Can passenger flow distribution be estimated solely based on network properties in public transport systems?
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Abstract
We present a pioneering investigation into the relation between passenger flow distribution and network properties in public transport systems. The methodology is designed in a reverse engineering fashion by utilizing passively measured passenger flow dynamics over the entire network. We quantify the properties of public transport networks using a range of centrality indicators in the topological representations of public transport networks with both infrastructure and service layers considered. All the employed indicators, which originate from complex network science, are interpreted in the context of public transport systems. Regression models are further developed to capture the correlative relation between passenger flow distribution and several centrality indicators that are selected based on the correlation analysis. The primary finding from the case study on the tram networks of The Hague and Amsterdam is that the selected network properties can indeed be used to approximate passenger flow distribution in public transport systems to a reasonable extent. Notwithstanding, no causality is implied, as the correlation may also reflect how well the supply allocation caters for the underlying demand distribution. The significance and relevance of this study stems from two aspects: (1) the unraveled relation provides a parsimonious alternative to existing passenger assignment models that require many assumptions on the basis of limited data; (2) the resulting model offers efficient quickscan decision support capabilities that can help transport planners in tactical planning decisions.
Keywords
Public transport systems Passenger flow distribution Network properties Topology Centrality Complex network scienceIntroduction
Estimation and prediction of passenger flow distribution is one of the most significant topics in the field of public transport (PT) research given its critical role in assisting planning and management. The conventional approach, like that in the road traffic research, is to develop passenger assignment models which take demand profiles—typically in the form of origin–destination matrices—as input and then distribute the demand across the network (Ortúzar and Willumsen 2011). These models are normally referred to as transit assignment models in the transport research community, and their core pertains to modeling travelers’ route choices in PT systems as functions of network conditions and travel preferences (Liu et al. 2010). Two types of static equilibrium transit assignment models have been mostly developed over the past decades, namely the frequencybased and schedulebased. The major distinction between them lies in the representation of public transport networks (PTNs) given their substantial impact on the passenger loading procedure (Gentile et al. 2016). More specifically, the frequencybased approach represents PTNs at the routelevel with corresponding frequencies (e.g., Nguyen and Pallottino 1988; Spiess and Florian 1989; Cepeda et al. 2006; Schmöcker et al. 2011), while the schedulebased one enables a more detailed representation of timedependent specific vehicle runs (e.g., Nuzzolo et al. 2001; Zhang et al. 2010).
Notwithstanding the continuous development of transit assignment models, other possibilities for understanding and further modeling the passenger flow distribution in PT systems to a networkwide extent have remained underexplored. Presumably, this is a result of the longstanding data scarcity in the field. Under this “datapoor and assumptionrich” situation (Vlahogianni et al. 2015), the conventional modeling approach has undoubtedly provided the most feasible solution to this challenging problem. Nonetheless, as the capability in measuring the PT passenger flow dynamics in a large spatiotemporal scale becomes increasingly available owing to emerging PT demand data sources, such as the automatic fare collection data (Pelletier et al. 2011), it is now worth investigating whether there can be alternative ways to model the passenger flow distribution in PT systems.
This study hence examines a research question: Can passenger flow distribution be estimated solely based on network properties in PT systems? While the answer to this question looks apparent, it has not been empirically investigated to a sufficient extent. One can make an underlying assumption that transport network properties should of course correspond to passenger flow distribution since networks are supposedly designed to efficiently accommodate prevailing demand patterns in PT systems (van Nes et al. 1988). However, it shall be stressed that a range of other factors, such as travelers’ behavior, historical network development and physical constraints, also have nonnegligible influences on demand and network structure in any transport systems. In fact, the discussion about whether traffic flows can be approximated by network properties in urban street networks has lasted for decades among urban planning researchers (e.g., Hillier et al. 1993; Penn et al. 1998; Turner 2007; Jiang and Liu 2009; Kazerani 2009; Gao et al. 2013). Recent evidence was provided by Gao et al. (2013) based on the traffic volume derived from the GPSenabled taxi trajectory data from a Chinese city. Their study concludes that the betweenness centrality, which has been commonly employed as a local indicator of network properties, is not a good predictive variable for urban traffic flow and the gap can be explained by the spatial heterogeneity of human activities and the distancedecay law. In addition, a limited amount of research attempts have also been made by scholars from various fields in the past few years to examine the relation between network properties—mostly limited to the betweenness centrality—and the traffic flows in urban road traffic systems. (e.g., Altshuler et al. 2011; Puzis et al. 2013; Ye et al. 2016; Zhao et al. 2017; Wen et al. 2017; Akbarzadeh et al. 2017). No such comparable effort, however, has been made in the context of PT systems, which therefore necessitates dedicated investigations into the proposed research question above.
To this end, we conduct this study with the methodology developed in a reverse engineering fashion, which unravels the correlative relation between passenger flow distribution and network properties in PT systems. Differing from previous studies, we examine a variety of network properties by considering the centrality indicators in different topological representations of PTNs. We show how concepts originating from the domain of complex network science can be applied and interpreted in the context of PT systems. We further apply the proposed methodology to two realworld tram networks in The Netherlands, i.e., The Hague and Amsterdam, where passenger flow observations are available. Regression models capturing the correlation between passenger flow distribution and several centrality indicators are first developed using the data from The Hague, and are then evaluated for both networks separately. Note that no causality is implied by the models. It is the correlation—rather than the causation—between PTN properties and passenger flow distribution that is essentially investigated. Moreover, the unraveled relation and developed models have the potential to serve as a complementary tool for PT operations management, while it is inappropriate to apply them to the longterm passenger flow forecasting.
The remainder of this paper is organized as follows: second section displays the proposed methodology. Third section describes the case study networks and experimental setup, which is followed by the presentation of the results and discussion in fourth section. The conclusions are drawn in final section with some remarks on the future research directions.
Methodology
Overview
Representation of public transport networks
Based on the fundamental representation of PTNs, we further apply two topological representations, the \(\mathbf {L}\) and \(\mathbf {P}\)space (von Ferber et al. 2009), to characterize the topology of PTNs’ two different layers, i.e., infrastructure and service. These topological networks, which can be represented by adjacency matrices, are suitable inputs for further analyses. As Fig. 2 illustrates, the \(\mathbf {L}\)space is a straightforward representation of PTNs’ physical infrastructure. Each node represents a stop, and a link between two stops is formed if two stops are adjacent on an infrastructure segment (i.e. road or rail). Moreover, duplicate connections between nodes are not allowed. The \(\mathbf {P}\)space is constructed solely based on the service layer designed by PT operators/agencies, i.e., routes. The nodes in this space also represent stops, and two nodes are linked if they are served by at least one common route. In this sense the neighbors of a node in this space are all stops that can be reached without performing a transfer. In order to make the use of these two topological representations more informative in the context of this study, we replace the terms “\(\mathbf {L}\)space” and “\(\mathbf {P}\)space” with “spaceofinfrastructure” and “spaceofservice” in the remaining of this paper.
Further enrichment of the topological networks of PTNs is performed by adding link weights related to PT service attributes. The spaceofinfrastructure is enriched in two ways, including the invehicle travel time as a type of link cost and vehicle frequency per time unit as a type of link importance. With common routes considered, the weight of a link’s ultimate frequency is determined by summing up the frequencies of all the routes traversing it, i.e., labeling the link with the respective joint frequency, which is consistent with the definition of spaceofinfrastructure representation. For the spaceofservice, the expected waiting time for a PT vehicle during a given time slice is considered as a type of link cost, which is defined as half of the planned headway with joint vehicle frequency between stop pairs considered. This definition is based on the assumption that (1) passenger arrival at stop is random in the context of urban highfrequency services, and (2) arrival times of vehicles serving different lines is independent, i.e. no systematic synchronization is performed in the context of urban highfrequency services. Both unweighted and weighted topological networks will be used in the following subsection.
Independent variables: timedependent centrality indicators of PTNs
Since the introduction of the “centrality” concept by Bavelas (1948), a variety of network centrality indicators have been proposed in the past decades. In principal, all these indicators are designed to capture distinct aspects of what it means to be “central” in a network for individual nodes. Based on this concept, this study employs several different centrality indicators for both spaceofinfrastructure and spaceofservice networks as the proxies of different properties of PTNs. The combination of different topological representations and centrality indicators enables a concise way to quantify a range of fundamental properties of PTNs. Moreover, some centrality indicators are computed in timedependent weighted networks, which correspondingly reflect timedependent characteristics of PTNs.
Summary of the centrality indicators used in this study
PTN representation  Notations  Centrality indicators  Weight  Weight attributes 

Spaceofinfrastructure  \({{\mathbf {d}}^{{\mathbf {L}},+/}}\)  In/outdegree  ✗  – 
\({\tilde{{\mathbf {d}}}^{{\mathbf {L}},+/}}\)  In/outdegree  ✓  Vehicle frequency  
\({{\mathbf {b}}^{{\mathbf {L}}}}\)  Betweenness  ✗  –  
\({\tilde{{\mathbf {b}}}^{{\mathbf {L}}}}\)  Betweenness  ✓  Invehicle travel time  
\({{\mathbf {c}}^{{\mathbf {L}},+/}}\)  In/outcloseness  ✗  –  
\({\tilde{{\mathbf {c}}}^{{\mathbf {L}},+/}}\)  In/outcloseness  ✓  Invehicle travel time  
Spaceofservice  \({{\mathbf {d}}^{{\mathbf {P}},+/}}\)  In/outdegree  ✗  – 
\({{\mathbf {b}}^{{\mathbf {P}}}}\)  Betweenness  ✗  –  
\({\tilde{{\mathbf {b}}}^{{\mathbf {P}}}}\)  Betweenness  ✓  Waiting time 
In/outdegree centrality

\({{\mathbf {d}}^{{\mathbf {L}},+/}}\): In/outdegree centrality in the unweighted spaceofinfrastructure network
This indicator corresponds to the number of road or rail links that directly lead in or out of a given stop. It thus directly relates to the underlying physical infrastructure of PTNs.

\({\tilde{{\mathbf {d}}}^{{\mathbf {L}},+/}}\): In/outdegree centrality in the weighted spaceofinfrastructure network
Links are weighted by the timedependent vehicle frequency between two adjacent stops with all the routes considered. Hence, this indicator quantifies the scheduled service intensity in terms of PT vehicle flows.

\({{\mathbf {d}}^{{\mathbf {P}},+/}}\): In/outdegree centrality in the unweighted spaceofservice network
This indicator measures the number of stops that can be reached without transfer for a given stop. It thus directly relates to the underlying service design of PTNs.
Betweenness centrality

\({{\mathbf {b}}^{{\mathbf {L}}}}\): Betweenness centrality in the unweighted spaceofinfrastructure network
The share of shortest paths that traverse a certain stop when path length is measured in terms of the number of stops traversed. Given some evidence (Guo 2011), this indicator may coincide with how travelers choose their routes in complex PTNs using the map provided by agencies/operators as a mean to approximate travel time.

\({\tilde{{\mathbf {b}}}^{{\mathbf {L}}}}\): Betweenness centrality in the weighted spaceofinfrastructure network
With the network weighted by the invehicle travel time, this indicator corresponds to the share of shortest paths in terms of onboard travel time that traverse the respective stop. Note that no regard is made to line configuration and thus the number of transfers induced.

\({{\mathbf {b}}^{{\mathbf {P}}}}\): Betweenness centrality in the unweighted spaceofservice network
This indicator relates to the interchange (hub) function of the respective stop. It therefore pertains to one of the most important and unique properties of PT systems, namely transfers.

\({\tilde{{\mathbf {b}}}^{{\mathbf {P}}}}\): Betweenness centrality in the weighted spaceofservice network
The share of shortest paths measured in terms of the average waiting time that traverse a given stop. The path cost consists of the waiting time at the first stop for the route chosen and the waiting time at all subsequent transfer locations.
In/outcloseness centrality

\({{\mathbf {c}}^{{\mathbf {L}},+/}}\): In/outcloseness centrality in the unweighted spaceofinfrastructure network
This indicator quantifies the phenomenon that passengers originating from the topologically central stops can reach the others in the network with fewer intermediate ones.

\({\tilde{{\mathbf {c}}}^{{\mathbf {L}},+/}}\): In/outcloseness centrality in the weighted spaceofinfrastructure network
The weight is determined by the scheduled invehicle travel time, thus making the shortest path more related to the PT service.
The closeness centrality in the spaceofservice network is not included in model development because it reflects a concept very similar to the one obtained through the degree centrality in the same space (\({{\mathbf {d}}^{{\mathbf {P}},+/}}\)), namely identifying the stops that are most reachable with the least number of transfers.
Dependent variable: timedependent passenger flow distribution
The timedependent passenger flow distribution at PT stops is leveraged as the dependent variable, denoted by \(\mathbf {q}\). Here we define the passenger flow at a stop in PTNs as the sum of inflow, outflow and throughflow at this stop during specified time slices. Specifically, inflow and outflow respectively represent the amount of passengers entering (boarding)/exiting (alighting) the PT system at a stop, while throughflow represents the amount of passengers that pass through a stop without leaving PT vehicles. This definition of the passenger flow sufficiently characterizes how intensively the stops are used across the network. In addition, the absolute passenger flows are converted into relative terms, i.e. flow share, at each stop divided by the sum of all stop flows across the network during the respective time slices. we do not attempt to directly predict absolute flow values based on scaled centrality indicators because the same centrality value may correspond to different contexts for different networks and time periods. Instead, we examine whether the distribution of passenger flows is correlated with service properties by considering each stop and timeperiod as a single observation. Absolute flow values are resorted by multiplying flow shares by the total passenger flow in the network.
Model development
The model development is performed in two steps, with the first being an exploratory analysis among variables based on the Pearson correlation coefficient, and the second being building regression models. The objective of the first step is to find out (1) which independent variables (centrality indicators) have higher correlation with the dependent variable (passenger flow distribution), and thus can be incorporated into the models to be developed; (2) the collinearity among independent variables. This is to ensure that variables that are mutually linearly correlated are not included in the models at the same time so that the developed models are as parsimonious as possible.
Model evaluation
Studied networks and experimental setup
Networks and data
Summary of the studied tram networks
Basic properties  The Hague  Amsterdam 

Nodes  232  192 
Directional routes  28  24 
Links in spaceofinfrastructure  520  418 
Links in spaceofservice  8901  6122 
Experimental setup
For the experimental setup, we selected 20 working days with normal demand patterns (out of 1 month) for The Hague. 15 working days were further randomly selected for the model development, with the data aggregated on an hourly basis from 6 a.m. to 12 a.m. (18 time slices). The rest 5day data set of The Hague and the 1day data set of Amsterdam were utilized for the model evaluation.
Results and discussion
The results of the exploratory analysis on the two employed networks are first shown in the first subsection. The second subsection then presents the results of model estimation, followed by the model evaluation in the final subsection.
Exploratory analysis
To gain more intuition about the spatial distribution of passenger flow and centrality indicators in the studied networks, the visualizations of them for the weekday morning peak (7 a.m.–8 a.m.) are performed and presented in Figs. 4 (The Hague) and 5 (Amsterdam). Both size and color are used to make the distinction in magnitude remarkable. Outdegree and outcloseness are omitted as they display the same pattern as their counterparts. Through the visualizations, it can be seen that considerable amount of passenger flows are loaded in the central area of both networks, though it is also observable that some corridors used by commuters also undertake a significant amount of flows, such as the one from center to the east in The Hague, and two horizontal corridors in Amsterdam with one on the middle of west and the other on the top of east. We can further notice that the indegree centrality in the weighted spaceofinfrastructure (\({\tilde{{\mathbf {d}}}^{{\mathbf {L}},+}}\)) and the betweenness centrality in both unweighted (\({{\mathbf {b}}^{{\mathbf {L}}}}\)) and weighted (\({\tilde{{\mathbf {b}}}^{{\mathbf {L}}}}\)) spaceofinfrastructure mostly match the flow distribution pattern with clear distinctions among nodes across the networks. Some indicators, including the indegree in the unweighted spaceofinfrastructure (\({{\mathbf {d}}^{{\mathbf {L}},+}}\)) and the incloseness in both unweighted (\({{\mathbf {c}}^{{\mathbf {L}},+}}\)) and weighted (\({\tilde{{\mathbf {c}}}^{{\mathbf {L}},+}}\)) spaceofinfrastructure, show rather plain patterns. Besides, the betweenness in the spaceofservice (\({{\mathbf {b}}^{{\mathbf {P}}}}\) and \({\tilde{{\mathbf {b}}}^{{\mathbf {P}}}}\)) makes the transfer locations in the networks really stand out.
According to Fig. 7, the in/outdegree centrality indicators in the weighted spaceofinfrastructure network (\({\tilde{{\mathbf {d}}}^{{\mathbf {L}},+/}}\)) show the highest positive correlation with \(\mathbf {q}\) in both The Hague and Amsterdam tram systems. This is consistent with the visual patterns from Figs. 4 and 5. It is also intuitive to interpret because the amount of passengers that is moved in the network depends on the PT vehicle flows. The following indicators are the in/outdegree centrality in the unweighted spaceofservice networks (\({{\mathbf {d}}^{{\mathbf {P}},+/}}\)). Note that these two indicators also show high correlation with the previous ones. They are thus not considered when the in/outdegree centrality in the weighted spaceofinfrastructure network are used in the model development.
The group of degree centrality indicators are followed by the betweenness ones. Note that in the case of The Hague (Fig. 7a), the values of betweenness centrality in both of the unweighted and weighted spaceofinfrastructure networks (\({{\mathbf {b}}^{{\mathbf {L}}}}\) and \({\tilde{{\mathbf {b}}}^{{\mathbf {L}}}}\)) are higher than those in the unweighted and weighted spaceofservice networks (\({{\mathbf {b}}^{{\mathbf {P}}}}\) and \({\tilde{{\mathbf {b}}}^{{\mathbf {P}}}}\)). This, nevertheless, is opposite in the Amsterdam system (Fig. 7b). In fact, the betweenness centrality in the spaceofinfrastructure does not seem to be a good proxy to the passenger flow distribution for the Amsterdam tram network. It performs even worse than the closeness centrality in the spaceofinfrastructure. The remaining centrality indicators are presented in the end as they do not show significantly high correlation with \(\mathbf {q}\).
Model estimation
The model estimation was performed using MATLAB, with the RE models estimated using the panel data toolbox developed by Álvarez et al. (2017). Note that the robust standard error estimation of the RE models was computed when accounting for heteroscedasticity. Moreover, the variance inflation factor (VIF), which quantifies the severity of collinearity in a regression model, was also computed for the parameters of Model 3 which includes several independent variables.
Estimation results of the selected models
Independent variables  Model 1 (OLS)  Model 2 (RE)  Model 3 (RE)  

Coef.  Std.Err  tstat  Coef.  Rob.Std.Err  zstat  Coef.  Rob.Std.Err  zstat  VIF  
CONST  \(\) 0.0003  0.0003  \(\) 1.3670  0.0022***  0.0004  5.8880  \(\) 0.0029***  0.0005  \(\) 6.1798  – 
\(b^L\)  1.0799***  0.0397  27.2241  –  –  –  0.5951***  0.0699  8.5133  3.7495 
\(\tilde{d}^{L,+}\)  –  –  –  0.4849***  0.1168  4.1511  0.1135***  0.0554  2.0491  5.1796 
\(c^{L,+}\)  –  –  –  –  –  –  0.8419**  0.1244  6.7693  1.8418 
\(b^P\)  –  –  –  –  –  –  0.1140***  0.0089  12.7630  2.5844 
Num. Obs.  232  4176  4176  
\(R^2\)  0.7632  0.85723  0.89954  
Adj \(R^2\)  0.7621  0.85720  0.89944 
Model evaluation
The estimated models are evaluated for the tram networks of The Hague (evaluation dataset) and Amsterdam. The results are summarized in Table 4. Note that the evaluation is performed based on the absolute flows obtained by multiplying the predicted relative flow shares by the total amount of flows in the network. Unsurprisingly, Model 3 largely outperforms Model 1 and Model 2 regardless of the metric used. This suggests that models based on a single centrality indicator that does not incorporate information also from the spaceofservice are not able to well capture the correlation. In addition, the discrepancy between weighted and unweighted metrics is striking, implying that significant predictive errors occur to stops with relatively low flows.
Results of the evaluation metrics for the selected models
Evaluation metrics  The Hague  Amsterdam  

Model 1  Model 2  Model 3  Model 1  Model 2  Model 3  
MAE (pax)  184  260  128  335  305  240 
WMAE (pax)  520  841  283  804  715  452 
MAPE (%)  77.6  248.3  70.9  71.4  155.7  68.8 
WMAPE (%)  42.0  58.7  29.1  55.6  50.6  39.8 
The spatial distribution of evaluation errors in both absolute and relative terms are also visualized and presented in Fig. 8. Both negative and positive values are considered in the visualizations, corresponding to underestimations (blue) and overestimations (red), respectively. Plots in Fig. 8b, e show absolute error terms, while those in Fig. 8c, f show relative error terms. In the case of The Hague, it can be observed from Fig. 8c that large relative over or underestimations occur at stops located further away from the center. However, these relatively large errors in relative terms are small in absolute terms as can be seen in Fig. 8b. In absolute terms, flows at stops in the core of the network tend to be underestimated, while flows along corridors that offer cycles between main parts of the network such as along crossradial lines are mostly overestimated. Similar overall patterns are observed in the case of Amsterdam, albeit with larger absolute deviations resulting from larger overall demand levels. Hence, flows in the very central core of the network around the central station and the key tourist attractions are underestimated while the flows along the two halfcircular infrastructure is overestimated (in both relative and absolute terms for both cases).
Conclusions
This paper presents a pioneering investigation into the relation between passenger flow distribution and network properties in public transport (PT) systems. Differing from the traditional approach that consists of demand estimation and assignment, this study is performed in a reverse engineering fashion by directly examining the relation between the observed flow distribution and network properties that are quantified by centrality indicators in various topological representations of public transport networks (PTNs). This research capitalizes on the capability to measure PT systems using passively collected PT data (e.g., AFC, AVL and GTFS). In addition, concepts and methods adopted from complex network science, including the topological representation of PT infrastructure and service networks and centrality indicators, also play a key role in a sense that the combination of them provides a systematic and concise way to quantify the network properties of PT systems. All the employed centrality indicators are also interpreted in the context of PT systems, which enriches the application of complex network science in the transport research.
The major conclusion drawn from the case study on the tram networks from The Hague and Amsterdam is that the selected network properties can indeed be used to approximate the global passenger flow distribution across the network to a reasonable extent of accuracy using solely regression models. This however does not imply causality as it is likely that supply provision has been designed to correspond to demand patterns and therefore the reflects the interplay between demand and supply distributions. Based on the evidence presented in this paper, several research directions can be further explored in the future. First, more realworld PT systems can be employed in order to further validate the finding. Second, the proposed approach can be instrumental in a range of PT applications. This includes conducting fullscan evaluations of the impact of planned disruption on the redistribution of passenger flows throughout the network, which can serve as a good complement to the prevailing tools, i.e., simulation models, at a much lower computational cost and with fewer assumptions. Third, the extent to which PT supply is well designed to reflect passenger flow distribution can be considered as a network performance metric for monitoring system performance over time as well as comparing alternative networks.
Notes
Acknowledgements
The authors thank HTM and Stichting OpenGeo respectively for providing the AFC and AVL data sets of The Hague. The provision of AFC and AVL data sets of Amsterdam by GVB and the help of Dr. Ties Brands during the process are also acknowledged. We acknowledge the support of the SETA project funded by the European Union’s Horizon 2020 research and innovation program (Grant No. 688082). An earlier version of this paper was presented on the Conference on Advanced Systems in Public Transport and TransitData 2018 (CASPT2018).
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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