Intra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace
aa r X i v : . [ phy s i c s . s o c - ph ] M a y Intra-City Urban Network and Traffic Flow Analysisfrom GPS Mobility Trace
Ian X. Y. Leung , Shu-Yan Chan , Pan Hui and Pietro Li`o Computer Laboratory, University of Cambridge, Cambridge CB3 0FD, U.K. Deutsche Telekom Laboratories, Ernst-Reuter-Platz 7, 10709 Berlin, GermanyE-mail: [email protected]
Abstract.
We analyse two large-scale intra-city urban networks and traffic flowstherein measured by GPS traces of taxis in San Francisco and Shanghai. Our resultscoincide with previous findings that, based purely on topological means, it is ofteninsufficient to characterise traffic flow. Traditional shortest-path betweenness analysis,where shortest paths are calculated from each pairs of nodes, carries an unrealisticimplicit assumption that each node or junction in the urban network generates andattracts an equal amount of traffic. We also argue that weighting edges based onlyon Euclidean distance is inadequate, as primary roads are commonly favoured oversecondary roads due to the perceived and actual travel time required. We showthat betweenness traffic analysis can be improved by a simple extended frameworkwhich incorporates both the notions of node weights and fastest-path betweenness.We demonstrate that the framework is superior to traditional methods based solely onsimple topological perspectives. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace
1. Introduction
Complex networks provide a natural abstraction of the structure and relationshipsbetween entities in real-world systems. From the study of social relationships tobiological systems, the study of Network Science has provided abundant tools andresearch opportunities for the understanding and modelling of such complex systems.Urban networks, those pertaining to city infrastructures, have been traditionally subjectto investigation from the fields of Urban Planning, Economic Geography, Economics,and Engineering. In a common node-link nomenclature, road junctions are consideredas nodes while roads are treated as edges connecting the nodes in the network. Typicalreal-world networks from the realms of social and biological systems are famouslyknown to exhibit properties such as scale-free degree distributions and the small worldphenomenon, where despite their large size, the average distance between any pairsof node remain relatively small. On the other hand, urban networks are unlike theaforementioned networks due to spatial and geographical constraints. They are knownto exhibit a smaller average degree and longer diameter, due to them being almostplanar [15, 7].Studies from both Urban Planning and Network Science have revealed interestingcorrelations between topological properties of urban networks to human related activitiesin a wide variety of city topologies [6, 30]. We are interested in further understandingthe relationship between topological properties and traffic flow with the support oflarge-scale mobility traces. Previous works have shed light into topological-based trafficprediction, but either focussed on unrealistic or incomplete traffic estimation or was donein a small and coarse scale. The development of cheap and portable Global PositioningSystem (GPS) has greatly advanced the state of the art of understanding humanmobility. By mining large-scale and real-time tracking datasets, one is able empiricallyevaluate important hypotheses. Such tools have for instance eased communicationnetwork deployment, e.g. network operators can choose to deploy more network facilitiessuch as Wi-Fi access points or relay nodes for mobile communication at areas with hightraffic flow [18].The purpose of this paper is to analyse various network-based metrics and theirability to predict traffic flow based on GPS mobility trace data in two major cities.We make use of two sets of GPS mobility trace data that were gathered from morethan 500 taxis in the San Francisco Bay area for 24 days and more than 4,000 taxisin Shanghai city for 28 days. We propose an effective centrality-based methodologycombined with minimal locale information in an attempt to predict the traffic flowin the road networks. We discuss four centrality metrics—namely, geodesic shortest-path betweenness, Euclidean shortest-path between, fastest-path betweenness, node-weighted fastest-path betweenness, and their relationships to traffic flow. We arguethat based solely on topological properties of urban networks, as supported by previouswork, provides a limited prediction power of traffic flow in a city. The accuracy can beenhanced given minimal information on location liveliness, for example by incorporating ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace
2. Related Work
Typical topology and patterns in spatial networks [15] such as the roads, Internet,and flight networks are unlike their well studied non-spatial counterparts (e.g. social,biological, and technological networks), which disregard the physical geography of thenodes and exhibit various important properties [1]. Urban spatial networks, due to thegeographical constraints, are typically planar, i.e., they can be drawn on a 2D planewithout any edges crossing due presumably to construction constraints. Planar graphscan be shown to have an average degree strictly less than 6 and a diameter which growsfaster than non-spatial real-world networks ( √ N as opposed to log N in a network ofsize N ) without a well defined low dimension (See [15]).L¨ammer et al. [27] reported the effective dimensions ranged from 2 to 2.5 in 20Germany city road networks as well as a power-law betweenness centrality distributionover the nodes. A deduction was then made that majority traffic volume is concentratedon a minority of roads but this was not supported by real evidence. Cardillo et al. [7]analysed unit-square mile tile samples of 20 different cities and reported that citiesexhibited meshedness , an alternative to the clustering coefficient used in non-planargraphs, given by the number of faces associated with the planar graph with N nodesand M edges over the maximum possible. Road networks were also reported to beglobally efficient [7], which meant that the actual distance required to travel from anytwo points in the city does not deviate too much from their straight line Euclideandistance. Attempts to characterise and compare different cities were done using theselocal and global properties specifically designed for spatial graphs.Researchers have studied human mobility for the understanding of the natureof human movement which is vital to aspects pertaining to epidemic spreading andcommunication network design [10, 16, 19]. For instance, analysis of human mobilitytraces has demonstrated power-law inter-contact time distributions with cut off [8, 23],levy-flight patterns consisting of lots of small moves followed by long jumps [32],etc. Vehicular mobility has also been known to be important for resource allocationand communication network optimization in a city [3]. Krings et al. [25] used the ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace integration (equivalent to closeness with restrictionof number of hops from the node). Traffic data was collected from official traffic censususing inductive loop and pneumatic tube counters but are restricted to only hundreds ofroads. Correlation as high as 0.8 were achieved but only when small sample areas fromthe entire map were selected. Attempts were also made in [22, 20] to correlate Google’sPageRank [28] on junctions to predict the respective traffic flow. Their results indicatedthat PageRank offered minimal advantage over simple measures based on degree orcloseness. PageRank assumes a random walk model over a network which is intendedto model human web browsing behaviour. However, we believe that, despite incompleteinformation on the road network structure, human tend to know the destination beforesetting off and hence the two processes are fundamentally different.De Montis et al. [12] studied a weighted inter-municipal commuting network in theSardinia region, Italy. Official statistics of traffic between major cities in the regionwas incorporated into the networks. High correlation between the connectivity of themunicipality and commuter traffic was found.Kazerani and Winter [24] argued that shortest-path betweenness centrality wasagainst human way-finding behaviour as most people find their way based on incompleteand inaccurate network knowledge. It was also argued that sources and sinks of trafficwere irregularly distributed in the networks, a fact that is not captured by betweennesscentrality which measures shortest paths between all pairs of nodes in the network. A ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace
3. Methodology
Since the San Francisco taxi traces were taken in 2008 [29], we have obtained the 2008US Government Census road shape files of the San Francisco Bay area [5]. The shape filecontains information of every road in the city as polylines defined by their correspondingGPS coordinates. Where the GPS coordinates of any segment of two different polylinematch, the coordinate is interpreted as a road intersection between the two roads.Each road is also conveniently classified by one of the ten possible road feature classesaccording to the US Bureau of the Census MAF/TIGER Feature Classification Code(MTFCC), of which we only keep three classes: type S1100—Interstate, type S1200—Major Highways and type S1400—Local Neighbourhood Road, Rural Road, City Street.The road network contains 26,049 nodes and 33,079 edges, spanning a total estimateddistance of 1,980km.For the analysis in Shanghai, we have obtained the network shapefiles fromOpenStreetMap ‡ . OpenstreetMap is a public mapping service which allows users toupdate and edit the map freely. We also note with caution that the discrepanciesbetween the map snapshot of Shanghai taken at the time of writing and in 2007 whenthe traces were taken are likely to introduce errors in terms of shortest-path calculationsas well as potential invalid snapping of GPS traces to the roads. The road network ofShanghai contains 54,151 nodes and 61,834 edges, spanning a total estimated distanceof 7,402km.We only take the largest connected component of the resulting networks. Thenetworks are undirected due to data constraint. While knowing the exact flowrestrictions on or between each individual road would be beneficial to our analysis,we leave it as a potential and important future work. For our analysis, we reduce thecomplexity and size of the network while keeping the maximum information using thefollowing scheme. A node/junction of degree two in the original network is either thesame road or a corner of two roads, which serves no other purpose than maintaining theroad’s correct shape on the map. Hence, for every such node B, we remove the nodeand the two corresponding edges (A,B) and (B,C), and establish a new link betweenA and C. We assign a weight to (A,C) equal to the sum of the weights of edges (A,B)and (B,C). If a link already exists between A and C, we assign the new weight to bethe minimum of the old and new weights. We simplify the network for the followingreasons:(i) It greatly speeds up the centrality analysis. ‡ ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace N K h l i W d h k i San Francisco 26,049 33,079 59.9m 1,980km 6.67km 2.54San Francisco (Simplified) 9,791 16,129 116.4m 1,877km 6.60km 3.29Shanghai 54,151 61,834 119.7m 7,402km 20.87km 2.28Shanghai (Simplified) 12,979 20,585 354.1m 7,239km 18.9km 3.17
Table 1.
Key statistics of the two road networks studied. All centrality-based analysesare carried on the simplified version of the networks as described in Section 3.1. Here, N denotes the number of nodes, K is the number of edges, h l i is the average edgelength, W is the total edge lengths, d is the characteristic path length (average lengthof all shortest paths) and h k i is the average degree. The trace for San Francisco consists of roughly 10 million GPS readings (latitude,longitude) from more than 500 taxis in the Bay area over a period of 24 days fromMay 17, 2008 to June 10, 2008 [29]. The trace for Shanghai contains more than 100million GPS readings from more than 4 000 taxis over the entire month of February,2007 [17].We first remove all the traces which were recorded when the taxis were unoccupied.We believe this provides a fairer measure of aggregated traffic flow given by the actualneed of passenger travels. It is believed that unoccupied taxis are highly incentive-oriented and tend to remain in certain areas where passengers are easier to be found(such as commercial areas, stations, airports, etc). Hence, including those traces in thetraffic flow estimate is likely to introduce bias. Secondly, by removing traces when taxisare unoccupied we also remove meaningless data, e.g. when they are waiting at the taxistops. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Figure 1.
The original network representation of the Shanghai road network( top )and the simplified network representation by removing all junctions with node degree2 ( bottom ). ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace trip to be a sequence of GPS traces ofa particular taxi in which each consecutive trace pair is less than 90 seconds apart andimplies a speed < = 120 km/h . Then, for every such valid trace in each trip, we locateits closest edge and increase the traffic count of the two corresponding junctions by one.By definition, each unique trip can only visit a junction once and hence traffic count foreach unique junction can only be increased once per trip.Due to the time resolution of the data, it is often the case that each consecutivetrace pair is several junctions apart. If we follow the simple scheme above, we risklosing valuable information of the entire trajectory taken by each taxi during the trip.We employ two methods to interpolate the trajectory of the traces when they are toofar apart. We first attempt to estimate the actual edgetraversed based on the fastest path between the two nodes. Where two nodes are morethan one hop apart, we run Dijkstra’s algorithm from either end to devise a fastestpath between them. The final path profile is a sequence of connected junctions whichapproximates the actual path of the taxi during that trip. Similarly, we increase thetraffic flow count of each junction in the interpolated path by one. This estimationtechnique is based on the assumption that taxis always take the shortest/fastest routefrom source to destination. Unfortunately, this interpolation is biased towards ourbetweenness-based prediction (Section 3.3.2) as betweenness is itself based on theshortest/fastest-path assumption. Since the average speed and time interval indicatean average distance of 500m between each consecutive trace, this estimation is highlyvulnerable to the bias especially in the case where the trajectory is not straight. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Our second technique involves interpolating thetrajectory of the entire trip by submitting each consecutive trace pair to the MapQuestDirections API service § with the default “Fastest Route” option enabled. Given thecomprehensive information on each individual road which online routing services suchas MapQuest possess, we believe that its fastest-path predictions can keep the errors toa minimal. Indeed, it is implicitly assumed that the route returned is the one picked bythe taxi driver which again is not always true. However, we believe that the resolution ofthe traces and the criteria we employ to merge the traces into individual trips keep thepotential errors to a minimum while retaining as much traffic information as possible.Here, the interpolated trajectory for each trip is subjected to further filtering as itis noted that some routes returned by MapQuest were affected by GPS snapping errors.In some cases, a route was suggested which would mean the taxi had travelled the entirea length of the road and to come back to a point nearby, just because the consecutiveGPS trace was snapped to the same road going in the opposite direction. Again, athreshold is set such that the interpolated trip could not imply a travel speed of morethan 120km/h by the taxi.Finally, for each valid trip, the traffic count on each of the distinct interpolatedjunctions is increased by one. This is similarly based on the assumption that in asingle trip, a taxi does not visit the same junction twice. Figure 2 depicts graphicallythe overall traffic flow in the key commercial districts of the two cities based on thisinterpolation technique.Figure 3 shows a distribution of traffic flow count per junction in the two cities. Apower-law decay trend with a cut off (which can be explained by the finite nature oftraffic flow count) is observed. Also, it indicates that only a tiny fraction of junctionscarry a relatively substantial traffic load in both cities. Both degree and closeness centralities were extensively used under the study of SpaceSyntax, and in many cases were found to significantly correlate to traffic flow, safetyagainst criminality, commerce activity, activity separation and pollution [11]. We shallpresent the experimental results including the aforementioned centrality measures in thenext section. Here, we focus our discussion on betweenness centrality as well as some ifits extensions for traffic networks which will be introduced in due course. refers tothe simplest definition of betweenness, i.e., the fractions of shortest paths between anypairs of nodes in the network which go through a particular node: C B ( v ) = X u,t ∈ V σ ut ( v ) σ ut , (1) § http://developer.mapquest.com/web/products/open/directions-service ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Figure 2.
Traffic flow (interpolated by MapQuest routing service) in the commercialdistrict of San Francisco ( top ) and Shanghai ( bottom ). For visualisation purpose,the traffic count for each edge is given by the average of its two respective ends.Red coloured edges correspond to a high number of traffic counts and blue colourcorresponds to a close to zero traffic count. where V is the set of nodes in the network, σ ut is the number of shortest paths betweennodes u and t , and σ ut ( v ) is the number of those going through node v . Here the networkis treated as a non-spatial network, where each edge corresponds to a single hop from onenode to another with equal weight. For its derivation, we employ the original Brandesalgorithm [4] which carries out a breath-first search from each node in the network andaccumulate the shortest path counts on each node that is on the shortest path. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Traffic Flow Count (x 100) P r obab ili t y D en s i t y ( x - ) San FranciscoShanghai
Figure 3.
Traffic flow distribution per junction in San Francisco and Shanghai, withtrajectories interpolated by MapQuest routing service.
The drawback of geodesic shortest-path is that it does not take into account the physical distance of the spatial networkwhen devising the shortest path. It is highly sensitive to short edges which are commonin spatial networks due to the large number of intersections of different roads. Traversingon such networks generally requires a relatively large number of hops compared to non-planar networks. A simple way to circumvent this problem is to take into accountthe Euclidean distance of each edge when calculating all the possible pairs of shortestpaths. We denote this as the
Euclidean shortest-path betweenness . The distributions ofthis betweenness of each junction in the two cities are shown in Figure 4.One can further improve the resemblance to traffic decision by including speedconsideration of different roads in the city. As with most route decision made by humanor computer, the fastest route is often favoured over the shortest route. We thereforeexperiment with replacing the weight of each edge by an estimate of the travel timerequired based on its pre-labelled feature class. In most modern cities, the speed limitin typical residential or city streets is 50km/h while that in highways normally doublesthe former. For the sake of brevity, we halve the weights of all edges specified asinterstate/highways/primary/trunk roads in the network to accommodate for an averagedriver decision on choosing the faster routes. Understandably, route decision should alsodepend greatly on the time of day, the width, permeability [31], or potential chargeson using the routes. A full analysis would therefore require a better understanding of ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Betwenness (x 10 ) P r obab ili t y D en s i t y ( x - ) Figure 4.
Distributions of Euclidean shortest-path betweenness centrality in the twocities. route decision made by every individual. We call this
Euclidean fastest-path betweenness .Similar attempts in capturing betweenness which the shortest-path assumption does notalways hold have been investigated in the realms of packet routing in communicationnetworks [13].For both the aforementioned measures, we employ the weighted version of Brandesalgorithm for its derivation, with all nodes set to the same weight (see Algorithm 1).
It has soon come to our attention thatpure topological measures which ignore locale information would fail to capture theactual traffic flow due simply to the fact that, after all, places or nodes which are moredensely populated or commercially active inevitably generate and attract more traffic.Without resorting to a full blown population or commercial density map and in thespirit of using as little accessible information as possible, we decide to choose restaurantdensity as a measure to give each node a corresponding weight. Restaurants and theirexact locations are readily available information on a lot of online or offline directories.While there is a potential bias to restaurants which publish themselves online, we believeit provides a fair estimate of traffic generation and attraction as restaurants are designedto serve in places where people are around.For each restaurant, we pre-assign to each junction within its 150m radius a weightinversely proportional to the number of junctions in that radius. This assumes thatevery restaurant carries the same traffic attracting and generating factor which is spread ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace node-weighted fastest-path betweenness framework follows closely the algorithmproposed in [9] which is based on a modification of the original Brandes algorithm totake into account of node weights. We present the full algorithm in Appendix A. The keyobservation is on line a of the algorithm, where we adjust the number of shortest-pathcounts based on the node weights during the back propagation step from destination tosource. For the sake of brevity, our assumption is that the overall traffic between twonodes can be estimated by the multiple of the two respective weights.As an example, we have located 4,066 and 53,748 restaurants that are within thearea of interest in San Francisco and Shanghai respectively. Figure 5 shows the densityof restaurants in the San Francisco Bay area. Note that again these restaurants arecurrent information and therefore may not reflect truly the distribution at the time thetraces were taken.
We evaluate the quality of predictions by devising the Pearson correlation coefficientbetween the traffic count and the predicted magnitude. The Pearson correlationcoefficient, or Pearson’s r , is a value ranged from -1.0 to 1.0 which measures the lineardependence of two variables. A value close to zero means that the two variables arenot correlated at all, i.e., knowing one does not tell much about the other. Conversely,a value close to -1.0 or 1.0 indicates that there is a perfect linear relationship betweenthe two variables. For each junction, we therefore calculate 7 different measures forcomparisons: its degree, closeness k , number of restaurants in its 150m radius, and thefour betweenness measures.While node-based correlation is simple and intuitive, it has several drawbacks.Firstly, it is susceptible to noise caused by routing inaccuracy due to insufficientinformation of the road structure. Consider a multi-lane highway which would typicallybe depicted as a number of parallel edges in the network. Disregarding the directional orturning restrictions, the shortest-path based estimation would often pick one and onlyone of these parallel edges to increase the traffic count. Also, routing errors mentionedin the last section due to insufficient time resolution of the GPS traces are bound to leadto noise and inaccuracies in node-based correlations. Finally, despite having removed alljunctions of degree 2 from the network, certain remaining junctions which are physically k The notion of closeness used in our analysis is different to space syntax’s global integration in thatdistance is calculated based on the fastest-path weight between nodes. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Figure 5.
Density of restaurants in the San Francisco Bay area. For visualisationpurpose, the restaurant count for each edge is given by the average of the numberof restaurants surrounding a 150m radius from its two respective ends. Red colourededges correspond to a high number of restaurants in its surroundings and blue colourcorresponds to insignificant number of restaurants. close to each other in the network might exhibit similar behaviours in terms of trafficflow as well as centrality. Therefore, blindly carrying out node-based correlations maybe biased to nodes which are located closely.To improve on these issues, we follow a similar scheme used in [30] by carryingout spatial smoothing on the predictions and traffic counts prior to the correlation, acommon technique known as Kernel Density Estimation (KDE). In essence, for anypoint on the map, a measure is obtained by summing up all the events occurred on thewhole 2D plane weighted inversely by their Euclidean distances from that point. Toachieve that, a kernel function κ , whose shape is symmetric and integral is one, is addedto an overall sum over the plane where an event occurs. The sum therefore forms asurface over the plane with a volume equivalent to the number of events. It is typicallynormalised such that it becomes a probability density function over the 2D plane. For agiven point on a 1-dimensional line with coordinate x , the measure by KDE is commonly ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace f ( x ) = 1 n n X i =1 κ (cid:18) ( x − x i ) h (cid:19) , (2)where n is the total number of events in the entire space, h is the bandwidth, and x i isthe coordinate of the event i . h acts as the smoothing factor—the larger h is, the morelikely a distant event has an effect on the point concerned.Since the coordinates are real-valued, it is common to discretise the 2D plane intopixels or squares prior to running KDE. We follow closely the settings used in [30],setting each pixel to be 10 m by 10 m , and h = 300 m . We employ the standard Gaussiankernel given by: κ ( x ) = 1 √ π e − x . (3)To ease the calculation, we pre-calculate the values of the six predictors, trafficcount, as well as restaurant count for each 10m by 10m pixel. Where there is more thanone node in the pixel, we take the sum of the node-based measures as an aggregatedestimate for the pixel. For restaurant count, the number of restaurants within the pixelis used instead. To further speed up the process, pixels which are beyond the bandwidth h from the pixel concerned are not considered in the summation.To summarise, we first pre-calculate the aggregated estimates for every pixel on themap. Then, for each pixel with coordinates x which contains any junction, we devisethe following metric:˜ f ( x ) = X { x i | d ( x , x i ) ≤ h } w i κ (cid:18) d ( x , x i ) h (cid:19) , (4)where d ( x , x i ) is the Euclidean distance between the concerned pixel with coordinate x and another pixel i with coordinate x i ; w i is the pre-calculated value (e.g. the aggregatedcentrality or traffic count) of pixel i .The correlation analysis only takes into account pixels which contain junctions.We do not correlate pixels that contain only edges (roads) since our knowledge of theedges, i.e., their traffic flows and centralities, is no more than that of their correspondingjunction ends. Hence, given the framework, correlating every pixel which contains anedge may yield an overall biased Pearson correlation.Note that the above measure transforms the effect of an event on the 2D space usinga 1-dimensional Gaussian as we assume an event’s effect is symmetric in any directionon the 2D plane. We do not normalise ˜ f ( x ) as it does not affect the correlations.
4. Results and Discussion
We present the results for our correlation analyses of San Francisco and Shanghai inTables 2 and 3 respectively. For each entry, both the node-based and KDE pixel-basedPearson’s r are included. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace San Francisco - Pearson’s r (Node-based/KDE)No Interpolation Non-Rush Hour Rush Hour OverallDegree 0.272/0.392 0.246/0.388 0.268/0.394Closeness 0.243/0.413 0.19/0.404 0.231/0.413Geodesic betweenness 0.236/0.189 0.191/0.142 0.226/0.177Shortest-path 0.273/0.333 0.201/0.271 0.256/0.318Fastest-path 0.264/0.315 0.203/0.249 0.249/0.298Node-weighted fastest-path 0.591/0.83 0.526/0.767 0.579/0.818Restaurant 0.445/0.717 0.458/0.709 0.453/0.72Fastest-Path InterpolationDegree 0.268/0.521 0.267/0.538 0.269/0.526Closeness 0.256/0.529 0.275/0.545 0.261/0.534Geodesic betweenness 0.204/0.266 0.222/0.314 0.208/0.277Shortest-path 0.252/0.417 0.274/0.469 0.257/0.429Fastest-path 0.386/0.456 0.437/0.531 0.398/0.473Node-weighted fastest-path 0.565/0.753 0.549/0.725 0.564/0.75Restaurant 0.419/0.622 0.381/0.568 0.412/0.613Routing-Service InterpolationDegree 0.218/0.486 0.231/0.46 0.223/0.48Closeness 0.239/0.501 0.273/0.484 0.25/0.498Geodesic betweenness 0.159/0.237 0.179/0.23 0.166/0.236Shortest-Path 0.224/0.392 0.259/0.397 0.235/0.395Fastest-path 0.223/0.383 0.25/0.381 0.232/0.383Node-weighted fastest-path 0.528/0.764 0.573/0.812 0.544/0.779Restaurant 0.465/0.65 0.463/0.66 0.467/0.654 Table 2.
Pearson product-moment correlation coefficients of 6 centrality predictorsand restaurant count against traffic flow in San Francisco. Traffic flow is furtherseparated into rush hour (6AM - 10AM and 4PM - 7PM) and non-rush hour forcomparisons.
A first glance at the results reveals that predictors work differently in the two citiesand across the different interpolation techniques. This is understandable given that thetwo cities have fundamental differences in terms of design and planning—San Franciscohas a well defined grid like structure while Shanghai road structure is more irregularand ad-hoc (self-organised). As explained earlier, the fact that fastest-path interpolationwould by definition favour the fastest-path betweenness predictor is also evident acrossboth tables. KDE-pixel based correlation has in general a much higher correlationcoefficient over the node-based counterpart. As discussed, due to imperfect informationon the road networks and from the raw traces, carrying out spatial smoothing on the ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Shanghai - Pearson’s r (Node-based/KDE)No Interpolation Non-Rush Hour Rush Hour OverallDegree 0.253/0.404 0.273/0.413 0.261/0.408Closeness 0.339/0.466 0.367/0.474 0.35/0.47Geodesic betweenness 0.153/0.505 0.173/0.516 0.16/0.51Shortest-path 0.322/0.642 0.35/0.637 0.333/0.641Fastest-path 0.262/0.684 0.29/0.676 0.272/0.683Node-weighted fastest-path 0.289/0.634 0.311/0.622 0.297/0.631Restaurant 0.256/0.44 0.276/0.436 0.263/0.44Fastest-Path InterpolationDegree 0.265/0.439 0.269/0.438 0.267/0.439Closeness 0.418/0.501 0.414/0.496 0.418/0.5Geodesic betweenness 0.167/0.568 0.179/0.577 0.171/0.572Shortest-path 0.387/0.646 0.388/0.631 0.389/0.642Fastest-path 0.481/0.744 0.476/0.724 0.48/0.738Node-weighted fastest-path 0.491/0.674 0.472/0.641 0.486/0.664Restaurant 0.329/0.39 0.321/0.377 0.328/0.386Routing-Service InterpolationDegree 0.313/0.505 0.309/0.507 0.312/0.506Closeness 0.552/0.583 0.55/0.586 0.552/0.584Geodesic betweenness 0.086/0.429 0.089/0.425 0.087/0.428Shortest-path 0.484/0.751 0.478/0.746 0.483/0.75Fastest-path 0.43/0.814 0.427/0.807 0.43/0.812Node-weighted fastest-path 0.528/0.832 0.515/0.817 0.525/0.828Restaurant 0.438/0.444 0.433/0.438 0.438/0.443 Table 3.
Pearson product-moment correlation coefficients between 6 centralitypredictors and restaurant count against traffic flow in Shanghai. Traffic flow is furtherseparated into rush hour (6AM - 10AM and 4PM - 7PM) and non-rush hour forcomparisons. observed traffic flow and predictions would remove some of the noise from these errors.Nonetheless, we are able to draw several important conclusions from our experiments.Degree and geodesic betweennesses offer minimal prediction power. This is asexpected as the variation of degree centrality in spatial networks is so small that itsdiscrimination power would be low. Geodesic betweenness does not take into accountedge weights and hence is inapplicable for flow prediction in spatial networks where thenumber of hops between source and destination carries little meaning. These coincidewith numerous previous findings that simple topological based measures seem to havean inherent limit in characterising traffic flow. Closeness centrality has a marginal ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace San Francisco (Node−based)
Traffic Count N ode − W e i gh t ed F a s t e s t P a t h B e t w eenne ss San Francisco (KDE−based)
Traffic Count N ode − W e i gh t ed F a s t e s t P a t h B e t w eenne ss Shanghai (Node−based)
Traffic Count N ode − W e i gh t ed F a s t e s t P a t h B e t w eenne ss Shanghai (KDE−based)
Traffic Count N ode − W e i gh t ed F a s t e s t P a t h B e t w eenne ss Figure 6.
Scatter plots of Node-weighted betweenness against the observed andinterpolated traffic count in the two cities. The plots are in log-log scale and clearpositive trends can be observed in all cases. The Pearson’s r from the correlationstudies are reported in Tables 2 and 3.
5. Conclusion and Future Work
In this literature, more than a hundred million GPS taxi traces in two intra-cityurban networks of San Francisco and Shanghai have been analysed with a weightednetwork perspective. We have reviewed relevant literature on spatial network analysis,applications of large-scale human mobility analysis, as well as existing work ontopological-based traffic flow analysis. We have discussed methodologies to effectivelysimplify the urban networks as well as estimate traffic flow from raw data. We haveargued that based on pure topological features, like a lot of previous work did, would beinsufficient to consistently predict traffic flow due to the inherently flawed assumptionwhich some centrality measures carry. A novel framework which allows node weight andtravel speed to be incorporated into traditional betweenness analysis has been shownto greatly improve traffic prediction performance. Based on the topological framework,we have also identified driving behaviours which potentially differ during rush hour andnon-rush hour traffic. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace
Acknowledgments
The Shanghai taxi data was obtained from Wireless and Sensor networks Lab (WnSN),Shanghai Jiao Tong University, for which we are grateful. The city graphs used in thismanuscript are generated using Tulip 3.5.0 [2]. ntra-City Urban Network and Traffic Flow Analysis from GPS Mobility Trace Appendix A. Node-weighted Brandes Algorithm
The detailed implementation of the modified Brandes algorithm which allows weightededges and nodes is given below in Algorithm 1.
Input : G = ( V, E ), demand [ ]( Node weight ), weight ( v, v ′ ) ( Edge weight of (v, v’) ) Output : C B [ ] ( The approximation of node-weighted betweenness of each node ) begin C B [ v ] ← , v ∈ V ; foreach s ∈ V do /*s is the source node in each iteration */ S ← empty stack; /* P[] is a list of predecessors in the shortest path */ P [ w ] ← empty list, w ∈ V ; /* counters for number of shortest paths */ σ [ t ] ← , t ∈ V ; σ [ s ] ← /* distances from source */ d [ t ] ← − , t ∈ V ; d [ s ] ← /* a Priority queue of nodes ordered with increasing d[node] */ Q ← empty queue; enqueue s → Q ; /* Dijkstra’s algorithm, also counting the number of equal distance shortest paths toreach each node */ while Q not empty do dequeue Q → v ;push v → S ; foreach neighbour w of v do /* w found for the first time? */ if d [ w ] < then d [ w ] ← d [ v ] + weight ( v, w );enqueue w → Q ; end /* shorter path to w via v? */ if d [ w ] > d [ v ] + weight ( v, w ) then d [ w ] ← d [ v ] + weight ( v, w ); σ [ w ] ← σ [ v ]; P [ w ] ← new list (); append v → P [ w ]; /* reorder w in Q with the value of d[w] */ Q.decreaseKey ( w ); endelse if d [ w ] = d [ v ] + weight ( v, w ) then /* one of the shortest paths to w via v */ σ [ w ] ← σ [ w ] + σ [ v ];append v → P [ w ]; endendend /* counters for number of shortest paths from s passing through each node*/ δ [ v ] ← , v ∈ V ; /* S returns vertices in decreasing order of distance from s */ while S not empty do pop S → w ; foreach v ∈ P [ w ] do a δ [ v ] ← δ [ v ] + σ [ v ] σ [ w ] · ( demand [ s ] · demand [ w ] + δ [ w ]); if w = s then C B [ w ] ← C B [ w ] + δ [ w ]; endendendendend Algorithm 1:
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