Exploring the association between network centralities and passenger flows in metro systems

doi:10.21203/rs.3.rs-2625640/v1

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Exploring the association between network centralities and passenger flows in metro systems

https://doi.org/10.21203/rs.3.rs-2625640/v1

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published 22 Sep, 2023

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Network science offers valuable tools for planning and managing public transportation systems, with measures such as network centralities proposed as complementary predictors of ridership. This paper explores the relationship between different cases of passenger flows at metro stations and network centralities of both metro and alternative public transport (substitute) networks; such an association can be useful for managing metro system operations when disruptions occur. For that purpose, linear regression and non-parametric machine learning models are developed and compared. The Athens metro system is used as testbed for developing the proposed methodology. The findings of this study can be used for deriving medium-term ridership estimates in cases of metro disruptions, as the proposed methodology can support contingency plans for both platform and rail track disruptions.

Network theory

centrality

metro system

passenger flows

linear regression

machine learning

XGBoost

Metro systems are crucial in metropolitan areas, since they offer fast, convenient and reliable transportation services. Information on ridership (often represented by passenger flows between metro stations) is essential for efficient planning and management of metro operations, such as the development of timetables, allocation of resources and so on. For that purpose, several econometric and machine/deep learning models have been developed for predicting passenger flows in metro systems (Han et al., 2021). The associated problem of estimating metro passenger flows is usually approached by a spatial-temporal perspective, while the temporal component is treated as time-series (Ou et al., 2020). Econometric time-series models are usually based on autoregressive integrated moving average (ARIMA) and seasonal autoregressive integrated moving average (SARIMA) (Zheng et al., 2020), while machine learning and deep learning-based efforts mainly include neural networks and support vector machine models (Sun et al., 2015; Guo et al., 2019; Li et al., 2019; Yang et al., 2021). Machine learning techniques have also been applied to long-term passenger flow estimates (Toqué et al., 2017) and to ridership predictions of new stations and lines (Gong et al., 2020; Wang et al., 2022).

Complex network theory (CNT) has been used for the analysis of traveler flows, including passenger flows in public transportation systems (Zhang et al., 2022). Since there exist correlations between topological properties and human activities, CNT can effectively be used for that purpose (Leung et al., 2011). Indeed, network properties can be useful passenger flow predictors, and network-based models can provide fast and efficient decision models for that purpose (Luo et al., 2020). In this context, complex network characteristics of road networks were recognized by the literature as determinants of traffic flow: their degree of association depends on network structure (Zhao & Zhao, 2016), topological and angular distance-based centralities seem to be more appropriate for predictions than metric distance-based ones (Omer & Jiang, 2015), and multiple centralities can demonstrate better performance than single-variable ones (Pun et al., 2019). Betweenness is the most frequently used centrality for traffic flow predictions, but most researchers agree that it is not sufficient on its conventional form (Kazerani, & Winter, 2009^b; Leung et al., 2011; Gao et al., 2013; Ye et al., 2016). As such, several modifications of betweenness centrality have been proposed (Kazerani & Winter, 2009^a; Galafassi, & Bazzan, 2013; Puzis et al., 2013; Henry et al., 2019; Zhang & Chen, 2020; Cogoni et al., 2023). Additional centralities used as traffic flow predictors include closeness (Jayasinghe & Sano, 2017), degree, PageRank (Zhao et al., 2017) and stress (Lowry, 2014). Similarly, conventional, or modified betweenness is the most common centrality used as a flow predictor in the cases of cycling (Cooper, 2017; Chan & Cooper, 2019; Hochmair et al., 2019) and pedestrian flows (Agryzkov et al., 2019; Cooper et al., 2021; Sevtsuk, 2021).

In public transportation, the association between network properties and passenger flows has been investigated, along with the possibility of using the former as efficient passenger flow predictors. Jayasinghe & Munshi (2014) used closeness, straightness and betweenness centralities, and developed linear regression models to estimate boardings and alightings at public transport stops. Senousi et al. (2022) explored the association between conventional and modified centralities with passenger flows in public transport networks, concluding that centralities can be used as passenger flow predictors only when specific network representations are concerned. Wang et al. (2022) explored factors affecting bus ridership considering spatial autocorrelation and found that betweenness centralities of bus and road networks can be significant determinants of bus ridership in specific days and timeslots. Dai et al. (2022) used degree, betweenness and closeness centralities to investigate the spatial relationship between bus line and temporal bus flow networks, and found some time-dependent differences among them. Liu et al. (2022) used machine learning models to explore the transfer ridership between bus and metro, with network density and closeness centrality being the most influential factors. When it comes to metro systems, He et al. (2019) suggested that network properties such as degree and betweenness centralities are associated with ridership at metro stations. Luo et al. (2020) showed that both infrastructure and service level-based network centralities can exclusively be used to estimate passenger flows in public transport systems. In this direction, Kopsidas et al. (2023) identified passenger flow predictors not only among centralities of the network of metro infrastructures, but also in their substitute network, which is the network of alternative public transport options that connect metro stations.

Although CNT-based models can incorporate nontrivial and dynamic properties of transportation networks, and thus bring crucial information to light (Zhang et al., 2022), their usage for predictions is still limited. Their accuracy, albeit increasing, is still questionable and the most efficient predictors are yet to be found. This is because the number of measures that can be used is vast, since there are endless capabilities in metro network specifications. Physical, traffic or service networks can be formed with stations, lines, intersections etc. as nodes, any form of possible interaction like direct or indirect connections as edges, attributes such as passenger flows and travel times as weights, as well as different network representations can be used (L-space, P-space etc.) (Lin & Ban, 2013). At the same time, although CNT measures have been incorporated into predictive models, they are still complementary elements. Only recently has the potential of exclusively network-based models been investigated.

In this direction, this study aims to further explore CNT-based predictors of passenger flows in metro systems, with focus on fast and reliable estimations in cases of metro disruptions. For this purpose, econometric and machine learning models are developed to associate network centralities with passenger flows of metro stations. Both the options of total flows and Origin-Destination (OD) flows are investigated, since the information extracted from each case is uniquely valuable for disruption management. OD flows account for passenger departures and arrivals from/to metro stations, and are mostly related to platform disruptions. Total flows are the sum of OD flows plus passthroughs, i.e., passengers who pass through a metro station without alighting, and are mostly related to rail track disruptions. Indeed, disruptions of the rail track infrastructure near a metro station would affect all passengers of that station, who would either board, alight or pass through. On the other hand, disruptions of the station platform would affect only the passengers who were willing to board or disembark the metro system, but not the ones passing through the station. On this occasion, the disruption could be bypassed by utilizing the closest metro station, but on the occasion of rail track disruptions it would lead to network segmentation.

The contribution of this work is that it extends the concept of extracting passenger flow predictors from network characteristics, by making a clear distinction between total passenger flows and OD flows. To the authors’ knowledge, although the association of the former with metro and substitute network centralities was already explored by Kopsidas et al. (2023), it is the first time that metro/substitute network-based predictors of the latter are explored in this paper. The structure of this paper is as follows: the methodology is given in the next section. The results emerging from the application of the methodology to a real-world metro network (the one of Athens, Greece) are subsequently presented, followed by a discussion on them. Last, the conclusions of the study are offered in the final section.

Overview

The factors associated with passenger flows are selected among network centralities of the metro and substitute networks. The graph of the metro network G is an unweighted, directed, L-space representation of the metro infrastructure, with stations as nodes, and connections between any two consecutive stations of the same line as edges. The graph of the substitute network G_s, as defined by Kopsidas & Kepaptsoglou (2022), is a weighted, directed P-space representation of the alternative options (usually bus routes) connecting any two metro stations outside the metro system. The nodes of the substitute network also correspond to stations, while the edges are alternative public transport travel options between them, weighted by their performance in terms of time. Alternative option performance is estimated as the reciprocal of the difference between the trip duration of the best alternative option and the duration needed for the same trip within the metro system. The idea behind exploring the correlation between the centralities of the substitute network and passenger flows is based on the assumption of a two-way causality vector: i) more passengers can reach a metro station to make a transfer between bus and metro when more efficient alternative bus services reach this station. ii) more alternative services departing from a metro station can be an incentive for passengers to use the metro as a first mode for reaching their destination. On both occasions, passenger flows at metro stations are expected to increase.

Centrality measures

The centrality measures used in this study are unweighted degree, closeness and betweenness, as well as weighted degree (also called strength) and weighted betweenness. Degree of a node $i$ in unweighted graphs measures the number of edges that $i$ belongs to and it is calculated by Eq. (1):

$$\begin{array}{c}{C}_{i}^{D}= \sum _{j}^{n}{e}_{ij}\#\left(1\right)\end{array}$$

where ${e}_{ij}$ denotes the edge formed by the nodes $i$ and $j$, and $n$ is the total number of nodes in the network. As degree is highly influenced by the size of the network, it is usually normalized divided by $\frac{(n-1)(n-2)}{2}$ (undirected graphs) or by $(n-1)(n-2)$ (directed graphs). In directed graphs, node degree is also equal to the sum of node indegree and outdegree. In weighted graphs, weighted degree or strength is calculated as the sum of the weights of the edges $i$ belongs to, and it is given by Eq. (2):

$$\begin{array}{c}{S}_{i}= \sum _{j}^{n}{w}_{ij}\#\left(2\right)\end{array}$$

where ${w}_{ij}$ denotes the weight of an edge ${e}_{ij}$.

Closeness centrality indicates the proximity of a node to all the other nodes of a network, and it is calculated by Eq. (3):

$$\begin{array}{c}{C}_{i}^{C}= \frac{n-1}{{\sum }_{i\ne j\in N}{d}_{ij}}\#\left(3\right)\end{array}$$

where ${d}_{ij}$ is the distance from any other node $j$ to node $i$.

In addition, betweenness centrality expresses how many of the total shortest paths of the network pass through a node $i$, and it is given by Eq. (4):

$$\begin{array}{c}{C}_{i}^{B}= \sum _{s\ne i\ne t\in N}\frac{{\sigma }_{st}^{i}}{{\sigma }_{st}}\#\left(4\right)\end{array}$$

where ${\sigma }_{st}$ is the number of shortest paths between any nodes $s$ and $t$, and ${\sigma }_{st}^{i}$ is the number of the shortest paths passing through node $i$. In weighted graphs, the calculation of shortest paths takes into account the weight and not the number of network edges.

Models

Two linear regression models are developed to explore the potential of network centralities as predictors of total passenger flows and OD flows, respectively. It is noted that in both cases the share of each station in the total flows is considered and not the respective absolute values. That is, passenger flows of a station are divided by the sum of the respective flows of all stations. This way, the proposed methodology can be applied to metro systems with different exogenous ridership determinants (e.g., population), by multiplying the estimated shares with the total passenger flows of each system. Henceforth, passenger flows will correspond to the share of a station with respect to total passenger flows.

The dependent variable of the first model is the one of total passenger flows (TPF), that is, the sum of inflows and outflows of a metro station. All boardings, disembarkations and pass-throughs are incorporated into this measure. Degree, betweenness and closeness of the nodes in the metro network are used as independent variables, along with the strength of the nodes in the substitute network, and a dummy variable $I$ of station importance.

The model specification is presented in Eq. (5):

$$\begin{array}{c}{TPF}_{i}={\beta }_{0}+{\beta }_{1}{C}_{i}^{D}+{\beta }_{2}{C}_{i}^{B}+{\beta }_{3}{C}_{i}^{C}+{\beta }_{4}{S}_{i}+{\beta }_{5}{I}_{i}+{\epsilon }_{i} , i=1,\dots ,n\#\left(5\right)\end{array}$$

Where ${TPF}_{i}$ is the total passenger flows of station $i$, ${\epsilon }_{i}$ is the error of the estimation $i$, ${\beta }_{0}$ is the intercept of the model, and ${\beta }_{1-5}$ are the coefficients of the independent variables.

When it comes to the OD flow model, the sum of only boardings and disembarkations is considered as the model’s dependent variable (ODF). Node strength and weighted betweenness centrality of the substitute network, along with the dummy variable of importance are used as covariates. The specification of the second model is presented in Eq. (6):

$$\begin{array}{c}{ODF}_{i}={\beta }_{0}^{{\prime }}+{\beta }_{1}^{{\prime }}{S}_{i}+{\beta }_{2}^{{\prime }}{C}_{i}^{WB}+{\beta }_{3}^{{\prime }}{I}_{i}+{\epsilon }_{i}^{{\prime }} , i=1,\dots ,n\#\left(6\right)\end{array}$$

where ${ODF}_{i}$ is the OD flows of station $i$, ${C}_{i}^{WB}$ denote the weighted betweenness centrality of station $i$ in the substitute network, ${\beta }_{0-3}^{{\prime }}$ are the intercept and covariate coefficients of the second model, and ${\epsilon }_{i}^{{\prime }}$ is the estimation error.

It is noted that a different variation of the substitute network is used for calculating node weighted betweenness centrality, because when calculating shortest paths, the higher the weight of an edge, the longer the path. As such, options of higher performance would be misleadingly related to longer paths. For this reason, the weights of the substitute network denote travel times (or inverse route performance) for the calculation of weighted betweenness. However, the conventional definition is used for the calculation of node strength.

Finally, on top of regression models, the same independent variables are fed to a machine learning model, based on the Extreme Gradient Boosting algorithm (XGBoost), to investigate possible non-linearities and evaluate the relative performance of the linear models.

Evaluation metrics

The accuracy of the models is evaluated through appropriate statistical metrics, including Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE) and Normalized Root Mean Square Error (NRMSE). The models are developed in training sets, consisting of the 75% of the initial datasets. The evaluation process takes place in test sets comprising the 25% of total data. The metrics are analytically given in Eq. (7–10).

$$\begin{array}{c}MAE=\sum _{i=1}^{n}\frac{\left|\widehat{{y}_{i}}-{y}_{i}\right|}{n}\#\left(7\right)\end{array}$$

$$\begin{array}{c}MAPE=\sum _{i=1}^{n}\frac{\frac{\left|\widehat{{y}_{i}}-{y}_{i}\right|}{{y}_{i}}}{n}\#\left(8\right)\end{array}$$

$$\begin{array}{c}RMSE=\sqrt{\sum _{i=1}^{n}\frac{{\left|\widehat{{y}_{i}}-{y}_{i}\right|}^{2}}{n}}\#\left(9\right)\end{array}$$

$$\begin{array}{c}NRMSE=\frac{RMSE}{\stackrel{-}{y}}\#\left(10\right)\end{array}$$

where ${y}_{i}$ denotes the observed passenger flows of station $i$, $\widehat{{y}_{i}}$ the predicted flows, and $\stackrel{-}{y}$ the average of the observed values.

Application

To test the applicability and performance of the above-described methodology, the Athens, Greece metro system is used as a testbed. Fare collection data are used for the estimation of total daily passenger flows and OD flows, while travel times between stations are extracted from Google Maps. The data on passenger flows and the performance of the alternative (substitute) public transport options correspond to a working-day, representative of the system’s activity. The Athens metro network (Fig. 1.) consisted of 3 lines and 60 stations; the graph of the Athens metro network G, consisting of 60 nodes and 122 edges, and the graph of the substitute network G_s, consisting of 60 nodes and 649 edges, are presented in Fig. 2. Moreover, trip data of the Athens metro system are presented in Table 1 and Fig. 3. In Table 1, both inter-line and intra-line daily ridership is reported, while Fig. 3 illustrates the five stations with the largest total passenger flows and OD flows.

Table 1

Daily ridership data of the Athens metro system.
	Internal			Transfers
	Line 1	Line 2	Line 3	Lines 1–2	Lines 1–3	Lines 2–3	Total
Stations	21	17	18	2	1	1	60
OD flows	99950	97908	89833	28961	34061	52789	403502
Total flows	1964300	2045332	1918772	-	-	-	5928404

Total flow model

The first linear regression model, presented in Table 2, explores the association between total passenger flows and different centrality measures of both networks. A positive correlation between all independent variables and total passenger flows is observed, while the level of statistical significance of the covariates is at least 0.05 (also 0.01 in most cases). In addition, the standardized coefficients suggest that betweenness centrality in the metro network and strength in the substitute network are the most influential predictors of total passenger flows. According to the model, stations of higher degree, betweenness and closeness in the network of metro infrastructure, higher strength in the substitute network, as well as important stations, are expected to carry more passengers, in terms of boardings, disembarkations and pass-throughs. For instance, important stations are related to a share of total passenger flows 0.4% higher than the one of regular stations. It is noted that a station is considered important, and the dummy variable takes the value of 1 when at least one of the following conditions is fulfilled (0 in any other case):

The station is a transfer station between metro and other rail networks (tram, suburban rail etc.).
The station is in large commercial areas.
The station is located near academic institutions.

When it comes to model fit, R² is equal to 0.911 indicating that a large amount of total variance is explained by the model. As for the linear model assumptions, first, there is no evidence of strong multicollinearity, since Variable Inflation Factors (VIF) are relatively low, and Pearson correlation coefficients do not exceed 0.80 (Fig. 4). Moreover, the Durbin-Watson statistic (1.912) indicates that there is no strong evidence of autocorrelation (a value near 2 indicates no autocorrelation). Furthermore, the Breusch-Pagan test suggests that there is no strong evidence of heteroscedasticity (p-value = 0.06 > 0.05). Last, the Shapiro-Wilk test (more appropriate for small samples) shows that the errors are normally distributed (p-value = 0.948 > 0.05). Therefore, a linear model is indeed appropriate for the association of total passenger flows with centrality measures of the metro and substitute network as predictors.

Table 2

Total passenger flow regression model.
Variable	Attribute		Coefficient (standardized)	P-value	VIF
Intercept			-0.012	0.001^*
Degree	Node degree centrality of the metro network		0.247 (0.229)	0.008^*	2.946
Betweenness	Node betweenness centrality of the metro network		0.027 (0.311)	0.001^*	3.189
Closeness	Node closeness centrality of the metro network		0.084 (0.200)	0.014	2.630
Sub_strength	Node strength of the substitute network		0.347 (0.299)	0.000^*	2.509
Important	1 if a station is important, 0 else		0.004 (0.148)	0.006^*	1.128
R²	0.911	Durbin-Watson statistic		1.912

Note: ^*0.01 level of statistical significance

In addition, an XGBoost machine learning model is also developed and fed with the same explanatory variables, so that valuable comparisons are made about the feature importance and model accuracy. Since tree-based models are widely believed to outperform statistical ones, when tabular data are concerned, the XGBoost algorithm can be utilized as a benchmark model, based on which the performance of the regression model is assessed. The model’s hyperparameters are set after a Grid search k-fold cross validation hyperparameter tuning (k = 5) as follows: “colsample_bytree” − 0.2, “eta” − 0.0005, “max_depth” − 1, “n_estimators” − 1100, “subsample” − 0.05. F-scores, and SHAP values (SHapley Additive exPlanations) are used to evaluate the feature importance in the XGBoost model, as depicted in Fig. 5. These values are widely used for measuring feature importance. Technically, the former count how many times a variable is used for splitting a decision tree (Chen et al., 2019), and the latter indicate the impact of the features on individual predictions (Spadon et al., 2019). Model F-scores suggest that node strength in the substitute network is the most important feature in this model, followed by closeness and betweenness in the metro network, while SHAP values suggest that node strength in the substitute network and betweenness in the metro network are the most influential factors. These findings are rather like the ones of the linear model, since metro betweenness and substitute strength are validated as two of the most critical model components.

The evaluation metrics of the models are presented in Table 3. According to them, both models demonstrate very satisfying levels of accuracy, and therefore, they can also be used to make predictions of total passenger flows at stations. The accuracy of the linear model, albeit lower, is comparable to the XGBoost’s, providing evidence that the linear models are appropriate for the purpose of this study.

Table 3

Evaluation metrics of total flow models.
	MAE	MAPE	RMSE	NRMSE
Regression	0,002422	28.18%	0,003121	18.73%
XGBoost	0,002469	15.74%	0,003425	19.12%

OD flow model

The second model explores the association between OD passenger flows and centrality measures of both metro and substitute networks. In this case, only departures and arrivals from/to metro stations are concerned, i.e., boardings and disembarkations. The model is presented in Table 4. All the explanatory variables are again positively correlated with the dependent variable. This means that stations with higher centrality measures are expected to be an origin or destination for more passengers than stations with lower centralities. Interestingly, only centralities of the substitute network can be significant determinants of OD flows (considering at least 0.05 level of statistical significance), along with the dummy variable of station importance, per Table 4. On the contrary, centralities of the metro network cannot stand as significant covariates, highlighting the different underlying mechanisms of this kind of passenger flows. As for feature importance, the standardized beta coefficients suggest that weighted betweenness in the substitute network is by far the most influential determinant of OD flows, followed by strength in the same network.

The fit of this model is quite lower than previous model’s (R² = 0.645), indicating that there are additional factors (other than centrality measures) that significantly contribute to OD flow generation. The assumptions of the linear model can be validated as follows: low VIF values and acceptable Pearson correlation coefficients (Fig. 6) validate the absence of strong multicollinearity. The Durbin-Watson statistic (1.884) lies in an acceptable range (between 1.5 and 2.5 as a rule of thumb). There is no statistically significant evidence of heteroscedasticity, based on the Breusch-Pagan test (p-value = 0.107 > 0). Last, the errors are normally distributed according to the Shapiro-Wilk test (p-value = 0.117 > 0). It can thus be inferred that a linear model associating network centralities with OD passenger flows can be valid, but its performance, in terms of the amount of total variance explained, is relatively low, due to the existence of other significant determinants (probably non-network-based) omitted by the model.

Table 4

OD passenger flow regression model.
Variable	Attribute		Coefficient (standardized)	P-value	VIF
Intercept			0.008	0.004^*
Sub_strength	Node strength of the substitute network		0.330 (0.303)	0.048	2.574
Sub_weighted_betweenness	Node weighted betweenness centrality of the substitute network		0.057 (0.481)	0.003^*	2.588
Important	1 if a station is important, 0 else		0.005 (0.195)	0.046	1.038
R²	0.645	Durbin-Watson statistic		1.884

Note: ^*0.01 level of statistical significance

The corresponding XGBoost model’s hyperparameters are: “colsample_bytree” − 0.2, “eta” − 0.0005, “max_depth” − 1, “n_estimators” − 150, “subsample” − 0.095. F-scores, and SHAP values are again used to evaluate the feature importance (Fig. 7). Both F-scores and SHAP values suggest that weighted betweenness centrality of the stations in the substitute network is the most important feature, followed by strength and station importance. The results are similar to those of the regression, and therefore, there is evidence that centrality measures of the substitute network are significant explanatory variables of OD passenger flows.

Finally, the evaluation metrics of the OD flow models are presented in Table 5; both the linear and XGBoost models demonstrate decent accuracy, but not as satisfying as the ones of the total flow model. The XGBoost seems to provide more accurate predictions, but the linear model’s accuracy is not far from it. Both models can thus be used to associate centralities of the substitute network with OD flows, but it would not be safe to use those models for predictions.

Table 5

Evaluation metrics of OD models.
	MAE	MAPE	RMSE	NRMSE
Regression	0.004415	55.82%	0.005417	42.38%
XGBoost	0.004446	45.21%	0.005284	34.81%

The models developed for associating passenger flows with network centralities of the metro network (network of infrastructure) and the substitute network (alternative public transport options) validate the existence of high correlations among those measures and highlight the potential of network-based predictive models, as first suggested by Luo et al. (2020). Although causation is not implied, the existence of correlation can be valuable regarding fast, cost-efficient, and reliable predictions during the operation of a metro system. In particular, measures such as node degree, betweenness and closeness of the metro network, and node strength of the substitute network can be appropriate predictors of total passenger flows at stations (all of boardings, disembarkation and pass-throughs). Moreover, node centralities of the substitute network, such as strength and weighted betweenness, can be used as predictors of Origin-Destination passenger flows (only boardings and disembarkations). Interestingly, node centralities of the metro network do not seem adequate predictors of OD flows. This finding is in line with the literature suggesting that conventional centralities, such as betweenness, are not appropriate predictors of traffic flows (Kazerani, & Winter, 2009; Gao et al., 2013; Ye et al., 2016). As such, further complex network formulations need to be considered when searching for centralities that can be used as OD flow predictors.

However, the results emerging from model application suggest that the case is not the same for both types of passenger flows, when it comes to model performance. The variance of total passenger flows among metro stations can be explained by centralities of metro and substitute networks to a great extent. At the same time, the evaluation metrics of model predictive accuracy are rather satisfying and thus indicate that network-based models can be used for predictions of total passenger flows. On the other hand, the amount of variance of OD flows explained by the proposed centrality-based model is significantly less, and the evaluation metrics are not good enough to support a satisfying predictive accuracy. Evidently, there are additional significant factors that affect the distribution of OD flows, rather than network structure. Demographic characteristics, such as place of residence, work area, place of education etc. are essential factors affecting travelers’ origin and destination at micro-level, as well as population and building density, land use etc. affect total departures and arrivals from/at metro stations at macro-level. For instance, He et al. (2019) suggested that except for network structure, land use, socioeconomics and intermodal transport accessibility are also significant determinants of metro ridership. As such, centrality measures cannot solely be used for OD flow predictions at this point, but they must be combined with other appropriate socio-economic variables, instead.

But what are the most appropriate centralities to begin with? The findings of this study suggest that node strength and weighted betweenness centrality of the substitute network can be the most appropriate predictors of OD flows, among centralities of the metro and substitute networks. In fact, the network of public transport alternative options, such as bus routes, can provide valuable insight about the volumes of departures and arrivals at metro stations, due to a reverse engineering association. Since bus route design, in terms of frequency, capacity and coverage, has already incorporated determinants of travel demand, the substitute network, which accounts for alternative route performance, succeeds in capturing the information about metro OD flows related to the same socio-economic determinants. This finding is in line with the literature, highlighting the superiority of modified centrality measures over conventional ones (Ye et al., 2016; Senousi et al., 2022). The same is also supported by the fact that it is weighted and not conventional betweenness which is significantly correlated with OD flows.

On the contrary, the findings suggest that total passenger flows can effectively be described by centralities of metro and substitute networks. This measure is different because it also includes the pass-throughs among metro stations. For a better comprehension of why these are highly correlated with network structure, one can imagine total passenger flows in a metro network as the equivalent of liquids in a system of tubes. It is reasonable that those flows are influenced by the exact structure of the network itself, as the flows of the liquids are influenced by the structure of tubes. Hence, the strong association between centrality measures of the metro network and total passenger flows can be justified in the same way. As for node strength of the substitute network, not only is it related to total passenger flows, but it is also the most important predictor among the centralities. A similar reverse-engineering justification like for OD flows can be proposed to explain this association.

In this study, more complex machine learning models (XGBoost) are also developed along with regression models to evaluate the relative performance of the latter through a constructive comparison. The results suggest that both models can be used for this reason, as well as for making predictions, when appropriate. Although the accuracy of the XGBoost models is higher, the difference is not big enough to exclude statistical models, which may be more appropriate for small datasets and more convenient to researchers. In fact, when small datasets are concerned, more complex machine learning models cannot unfold their full potential, and thus, statistical models can be equally reliable. However, the proposed methodology can also be scalable through more complex machine learning techniques.

Total passenger flows and OD flows are treated separately in this study. According to results, different explanatory variables are significant for each type and different model fits are encountered. This suggests that there are different mechanisms behind the birth of each flow type. From a disruption management perspective, there are also distinct implications attached to each flow type. On the one hand, rail track disruptions at/near metro stations would create a network segmentation. That is, both the upstream and downstream passenger flow would entirely be disrupted and each of the new segments could not be reachable by the other. Practically, a trip would violently be terminated at the point of the disruption. Evidently, all metro passengers of the disrupted station would be affected by this situation, since one could not board, alight, or pass through that station. On the other hand, station platform disruptions would affect only the exact station, that is, only a node of the network, but the rest of the network would remain unharmed. On such occasions, only passengers who would be willing to board or alight from/to this station would be affected. The passengers passing through the station would continue their trip freely, since the rail would operate normally. Total passenger flows would be affected in the first case, but only OD passenger flows would be affected in the second case. As such, the determinants of each flow type must be researched separately, so that predictions can be customized depending on the needs of the operator.

The policy implications of this study mainly focus on ridership estimations in cases of disruptions. Through network theory, valuable information about metro system operation can effectively be captured. CNT-based models can be much faster and economical for making reliable ridership estimations. Public transport operators are expected to be supported by such models during daily operations management, especially when disruptions occur. In cases of disruptions, ridership estimates are essential for assessing the potential impacts of it, as well as for designing, sizing and budgeting for mitigation and contingency plans. For instance, platform disruptions at metro stations could be addressed by enhancing the capacity of bus networks serving the nearest operational stations, or rail track disruptions could be faced by bus-bridging services connecting the segmented parts of the metro network. The different possibilities in handling disruptions highlight the importance of treating different flow types separately.

As far as study limitations are concerned, the passenger flows considered in the analysis are daily, representative of a workday. Their static nature constitutes a limitation of this work since time-dependent dynamic data of passenger flows would further enhance the analysis and would be expected to provide more valuable insights. The size of the available dataset is also a limitation, since larger datasets (corresponding to longer time periods) would provide safer conclusions. Last, daily data are appropriate for medium-term estimations, for example, when disruptions of 1–3 days are concerned. Narrowing down the temporal horizon in smaller timeslots would increase the applicability of the method, including short-term estimations, as well.

In this study, network theory-based determinants of metro ridership are explored among centralities of the network of infrastructure (metro network) and the network of alternative public transport options (substitute network). For this purpose, both linear regression and non-parametric machine learning models are developed. Regarding metro ridership, total passenger flows and Origin-Destination flows are treated separately since they exhibit distinct characteristics and are related to different implications. According to modeling results, a centrality-based model can be rather accurate for total passenger flow prediction, but for OD flows, the accuracy is limited. Node degree, betweenness and closeness centralities of the metro network, node strength of the substitute network, as well as a dummy variable of station importance are significant explanatory variables of total passenger flows. On the other hand, node strength and weighted betweenness centrality of the substitute network, along with station importance are significant covariates regarding the OD flow models. Metrics of feature importance indicate that node strength and weighted betweenness centralities of the substitute network are the most influential predictors of total and OD passenger flows, respectively. Furthermore, according to the evaluation metrics, the performance of regression models can be like the XGBoost models for passenger flow estimation in metro systems. The fact that only centralities of the substitute network are significant OD flow predictors highlights the importance of utilizing the concept of the substitute network in similar modeling efforts. In addition, the fact that different variables are found significant for each passenger flow type indicates different patterns and validates the need for separate handling. The findings of this study can facilitate public transport operators, when fast, cost-efficient, and reliable ridership estimations in metro systems are needed, especially in cases of disruptions. The proposed methodology is optimized for ridership estimations regarding both rail track and platform disruptions. For future research, the association between public transport ridership and additional network theory measures should be explored to discover the most appropriate network-based ridership predictors. Network-based predictive models are still at an early stage, and thus, further research on the field is necessary. Dynamic travel demand data, as well as data from different metro systems should also be utilized for the calibration and enhancement of the proposed methodology.

CNT: Complex Network Theory

OD: Origin-Destination

TPF: Total passenger flows

ODF: Origin-Destination passenger flows

XGBoost: Extreme Gradient Boosting

MAE: Mean Absolute Error

MAPE: Mean Absolute Percentage Error

RMSE: Root Mean Square Error

NRMSE: Normalized Root Mean Square Error

VIF: Variable Inflation Factors

SHAP: SHapley Additive exPlanations

Availability of data and materials

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare that they have no competing interests.

Funding

This work was supported by the Basic Research Program, PEVE 2021, National Technical University of Athens.

Authors’ contributions

AK contributed to the conception and design of this work, data analysis, visualization and interpretation, and writing of the manuscript. AD contributed to the data analysis, visualization and interpretation. KK contributed to the data acquisition, general supervising and revision of the manuscript. All authors read and approved the final manuscript.

Acknowledgements

Not applicable.

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Exploring the association between network centralities and passenger flows in metro systems

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Abstract

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Introduction

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Centrality measures

Models

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Results

Application

Total flow model

OD flow model

Discussion

Conclusions

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