As a popular and efficient mode of public transportation in urban areas, subways play crucial roles in modern urban transport systems. Their ability to transport large volumes of people quickly and comfortably makes them an integral part of sustainable and environmentally friendly urban mobility. Powered by electricity, metros reduce traffic congestion and improve air quality, addressing the challenges of densely populated urban centers. These metro systems offer many benefits to commuters and travelers around the world. With faster speeds than buses or cars, fixed schedules, and frequent service, subways provide reliable transportation options. With a global presence in cities ranging from sprawling metropolises like New York City to emerging urban centers, subways have proven their adaptability and indispensability.

As a key component of modern urban transportation systems, metros are relied upon by millions of people every day for commuting and travel. However, as urbanization and population growth continue, the demands and expectations placed on transit systems are escalating. This shift requires continuous improvement and optimization of the efficiency of metro networks. In megacities such as Seoul, the efficient operation of metro networks has a direct impact on the flow of urban traffic and the overall quality of urban life.

Analyzing the efficient operation of subway networks is inherently intricate and challenging due to the multitude of variables involved. However, the advent of electronic fare payment systems (EPSs) has significantly streamlined the process of monitoring and researching passenger movement within transportation networks, including subways and buses. EPSs operate by automatically deducting transit fares when passengers touch their smart cards, embedded with electronic chips, upon entry. These systems have gained widespread adoption in modern transportation networks across the globe, with notable implementations in countries such as the United Kingdom, France, Finland, Italy, China, and Korea[1]. In recent years, smart card data has become a valuable resource for a wide range of research efforts. It has been used in various research projects, including the analysis of subway passenger commuting patterns and movement behavior[2-6], investigations of the directional flow of public transit users[7], and investigations of congestion issues in subway systems[8, 9]. In particular, the physics community has used smart card data for in-depth analyses of subway systems. Recent physics-oriented research focused on the Seoul Metropolitan Subway System has provided important insights into its operational dynamics. These studies have used advanced mathematical and statistical models to analyze passenger flow and network efficiency. Notable discoveries include the identification of asymmetric passenger distribution patterns[10], deviations from conventional models in inter-station passenger flow[11], and the observation of power-law behavior in network connectivity and passenger dynamics[12]. This body of research provides invaluable perspectives for urban transportation planning and emphasizes the potential of physics-based approaches in understanding and improving the efficiency and functionality of complex urban transportation systems.

People choose a mode of transportation based on factors such as travel time, cost, convenience, and reliability. Speed is a critical factor that affects the overall efficiency and economy of a travel experience. Planned speed, which is calculated by dividing the distance traveled by the total time, is a useful metric for evaluating the true efficiency of a mode of transportation. According to statistics[13, 14] on vehicle traffic speed in Seoul, the average speed of cars in downtown Seoul is 26.4 km/h, while the average speed of city buses is 19.8 km/h. In contrast, Seoul’s subway system has an average speed of 33.7 km/h. This is because subways are not affected by traffic congestion, which can significantly slow down cars and buses. During rush hours, when road traffic is heaviest, the speed of cars and buses can be reduced by as much as 50%. However, subways can maintain their speed, making them a more efficient and reliable way to travel during these times.

This study aims to evaluate the operational efficiency of the Seoul subway system and explore improvement measures by analyzing one week of subway usage data from July 10 to 16, 2013. Human mobility patterns exhibit repetitive characteristics over a weekly cycle, making one week’s worth of data representative of consistent patterns[15]. This study focuses on smart card data. Specifically, the relationships between distance, travel time, and speed are analyzed based on subway station pairs. In particular, we consider two types of distance. The first is the geodesic distance, the shortest straight-line distance between two stations, which does not consider geographical or structural obstacles. The other is the actual distance traveled along the subway line, which considers the curves and turns of the route. The travel distance reflects the actual shape and path of the subway. We also examine the corresponding speeds associated with these distances. This allows us to gain a comprehensive understanding of the speed of subway travel in Seoul.

This study examines the efficiency of the Seoul subway network, focusing on three crucial factors: travel time, distance, and speed. The time it takes to reach a destination is paramount for any transit system’s effectiveness. We analyze average travel times from origin to destination across the network.

We employ a dual approach considering both geodesic distance (the shortest straight-line distance between stations) and traveled distance (the actual distance traveled along the subway line). This combined analysis provides a holistic understanding of the spatial efficiency of the network. Average speed across the network serves as a key indicator of operational efficiency. We assess factors impacting speed, such as station dwell times, line speeds, and transfer connections. By analyzing these factors, this study seeks to evaluate the operational efficiency of the Seoul Subway in 2013 and identify areas for improvement.

This study may provide valuable insights into the operational efficiency of the Seoul subway system. By analyzing the interactions between travel distance, time, and speed, the research sheds light on key factors influencing passenger experience and network performance. The findings offer valuable tools for planning future subway expansions and optimizing station layouts. By prioritizing passenger comfort and system efficiency, the research directly contributes to informed urban planning and transportation policy, ultimately leading to a more robust and user-friendly public transportation system.

Figure 1(a) shows the Seoul Metropolitan Subway system, which covers Seoul, Incheon, and Gyeonggi-do. Each colored dot represents a station[16], while black dots indicate major transfer points connecting different lines. Note that Line 1 (in blue) is the longest line in the network[17]. We can imagine the subway system as a dynamic network of stations and lines, where the movement of passengers resembles a flowing stream. Over a week, passengers traveling from station i to station j can be likened to particles in the flow through this network. We denote the total number of passengers on the particular route as fij, and treat them as a single “particle” representing travel volume. To quantify this flow, we measure the passenger volume (the total number of passengers) between certain nodes over a given period of time, thus revealing the passenger dynamics within the network, similar to fluid particles within a complex system.

Figure 1. (Color online) (a) Map of the Seoul Metropolitan Subway system, showing the station locations on each line (colored dots) and transfer stations (black dots). On this map, the Seoul area is colored in red. (b) Diagram comparing the shortest straight-line distance (blue arrow) between stations i and j with the actual subway route (red arrows). This shows the effect of the network’s curves and turns on the travel distance.

We define the average travel time between stations i and j, denoted as t¯ij, as the typical time it takes passengers to complete the travel. It is calculated by averaging the travel times of all passengers on the route:

Here, tij(k)represents the travel time for the k-th individual passenger traveling from station i to station j. This is calculated for each user who travels on this route, and the sum of all the individual travel times is then divided by the total number of passengers fij. Therefore, t¯ij serves as a representative value for the time it takes to travel from station i to station j. However, this includes more than just the time spent on the train itself[18]. In addition to just the time spent on the train, our travel time data includes all the walking done within the station: getting to and from platforms, boarding and alighting trains, and navigating transfers between lines. As Fig. 1(b) shows, this can add significantly to your travel compared to the straight-line distance between stations. This distance for the blue arrow is called the geodesic distance and is denoted by Lij. The speed along the straight line is called the “geodesic speed”, labeled vij.

On the other hand, the distance for the red arrows from station i to station j is calculated by the sum of the distances using information about the distance between stations from Seoul Metro[19]. It is called the “travel distance” and is denoted by Lij′. The speed along the actual route is called the “travel speed” and is denoted by vij′.

To calculate the geodesic distance between subway stations i and j, we used their latitude and longitude. The travel distance between the stations, Lij′, was calculated differently depending on whether there was a transfer involved.

With the development of metro networks in major cities around the world, passengers have more options when planning their travels[20]. However, passengers do not always prefer the shortest route, as they may be willing to make detours to avoid transfers[21]. Transfers are an inherent part of using public transport and can inconvenience passengers by adding time and complexity to their travels. In particular, subway passengers often prefer routes with minimal transfers, even at the cost of longer travel times, due to their aversion to transfers[22].

Passenger preference for minimizing transfers was incorporated into the calculation of travel distance between stations (i and j) in this study. For stations located on the same subway line, direct travel was assumed even when shorter routes involving transfers were potentially available. Consequently, the total distance was computed by simply summing the distances between consecutive stations. In contrast, for stations located on different lines, considering transfers became crucial. To determine the shortest path among these options, we used a depth-first search algorithm[23, 24].

We used a dataset of 333,506 pairs of 578 subway stations, representing 45,903,171 passenger trips over a week (July 10–16, 2013). To estimate average travel time between stations i and j, we employed the arithmetic mean of trip durations. However, the initial range of arithmetic means ranged from 6 minutes to a staggering 2,234 minutes (approximately 37 hours). Although intriguing, this outlier, likely coming from system error, unusual passenger behavior, or exceptional circumstances, was significantly different from typical travel patterns and was therefore excluded from our analysis.

To ensure the representativeness and robustness of our findings, a systematic filtering process was implemented. We sought to identify a threshold for data inclusion that reflected consistent and representative travel patterns. Therefore, we calculated the average number of passengers per route pair by dividing the total number of passengers (45,903,171) by the total number of station pairs (333,506), resulting in approximately 138 passengers per weekday pair. To focus on routes with typical travel patterns, we then included only pairs exceeding this average, resulting in a refined dataset of 52,920 pairs. This filtering approach effectively narrowed the travel time range to a more realistic 6–156 minutes, enhancing the confidence and reliability of our subsequent analysis.

Figure 2 shows the temporal distribution of subway usage in Seoul, Korea, which serves as a key visualization to evaluate the overall travel time performance of the subway network. It is important to note that “travel time” in this context refers to a passenger’s entire travel within the system, including all stages: entering the station, boarding, transferring, waiting, and exiting[25]. Our analysis prioritizes total travel time as a holistic measure of system efficiency, rather than examining individual time components separately.

Figure 2. (a) Number of passengers during subway usage. (b) Average number of subway routes based on the average travel time for the i→j route. (c) Average number of passengers during the average travel time for the i→j route. (d) Average (number of passengers/OiDj) of the average travel time for the i→j route. The subway travel time was given in minutes. The size of the bin is one minute.

Figure 2(a) shows the travel time distribution of passengers in one-minute intervals without considering their routes. The horizontal axis denotes the travel time for each passenger, corresponding to tij(k) from Eq. (1), excluding route information. The vertical axis represents the number of passengers with identical travel times, measured in minutes. This analysis elucidates the temporal patterns of metro usage, revealing the actual travel durations of passengers. Interestingly, the predominant travel duration is 16 minutes.

Figure 2(b) shows the number, N, of subway routes of average travel time, t¯ij, for the i→j route. The number of routes with an average travel time of 33 minutes is highest, suggesting a substantial portion of travels within the Seoul subway network fall within this duration. This finding suggests that a considerable proportion of travel routes within the network fall within this duration. The result means that travel times are influenced by station locations and network topology. However, it does not consider the behavior of passengers and the characteristics of the surrounding area of the subway station. While the majority of trips fall within the 30–50 minute range, real-world travel times can vary, influenced by variables such as passenger origin-destination density and transfer frequency. In particular, high-density residential areas or commercial centers with heavy pedestrian traffic can significantly affect travel times due to longer boarding and alighting times.

Figure 2(c) provides insight into the average number of passengers per route based on the average travel time, <fij>t¯ij, for the i→j route:

Here, t¯ij is represents the average travel time for all passengers on the i→j route. To calculate the average passenger volume per route, <fij>t¯ij, we divide the total number of passengers on that route by the number of routes with the same average travel time interval (1 minute) across the network. Analyzing this metric provides insight into the average passenger density for each minute of travel time on the i→j route. This analysis explores the distribution of passengers across routes, and provides valuable insight into preferred travel options, network efficiency, and station importance. In Fig. 2(c), we observe a distinct pattern: routes with an average travel time of 6 minutes see the highest passenger volume. Notably, Gangnam Station dominates the network, representing 24.7% of the total passenger flow across its four most common route pairs[26, 27]. This concentrated usage highlights its crucial role and allows us to examine differences in passenger distribution across routes. By examining origin and destination station characteristics, we uncover factors impacting passenger distribution. Stations in densely populated areas such as residential zones, business centers, or commercial areas with many transient residents experience high passenger inflows and outflows.

The average passenger volume can be distorted by the influence of stations with a high number of passengers boarding and alighting, such as Gangnam Station. Therefore, different measurement methods or the application of weights may be necessary to better understand the travel time within the network. While the sheer congestion at Gangnam Station prevents direct visual access to the underlying infrastructure, understanding the dynamics of these high-volume routes remains critical. To address this challenge, we draw on the concept of the interstation passenger capacity, inspired by the notion of flux in physics and provides a quantifiable metric for studying the characteristics of these routes. Interstation passenger capacity is denoted by the product of the total number of passengers departing from station i, denoted as Oi, and the total number of passengers arriving at station j, denoted as Dj.

We devised a metric called normalized passenger volume to assess route usage regardless of station size or popularity. This metric divides weekly ridership by interstation passenger capacity, adjusting for differences in station characteristics and surrounding areas. Averaging these values across the network yields normalized passenger volume, fijOiDjt¯ij. Figure 2(d) plots this average, revealing a key insight: the average time spent within the subway system, considering passenger capacity, is very similar to the overall average travel time, shown in Fig. 2(a). This explains the similar peak at 16 minutes in both figures. Notably, longer travel times show some fluctuations due to smaller passenger volumes at those stations, causing the graph to become less smooth beyond 80 minutes. By incorporating the concept of interstation passenger capacity, similar to physical flux, this analysis offers a deeper understanding of subway usage patterns. It shows that the Seoul Subway effectively facilitates fast and efficient passenger transportation throughout the network by considering not only average travel times, but also the capacity of stations to handle passenger volumes.

Figure 3 shows a scatter plot comparing the traveled distance with the geodesic distance within the Seoul subway system. This comparison makes it possible to evaluate the efficiency of the Seoul subway network in terms of the distance between stations. The efficiency of the network increases as the travel distance approaches the geodesic distance, indicating that the routes of the network are optimized for distance. The similarity between the actual and geodesic distances suggests that the subway network focuses on efficiency by minimizing travel time for passengers.

Figure 3. (Color online) (a) Scatter plot of the travel distance Lij′ and the geodesic distance Lij between two subway stations on the same line. (b) Scatter plot of Lij′ and Lij between two subway stations, considering all possible transfer scenarios. The red lines are the linear regression lines between Lij and Lij′. The blue dashed lines are when Lij and Lij′ are equal.

We employed simple linear regression analysis, a basic statistical method[28, 29, 30, 31] to analyze the correlation between the travel distance (Lij′) and the geodesic distance (Lij) within the Seoul Metropolitan Subway system. The term “circuity factor", defined as the ratio of the travel distance Lij′ on the subway network to the geodesic distance Lij, serves as an indicator of the organization of the network[32]. A circuity factor with a slope close to 1 on this scatter plot indicates that the actual travel path of subway network is close to the geodesic distance, indicating an efficiently structured network. Figure 1(a) reveals that certain areas within the Seoul metropolitan area boast nearly straight subway lines, implying minimal discrepancies between the geodesic distance and the actual travel distance between stations in the regions.

Figure 3(a) compares the geodesic distance Lij with the actual travel distance Lij′ between stations on the same line. Figure 3(b) expands this analysis to the entire network, including transfers. The inclusion of transfers reveals the additional complexity they introduce to travel within the network. This broader picture allows the slope of the best-fit line to provide deeper insight into the overall planning and operational efficiency of the subway system. In this study, we used simple linear regression analysis to investigate the relationship between the actual travel distance and the geodesic distance between pairs of subway stations. The readily available algorithms associated with machine learning techniques have proven to be widely applicable in a variety of data-driven studies[33-37]. Linear regression uses scatter plots to assess whether there is a linear relationship between two variables[38].

The slopes of the regression lines in Figs. 3(a) and 3(b) provide insight into network efficiency. In Fig. 3(a), which focuses on individual lines, the slope of 1.186 suggests that stations are generally well distributed along each line. When transfers are considered, as in Fig. 3(b), the slope of 1.169 remains close to 1, indicating that even with transfers, actual travel distances do not deviate significantly from geodesic distances. The slopes close to 1 in Figs. 3(a) and 3(b) suggest good overall network planning and operational efficiency from a distance perspective.

Faster travel not only benefits passengers by getting them to their destinations faster, but also improves network efficiency by reducing overall travel time. By examining this distribution, it is possible to identify areas for improvement and increase the speed and overall smoothness of the transit. Figure 4 plots two speeds: geodesic speed (vij) and travel speed (vij′). The geodesic speed, shown on the horizontal axis, is calculated as the geodesic distance between stations i and j divided by the travel time. This reflects the fastest possible travel time if a passenger could travel in a geodesic line between stations. In contrast, the travel speed is calculated using the actual travel distance. For this calculation, we used a bin size of 1 km/h.

Figure 4. (Color online) (a) Average number of subway routes of average geodesic speed and travel speed for the i→j route. (b) Cumulative proportion of the number of passengers of average geodesic speed and travel speed for the i→j route. (c) Cumulative proportion of average number of passengers of average geodesic speed and travel speed for the i→j route. (d) Cumulative proportion of average (number of passengers/OiDj) of average geodesic speed and travel speed for the i→j route. The red and black lines indicate the average geodesic and actual travel speeds, respectively. The size of the bin is 1 km/h..

Figure 4(a) shows the number of routes, N, as a function of vij or vij′. By looking at the peaks, we can find that most passenger travels occur at about 21 km/h for vij′ This is slightly higher than the 17 km/h peak for the geodesic speed vij, suggesting that actual routes often deviate from straight lines, adding to travel time. As mentioned earlier, in some parts of the Seoul metropolitan area, the subway lines are almost straight, so for two stations in these areas, the geodesic distance is almost equal to the distance traveled. As a result, the geodesic speed and the travel speed are almost the same.

Figures 4(b)–4(d) show the cumulative percentage of travel within different speed ranges. As the cumulative ratio on the vertical axis rises sharply with steeper slopes, it indicates a concentration of travels within specific speed ranges. In particular, the graphs show that most travel falls within the 10–30 km/h range, suggesting a preference for routes offering moderate speeds. The formula for the cumulative fij is

where

For both vij and vij′, Fig. 4(b) shows a clear pattern: gentle slopes for speeds below 10 km/h, followed by a sharp increase in the 10–30 km/h range before flattening out again above 30 km/h. This suggests that passenger traffic strongly favors routes providing moderate speeds between 10 and 30 km/h, which is in good agreement with the peak found in Fig. 4(a).

Figure 4(c) shows the cumulative ratio of the average travel volume, <fij>vij(vij′) for vij and vij′. The formula for cumulative <fij>vij(vij′) is as follows:

where

The graph resembles a linear function, with an almost constant slope. In the range of 0–30 km/h, the cumulative ratio of the average travel volume (cumulative <fij>) for vij′ is slightly larger than for vij, almost the same in the 30–40 km/h range. For each speed range, the graph resembles a linear function, with a slope that remains nearly constant, 40 km/h, and above 40 km/h, the trend reverses.

Figure 4(d) shows the cumulative ratio of the average normalized passenger volume <fij(OiDj)>vij(vij′). This value is considered to be the normalized value for that route, obtained by dividing the number of passengers fij by the interstation passenger capacity OiDj for a one-week period. The graph for vij′ exhibits three distinct slopes in three speed ranges: 0–14 km/h, 15–32 km/h, and over 33 km/h. At speeds between 15 and 32 km/h, the slope becomes 0.62 times steeper than at speeds between 0 and 14 km/h. Above 33 km/h, the slope becomes 4.5 times steeper than the 0–14 km/h range. Travel times in and around Seoul can feel slow due to platform access and waiting times that exceeds the actual travel time between stations. In contrast, route pairs with faster perceived speeds are likely to connect more distant areas with fewer stations, minimizing station access and waiting time.

We used smart card data from 2013 to track the movement of passengers in Seoul's subway system. We explored the subway network to find the shortest possible travel paths with transfers. Our analysis included a robust dataset of 333,506 subway station pairs, which allowed us to produce results that reflect real-world subway usage patterns. We examined the relationship between distance, travel time, and speed between station pairs, considering both geodesic and travel distances.

Using extensive data on various subway lines and passenger flows, we evaluated the operational efficiency of Seoul's subway network. The time analysis (Fig. 2) shows that the average travel time for passengers is about 15 minutes. This is likely to be influenced by the station proximity and the network structure. Analyzing the distribution of passengers across routes further reveals the influence of surrounding areas (origin and destination stations) on travel patterns. By employing normalized passenger volume, we can isolate route usage independent of station size or popularity, allowing for a more focused comparison. Analysis using linear regression (Fig. 3) shows that the actual travel distance is about 1.2 times the geodesic distance between subway stations. These results show that the subway network in the Seoul metropolitan area is relatively well planned and constructed. In addition, the speed analysis (Fig. 4) shows that most travels are made at speeds in the 10–30 km/h range, which balances fast travel with passenger comfort and safety. These results provide a valuable framework for future improvements in urban transit planning and management. We fully recognize the limitations of using 10-year-old smart card data. We plan to conduct further research using more recent data as it becomes available. This will allow us to validate our findings, assess whether they remain relevant, and gain a deeper understanding of how commuting patterns have evolved over time.

The authors gratefully acknowledge financial support from the faculty research fund of Sejong University, Korea in 2022 (Grant No.: 20220436), and also thank Professor Hyeong-Chai Jeong for constructive discussions.