as (cid:126) w ( e ) = (cid:126) w ( e )[0 . . ,(cid:126) w ( e ) = (cid:126) w ( e )[0 . . , and so on. Hence, for each time instant in which d ( p , p ) < .
5, the counter (cid:126) w ( e )[0 : 0 .
5] is incremented of one unit. Simi-larly, whenever 0 . < d ( p , p ) < .
0, the weight (cid:126) w ( e )[0 . . etc . Note that updating the data structure re-quires only all the pairwise distances (smaller than D (cid:48) ) at eachframe. The choice of bin size, which regulates the quality ofthe approximation of the RDF of p and p , is clearly arbitraryand needs to be a trade-o ff between the required resolution onthe RDF and memory allowance.Consistently with Eq. (4), scaling the counts (cid:126) w ( e ) by the(inverse of the) sensors sampling frequency, f , yields the timeduration, in seconds, pedestrians p and p maintain a given6 c • g (r) r [m] a. Regulations (D)Platform width D’Fit ~r Fit ~r Covid-19 weekend day Covid-19 weekendw/o families 0 0.05 0.1 0.15 0.2 0.25 0 1 2 3 4 5 6 c • g (r) r [m] b. Figure 6: Radial distribution functions (RDF), g ( r ), for a typical weekend day in case of (a) 40-50 pedestrians on the platform, (green domain in Figure 2) and (b)70-80 pedestrians on the platform (purple domain in Figure 2). The same conventions of Figure 2 hold. The presence of family-groups determine a peak in theRDFs around r ≈ . ∼ r growth rate at small r values. (quantized) distance (i.e. f − (cid:126) w ( e )[0 . .
0] is the amount ofseconds p and p had a distance between 0 . . r , weighted by f − (cid:126) w ( e ) enable tocalculate the total contact time of p and p , their average dis-tance and fluctuations. In all cases, the statistics are restrictedto r ≤ D (cid:48) (insights on the relevant statistical properties of thegraph are left to the next subsection). Operationally, we buildthe graph as reported in Algorithm 1. Additionally, in Fig-ure 4a, we provide a visual description of the graph in the caseof a subsection of our train platform, while in Figure 4b weshow examples of typical graphs built in time windows about10 minute-long around the train arrivals. Data:
Trajectories dataset, possibly live-streaming
Result:
Distance-interaction graph HH = empty graph; for t in time doadd any trajectory, p i , starting at time t as a node in H , store origin; update persistence time τ p i , destination of allobserved trajectory-nodes p i ; for < i < j ≤ doif d ( p i , p j ) < D (cid:48) then (cid:126) w ( p i , p j )[ q ( d ( p i , p j ))] += endendendAlgorithm 1: Graph construction algorithm in pseudo-code.The data structure is built streaming the trajectory data once.The function q = q ( d ) returns the distance bin to which d belongs. Hence, given the quantization in Eq. (5), it holds q (0 < d < . m ) = q (0 . m < d < . m ) =
1, etc.
Extensions and variations.
In settings as train platforms, notall the areas come with the same importance or criticality. The so-called “danger zone”, the last 80 cm-wide bu ff er region onthe platform that is stepped on just before boarding a train, isan example. For our use case it is imperative to have the capa-bility of discriminating between events happening inside andoutside such an area. To achieve this, we consider two separatesets of weights on each edge: (cid:126) w dz ( e ) and (cid:126) w c ( e ), which count,respectively, the time instants a given distance below D (cid:48) occurswhen the centroid between the two pedestrians lies in the dan-ger zone, and otherwise. According to our previous definition, (cid:126) w ( e ) = (cid:126) w dz ( e ) + (cid:126) w c ( e ) holds. Figure 4a reflects this aspect byrepresenting (cid:126) w dz ( e ) and (cid:126) w c ( e ) stacked (and with di ff erent colorshade). The graph is a collection of RDFs functions restricted topairs of pedestrians. As such, within the limits of the quantiza-tion, it is richer in information than the “global” RDF (Eq. (1)).The latter, in fact, can be recovered by averaging the edgeweights over the entire graph, i.e. by combining each pairwisecontribution. Restricting to a graph describing conditions withequal density, the global RDF can be approximated as (cid:90) r d + r d g ( r ) dr ≈ c (cid:28) (cid:126) w d ( e ) (cid:80) j (cid:126) w j ( e ) (cid:29) e , ∀ d , (6)where c is a constant scaling depending on the normalizationconsidered for g . In words, the integral of the RDF in the bin[ r d , r d + ] can be approximated by the d -th (enseble-averaged)edge weight. Averaging over a graph including non-homogeneousdensity levels, yields the RDF averaged among such densities. In this subsection we leverage on the graph topology andedge data to deduce relevant information about pairwise dis-tance, family-group relations, exposure times, and physicaldistance o ff enders.7. A v g . i nd i v . e x po s . t i m e w / o f a m ili e s T dp ,f [ s ] Week number [0.0 : 0.5] m[0.5 : 1.0] m[1.0 : 1.5] m[1.5 : 2.0] m[2.0 : 2.5] m0 k5 k10 k15 k20 k25 k 17 19 21 23 25 P ed s . N da il y b. γ • T / N da il y Week numberActual growth relative to the passengersContact time growing as the number of passengers
Figure 7: (a) Average individual exposure time without family contributions (Eq. (9); weeks 17-26, working days only. Corona lockdown measures in TheNetherlands started around week 13). Each line reports average data from bins (cid:126) w ( e ) , (cid:126) w ( e ), etc. The inset on the left shows the average daily passenger count, N daily . The inset on the right reports the same individual exposure time data for week 21 in histogram form. A change in the train schedule on the 2 nd of June (week23, indicated with a vertical black dashed line) increased the train frequency by a factor γ ≈ . γ , Eq. (14) (dashed green line, bin (cid:126) w ( e ) only, i.e. r ∈ [1 . , . m ≈ D ). We notice that the compensated exposure time grows steadily in timegaining a factor 3 . × . This is a combined e ff ect of the passenger growth and a reduction in attention and / or di ffi culty in adhering to physical distancing regulations.In panel (b) we scale the exposure time by the number of passengers, i.e. we report γ T dp , f / N ( d = ≈ Pairwise exposure time and pairwise distance statistics.
Pedes-trians p and p , whose distance satisfied r = d ( p , p ) ≤ D (cid:48) atleast in one frame, have their interaction recorded on the graphedge e = ( p , p ). The weight (cid:126) w ( e ) allows us to character-ize their distance properties restricted to the instants in which r < D (cid:48) . In particular, the contact time T e between p and p satisfies T de = f − d (cid:88) j = (cid:126) w j ( e ) , [Contact time] (7)where the index d selects the farthest relevant distance bin.According to our quantization in Eq. (5), d = r ≤ . d = d max = e , i.e. r ≤ D (cid:48) .Similarly, the average pairwise distance reads (cid:104) r (cid:105) de = f − (cid:80) dj = r j + (cid:126) w j ( e ) T de , [Avg. pairwise dist.] (8)where r j + identifies the mid point between bin j and j + r weighted by (cid:126) w ( e ) canbe used to estimate the fluctuation (variance) of the distancein time. In particular, if the variance σ ( r ) = (cid:104) r (cid:105) de − ( (cid:104) r (cid:105) de ) issmall, then the pedestrians kept an almost fixed distance duringtheir trajectories. Notably, small or possibly zero r variance, σ ( r ), are necessary consition for non-positive (Finite Time)Lyapunov Exponent of the distance between p and p [40]. Total individual exposure time.
The total time an individual p has been exposed to contacts can be computed by summing thepairwise contact times T de (Eq. (7)) for all pedestrians, p j , thatentered into contact with p , i.e. for all the edges e = ( p , p j ) that converge to p , in formulas T dp = (cid:88) p j ∈ N ( p ) T d ( p , p j ) , [Individ. expos. time] (9)where N ( p ) is the list of the first-neighbor of p (nodes con-nected to p through at least a single edge). Equation (9) pro-vides a counterpart to Eq. (3) in which we consider a specificpedestrian, p , rather than averaging over all pedestrians. No-tice that the index d is the discrete analogue of the cuttingthreshold r c . Family-group relations.
We determine whether two pedestri-ans, p , p , belong to the same family-group on the basis oftheir contact time T d ( p , p ) and their persistence time in the track-ing area, τ p and τ p . In particular, if the symmetric relationhenceforth indicated as p ∼ p holds p ∼ p ⇐⇒ [Fam-group condition](10)min T (1) e τ p , T (1) e τ p > λ (1) and min T (2) e τ p , T (2) e τ p > λ (2) , we consider p and p as belonging to the same family-group.We set λ (1) = λ (2) =
90% which, in words, translatesto people who have a pairwise distance of less than 1 . r ≤ T dp (Eq. (11)) after discounting family-groupcontacts: T dp , f = (cid:88) p j ∈ N ( p ) p j (cid:28) p T d ( p , p j ) . [Individ. expos. time w / o families](11)Analogously, we can consider a RCDF discounted of family-group contributions, say G f ( r ) (cf. Eqs. (1), (3)), such that T r c f = ∆ T ρ G f ( r c ) (12)is the total exposure time with non-family individuals and up toa spatial threshold r c . By di ff erentiation (as in Eq. (1)), we cansimilarly define the RDF g f ( r ) discounted by family-groups. Family sub-graph transitive closure.
The relation “ ∼ ” can benon-transitive, i.e. p ∼ p , p ∼ p do not imply p ∼ p . Onemay thus consider the transitive closure of “ ∼ ”, say “ ¯ ∼ (cid:48)(cid:48) whichis defined as p and p belong to the same family-group ( p ¯ ∼ p )either if p ∼ p or if they have common family-group mem-bers.In the data we analyze in Section 6, we do not considersuch transitive closure. In other words, family-group relationsare exclusively defined by Relation (10). On one hand, wedeem rare the event that a family remains not represented by aclique. Even in this case, we expect contributions to the overallRDF statistics be minimal. On the other hand, in real-life datacollected with sensors similar to ours, short / broken trajecto-ries may appear. We observed that these, in combination withRelation (10) would unrealistically increase the probability ofobserving large family-groups. Hence, to avoid excessive andunjustified complications in the heuristics we restrict to Rela-tion (10). Relevant interactions, family-discounted graph and o ff enders. Combining the previous elements we can now identify dis-tance o ff enders as those pedestrians that violate physical dis-tancing while not being part of a family. We consider thesub-graph H (cid:48) ⊂ H obtained after pruning H of family-groupedges. The connections left in H (cid:48) must indicate sporadic (i.e.not time-consistent) distance infringements.As exposure time is deemed a critical parameter for conta-gion [41], we apply a further time requirement to discriminateactual o ff enders. Specifically, we introduce the set P (cid:48) α definedas P (cid:48) α = { p ∈ H (cid:48) : T dp , f > α } ; (13)in words, elements of P (cid:48) α are pedestrians who violated phys-ical distancing with non-family members for an overall timelonger than α . The number of first neighbors of a node in P (cid:48) α identifies how many contacts such pedestrian had: we label asrepeated o ff enders those with more than 10 first neighbors (i.e.pedestrians that violated physical distancing with more than10 di ff erent people and for an overall time larger than α ). Weremark that this classification can be run in real-time as all the aforementioned requirements can be constructed in addi-tive manner.
6. Physical distancing at Utrecht Central Station,platform 3
In this section we employ the graph H to analyze trajectorydata collected in Utrecht station (see Section 3) and we com-pare statistics from before and during the Covid-19 pandemic. Typical graphs and qualitative aspects.
In Figure 4b, we re-port examples of the graphs acquired during a typical morning(time interval 4AM - 8.30AM). Train arrivals are the most crit-ical conditions when it comes to respecting physical distanc-ing, thus we create a new graph two minutes after each traindeparture, when the platform is almost empty (this step is notstrictly necessary, but increases computational e ffi ciency). Inthe figure, nodes size and color follows the node degree, i.e.the number of first neighbors and, thus, the distance o ff ensescommitted by that node.As a qualitative example of the capability of the method toextract relevant data, we showcase two antipodal conditions inFigure 5. In the first case (left panel), we report two pedestri-ans in a family-group relation that remain together throughouttheir entire trajectories: from the escalators to the boarding. Inthe second case, we have a repeated o ff ender: the associatednode exhibits 28 first-neighbor connections. Interestingly, asignificant part of the o ff enses happens while the pedestrianwaits in proximity of the escalator. This, therefore, rathermarks a waiting area to be disallowed, than a willing o ff ender. Family-group discounted RDFs.
In Figures 3 and 6 we reportRDFs prior and after excluding family-group interactions. TheRDFs for r < D are non-vanishing, even after discounting thecontributions of family-groups, most significant in the week-ends (when the presence of workers and commuters is lower;cf. short-distance “bump” in Figure 6). We expect these re-maining contributions at r < D to be to Corona events by dis-tance o ff enders. Notably, once removed of family-groups con-tributions, the RDFs at small r values recover a linear growthrate, as expected by a random spatial distribution of passengers(scaling as the derivative of Eq. (2)). Exposure time, node degree and evolution through the pan-demics.
In Figure 7a we report, week-by-week, the averageedge weight as introduced in Eq. (6), pruned of family-groupcontributions and scaled by the sampling frequency. This pro-vides an approximation of how the (family-discounted) av-erage individual exposure time has been changing over time(weeks 17 to 26 in 2020). As the usage of the platform grewafter a drop at the beginning of the outbreak, so did the ex-posure time for distances between 0 . . . - -
27 2020 - -
04 2020 - -
11 2020 - -
18 2020 - -
25 2020 - -
01 2020 - -
08 2020 - -
15 2020 - -
22 2020 - - % o f t o t a l b.
100 1000 P D F Individual expos. time T dp [s]1 min 5 min 10 minr = [0.0 : 0.5] mr = [0.0 : 1.0] mr = [0.0 : 1.5] mr = [0.0 : 2.0] mr = [0.0 : 2.5] mfit r = [0:1.5] m with A * x p where p = -2.28 Figure 8: (a) Distribution of node-pedestrian degree per day as a percentage of the total number of passengers. The degree of a node counts the number of peopleencountered with a mutual distance smaller than 1 . ff enders, increased steadily until the train schedule change (e.g. nodes with 10 + contacts grew from ≈
1% to ≈ ff ender percentage after which it started increasing again. This can be a sign of warning towards the relaxation in the complianceof physical distancing rules. (b) Probability density function of the individual exposure time discounted of families, T dp , f considering di ff erent maximum distances(Eq. (9)). Exposure times show a power-law behavior. The PDF depletion after T = ff for large times. a temporary reduction of the load on the platform, making eas-ier to respect physical distancing. From week 24 onward, in-dividual exposure times showed again a growing trend. Torender the data comparable, we consider also exposure timescompensated for the new train schedule (Figure 7a, only forthe case r ≈ D ) , i.e. corrected by a factor γ = ≈ . . (14)This shows a more stable growth pattern and an increase offactor 3 . γ is an esti-mate, considering the presence of trains of di ff erent kind andlengths). Scaling the corrected exposure times with the num-ber of passengers, which is itself growing, we additionally no-tice, that the former is growing faster (i.e. exposure time growsuper-linearly with respect to the passengers). This suggests apossible relaxation or an increased di ffi culty in following phys-ical distancing regulations.We report in Figure 8a an in-depth breakdown of the dis-tribution of node degrees, i.e. the number of first neighbors ofeach node and thus the number of contacts with di ff erent indi-viduals the node had (including both o ff enses and families).Consistently with our previous observations, the fraction ofhigh degree nodes (5 + or 10 + ), i.e. repeated o ff enders, has alsobeen growing steadily, but a temporary drop following the trainschedule change. In Figure 8b we display the distribution of in-dividual exposure times pruned of family contributions, and upto the distance thresholds r d (Eq. (5)), i.e. the pdfs of T dp , f (cf.Eq. (11)). Similarly to what discussed in [42], and consistentlywith the model in [43], we observe a power-law distribution inthe exposure times (exponent p < − ff ect and, possibly, it also weakly influences the exponent. Itis worth remarking that our largest observation times are bound by the fact that we limit our graphs to time intervals of about10-15 minutes around each train arrival. This reduces our res-olution at large time scales and thus yields the exponential-likedrop in the distribution tails. Crowd density: incidence of family-groups and e ff ective of-fenders. The passenger density on the platform has an influ-ence on the o ff enses: the higher the density the easier it gets toviolate physical distancing. In Figure 9a we report for a sampleday (12 th of June 2020), how the percentage of “family” nodesand o ff enders scales with the density within the global densityinterval [0 , .
5] ped / m . As in Eq. (13), we also include mini-mum contact time thresholds, α , for tagging o ff enders. We ob-serve that the percentage of nodes belonging to family-groupsremains stationary (value ≈ ff enses assuming no min-imum time threshold ( α = α = α =
30 s. In particular, such percentageremains stationary (value ≈ . / m and thensuddenly increases. This can suggest an increase in di ffi cultyin following distancing rules around this density level. We re-port the coe ffi cients of the linear fitting of such data in Table 1.
7. Discussion
We have presented an highly e ffi cient and accurate approachto the problem of studying, in real-time, the distance-time en-counter patterns in a crowd of individuals. Our approach al-lows us to identify social groups, such as families, by imposingthresholds on the distance-time contact patterns. In the contextof the currently ongoing Covid-19 pandemic, we demonstratethis as an e ff ective and promising tool to monitor, in a full10. N ode pe r c en t age [ % ] Density [p/m ]Nodes with any contactIndiv. nodes exposedIndiv. nodes with exposure > 10sIndiv. nodes with exposure > 30sNodes part of family-group b.
60 65 70 75 80 85 90 95 100 - -
27 2020 - -
04 2020 - -
11 2020 - -
18 2020 - -
25 2020 - -
01 2020 - -
08 2020 - -
15 2020 - -
22 2020 - - % o f t o t a l Solo’s Duo’s Trio’s Groups
Figure 9: (a) Percentage of pedestrian nodes exposed to contacts as a function of the global density on the platform (density calculated as number of people in aframe divided by the total sensor area, 450 m , discounted of the danger zone, 96 m ). Exposed nodes that have at least one contact, of any duration, with anotherpedestrian (within or outside their family-group or not) are in blue. This percentage if further broken down into nodes part of a family group (red) and actual distanceo ff enders (green). The purple and orange lines restrict, respectively, to nodes with a minimum exposure time of 10 s and 30 s. Linear fitting parameters are reportedin Table 1. (b) Distribution of individuals and cliques day-by-day as a percentage of the total number of nodes. Between 80% and 85% of the nodes do not belongto cliques, i.e. they travel alone and their contacts are all distance infringements. Family-groups of two people cover about 12% −
15% of the remaining nodes;family-groups of three and more provide a minimal ≈
3% contribution.Table 1: Linear fit parameters for the node classification (percentage data) inFigure 9a. As a measurement unit for the pedestrian density we employ tenthsof pedestrians per square meter : δ = − ped / m . Thus the fitting interceptis at δ = δ unit. As an example, pedestrians that do not belong to a family-group andare in contact with someone else grow about 13% when the density increasesbetween 0 . . / m and similarly between 0 . . α =
30 s as the growth isnon-linear. δ ≈ /δ Ped.-nodes exposed 17.1 % 12.9 % /δ Ped.-nodes with exposure α >
10 s 9.56 % 12.4 % /δ Ped.-nodes part of a family-group 11.5 % 0.48 % /δ privacy respectful way, the observation of physical distancing.The outcome of the analysis can provide early warnings in re-spect to an average relaxation towards the compliance of phys-ical distance rules, can allow to identify spots where physicaldistancing most frequently is violated and it may, as well, al-low to identify in real-time the presence of distance o ff enders.We observed, besides, a super-linear dependence between con-tact times and passenger number. This can be caused both by areduction of attention towards social distancing rules but alsoto an intrinsic increase in di ffi culty in complying to regula-tions. The investigation of this aspect is left to future research.The proposed algorithm is simple and can be easily imple-mented using existent graph code libraries. In our case, wecould process a day of data in few minutes using the pythonNetworkX library [44]. Libraries sporting higher performanceand / or scalability exist in case of even more demanding situa-tions.It is worth mentioning that the approach here proposed canbe applied to any type of tracing trajectories and possibly to study the collective dynamics of large groups of active or pas-sive particles making it a tool capable of going well beyondthe application to crowd dynamics and physical distancing dis-cussed here. References [1] World Health Organization, COVID-19: physical distancing, Tech. rep.(2020).URL [2] H. Tian, Y. Liu, Y. Li, C.-H. Wu, B. Chen, M. U. G. Kraemer, B. Li,J. Cai, B. Xu, Q. Yang, B. Wang, P. Yang, Y. Cui, Y. Song, P. Zheng,Q. Wang, O. N. Bjornstad, R. Yang, B. T. Grenfell, O. G. Pybus, C. Dye,An investigation of transmission control measures during the first 50days of the COVID-19 epidemic in China, Science 368 (6491) (2020)638. doi:10.1126/science.abb6105 .[3] B. Rader, S. Scarpino, A. Nande, A. Hill, R. Reiner, D. Pigott, B. Gutier-rez, M. Shrestha, J. Brownstein, M. Castro, H. Tian, O. Pybus, M. Krae-mer, Crowding and the epidemic intensity of COVID-19 transmission,medRxiv doi:10.1101/2020.04.15.20064980 .[4] R. M. Anderson, H. Heesterbeek, D. Klinkenberg, T. D. Hollingsworth,How will country-based mitigation measures influence the course ofthe covid-19 epidemic?, Lancet 395 (10228) (2020) 931–934. doi:10.1016/S0140-6736(20)30567-5 .[5] M. Chinazzi, J. Davis, M. Ajelli, C. Gioannini, M. Litvinova, S. Merler,A. Pastore y Piontti, K. Mu, L. Rossi, K. Sun, et al., The e ff ect of travelrestrictions on the spread of the 2019 novel coronavirus (covid-19) out-break, Science 368 (6489) (2020) 395–400. doi:10.1126/science.aba9757 .[6] Sensormatic, Privacy-Preserving Contact Tracing, Tech. rep. (2020).URL [7] E. Cristiani, B. Piccoli, A. Tosin, Multiscale Modeling of Pedestrian Dy-namics, Vol. 12 of Modeling, Simulation and Applications, Springer,2014.[8] D. Helbing, Tra ffi c and related self-driven many-particle systems, Rev.Mod. Phys. 73 (4) (2001) 1067. doi:10.1103/RevModPhys.73.1067 .[9] J. Adrian, M. Amos, M. Baratchi, M. Beermann, N. Bode, M. Boltes,A. Corbetta, G. Dezecache, J. Drury, Z. Fu, et al., A glossary for researchon human crowd dynamics, Collective Dynamics 4 (A19) (2019) 1–13. doi:10.17815/CD.2019.19 .
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