1 Introduction

The urban function refers to various roles and activities that are performed within a certain urban area or an urban facility like a hospital, restaurant and park (Batty, 2013; Gao et al., 2017; Tu et al., 2017; Yuan et al., 2015). Regarding the significance of the urban function, in addition to the qualitative description, that is, the type information, there exists another question that merits further exploration: how to differentiate the urban functional disparities between two locations with similar urban function type. We can summarize this question as the quantification of urban function. In this paper, we dissect the quantification of urban function into two aspects: one is the intensity of the urban function, which means the level of the activity related to the urban function, and the other is the influence on the surrounding neighborhood. In other words, the effects of the same type of urban function can significantly vary, leading to differences in their impacts. For instance, two places may share the same commercial function, but they may attract different numbers of people and have different ranges of influence; two hospitals may vary in their patient accommodation capacities and the ranges of service they provide. As a result, revealing the intensity and influence of urban function is crucial to comprehend the essence of urban function. Despite its significance, previous studies have mainly focused on the identification of the type of urban function, and the quantification of the intensity and influence of urban function still lacks effective methods. In recent years, as the utilization of geographical flow data delves deeper, the flow data analytics may provide a new perspective for addressing this issue (Kong et al., 2017; Wang et al., 2021; Yin et al., 2023). In this paper, we try to emphasize the importance of the quantification of the urban function and provide a summary of how to implement it with flow data.

Quantifying the urban function of a subregion with flow data is based on the notion that the urban system can be conceptualized as a network, consisting of interconnected subregions that are linked through various urban flows (Batty, 2013; Castells, 1999; Lai & Pan, 2020). Here, urban flows refer to the urban elements moving from the origins (O) to the destinations (D) within a city (Pei et al., 2020; Shu et al., 2022; Shu et al., 2021), such as commuting citizens and vehicles traversing between locations. The urban function and urban flows are mutually causal. On one hand, the urban function of a subregion can affect the volume of inflow/outflow and their distributions (Wolday, 2023). For example, an urban center may attract many flows of people and vehicles from a larger neighborhood area due to its function as a hub. On the other hand, an increase in flows and their coverage may continuously reinforce the urban function as an urban center (Liu et al., 2014; Zhang et al., 2018; Zipf, 1946). The cause-and-effect dynamics between urban functions and urban flows (specifically, the number and distribution of flows), serve as the basis for quantifying the urban function of a region in terms of intensity and influence. To realize the quantification of urban function with urban flows, we will begin by introducing the concept of flow and flow space. Subsequently, we will summarize the progress of evaluating the intensity and influence of urban functions using flow data.

2 Conception of geographical flow and flow space

To clarify the utilization of geographical flows for the quantitative characterization of urban functions, we proceed to introduce the definition of the flow. Then, we explain the conception of flow space, followed by the essential flow measurements. Lastly, we will provide a brief introduction to the aspects to consider when assessing the intensity and influence of urban functions.

2.1 Definition of the geographical flow

A geographical flow can be modeled by its O point and D point (Pei et al., 2020; Shu et al., 2021). Let the coordinate of the O point of a flow be \(({x}^{O},{y}^{O})\) and the coordinate of the D point be \(({x}^{D},{y}^{D})\), a flow \(f\) can be modeled as a tetrad of \(f=\left(\left({x}^{O},{y}^{O}\right),\left({x}^{D},{y}^{D}\right)\right)\). Note that the inner brackets of the tetrad indicate that the O point and D point have their own unique geographical locations, in other words, the x-coordinate and y-coordinate of the O point must be paired and cannot be separated when analyzing a flow, so does the D point. Moreover, the O point and D point are ordered to describe the direction of a flow.

2.2 Basic measurements of geographical flows

When it comes to geographical flows, there are three fundamental measurements to consider. The first pertains to the flow properties, the second concerns the relationship between flows and the third relates to the basic calculators of flows. Here, we focus on the buffering zone analysis. Within the first type, there are two key measurements to consider: length and direction. The length of a flow refers to the distance between the O point and D point, while the direction is defined as the angle between the east direction and the flow (Fig. 1a) (Murray et al., 2012; Tao & Thill, 2016).

Fig. 1
figure 1

Length, direction, and distance of geographical flows

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In the second type, there are also two important measurements to consider: distance and angle. For two geographical flows, the distance between them depends on both the distance between their O points and that between their D points (Gao et al., 2018; Pei et al., 2020; Tao & Thill, 2016; Yan et al., 2023). Based on this principle, let \({f}_{i}=(({x}_{i}^{O},{y}_{i}^{O}),({x}_{i}^{D},{y}_{i}^{D}))\), \({f}_{j}=(({x}_{j}^{O},{y}_{j}^{O}),({x}_{j}^{D},{y}_{j}^{D}))\), \({d}_{ij}^{O}\) be the Euclidean distance between the O points of \({f}_{i}\) and \({f}_{j}\), \({d}_{ij}^{D}\) be the Euclidean distance between the D points of \({f}_{i}\) and \({f}_{j}\), and \({d}_{ij}\) be the distance between \({f}_{i}\) and \({f}_{j}\) (Fig. 1b). We can define 3 types of flow distances as follows.

  1. (1)

    Maximum distance: The larger one of the distances \({d}_{ij}^{O}\) and \({d}_{ij}^{D}\) (Shu et al., 2021; Song et al., 2019).

$${d}_{ij}={\text{max}}\left({d}_{ij}^{O},{d}_{ij}^{D}\right)$$
(1)
  1. (2)

    Additive distance: The sum of the distances \({d}_{ij}^{O}\) and \({d}_{ij}^{D}\) (Shu et al., 2021; Song et al., 2019).

$${d}_{ij}={d}_{ij}^{O}+{d}_{ij}^{D}$$
(2)
  1. (3)

    Weighted distance: The weighted average of \({d}_{ij}^{O}\) and \({d}_{ij}^{D}\) (Tao & Thill, 2016, 2019).

$${d}_{ij}=\sqrt{\frac{\alpha {\left({d}_{ij}^{O}\right)}^{2}+\beta {\left({d}_{ij}^{D}\right)}^{2}}{{L}_{i}{L}_{j}}}$$
(3)

In Eq. (3), \(\alpha\) and \(\beta\) are weight coefficients, by default, \(\alpha +\beta =2\), \(\alpha\) and \(\beta\) can control the effects of \({d}_{ij}^{O}\) and \({d}_{ij}^{D}\) on \({d}_{ij}\); \({L}_{i}\) and \({L}_{j}\) are the lengths of \({f}_{i}\) and \({f}_{j}\), respectively. When \(\alpha =\beta =1\) and \({L}_{i}={L}_{j}=1\), \({d}_{ij}\) will be the Euclidean distance in the 4-dimensional Euclidean space.

The third type of measurements is related to the buffering zone analysis and the central measurement is the volume for flows, which can be derived from the flow distance. If we treat a flow as the basic unit of a space, then the space is known as the flow space (Pei et al., 2020; Song et al., 2019). The flow space can be seen as the Cartesian product of two 2-dimensional Euclidean space. As such, the volume of a flow sphere can be calculated as (Song et al., 2019):

$${V}_{\Omega ({\it\text{r}})}=\underset{\Omega ({\it\text{r}})}{\overset{}{\iint }}{\rho }^{O}{\rho }^{D}{\text{d}}{\rho }^{O}{\text{d}}{\rho }^{D}\iint {\text{d}}{\theta }^{O}{\text{d}}{\theta }^{D}$$
(4)

where \(\Omega\) represents a sphere with a radius of \(r\) in flow space; \(({\rho }^{O},{\theta }^{O})\) and \(({\rho }^{D},{\theta }^{D})\) are the polar coordinates of the O and D, respectively. The flow volume varies with the definitions of flow distances. For instance, the volume is \({\pi }^{2}{r}^{4}\) with the maximum distance, and \(\frac{{\pi }^{2}{r}^{4}}{6}\) with the additive distance. A sphere is the buffering zone of a flow in the flow space, with its volume corresponding to that of the sphere.

2.3 Indices for quantifying the intensity and influence of urban function

As discussed above, the quantification of the urban function includes two aspects: intensity and influence. In the context of urban studies, the intensity of urban function here can be treated as the capacity to accommodate people engaging in activities related to its function. The influence of urban function refers to the area served by the urban function provided by a region or a particular facility. The properties of geographical flows, such as the locations of O/D points, their lengths and directions, combination of them, and basic measures in flow space, e.g., the volume of flow buffer, as mentioned in section 2.2, can be used to quantify the intensity and influence of urban function. In the following sections, we will summarize the indices that were developed for evaluating the intensity and influence of urban function.

3 Evaluation of the intensity of urban function

The meaning of urban function not only includes the type of service provided by a place for residents, but also includes the capability of providing the service. Previous studies on quantifying the intensity of urban function provided by a place were mainly based on the information in situ, such as employment counts (Giuliano & Small, 1991), population size (Liu & Wang, 2016), land use diversity (Escamilla et al., 2016), and built-up volume (Taubenböck et al., 2017). However, these measures are relatively static and ignore how people actually use the place. For example, a newly established center with high building density may not be able to attract enough labor due to inadequate infrastructure nearby. Regarding the public facility, traditional research mainly focused on internal characteristics, such as the number of beds and doctors in hospitals (Green, 2005), the total area of urban parks (Wang et al., 2015), and the size of transportation hubs (Burghouwt & Redondi, 2013). However, these characteristics do not explicitly reflect the actual number of users served. For example, suppose a hospital has adequate medical resources; however, patients rarely seek medical treatment there due to its low service level. The above examples indicate that the type information alone is not enough to determine whether the place or the facility provides a satisfactory urban function. To evaluate the intensity, urban flows, which have a causal relationship with the urban function, can be used. The methods of quantifying the intensity with flows mainly rely on the indices based on the volume of flows.

Previous research showed that two sets of indices were used to quantify the intensity of urban function (Gu et al., 2023; Lee & Seo, 2021; Wu et al., 2022). The first set includes the degree centrality indices, which are based on the volume of inflow or outflow and the ratio between them (Hong et al., 2015; Wei et al., 2020). Here, the inflow refers to the flow flowing into a region and the outflow refers to the flow flowing out of a region. On the one hand, this set of indices can provide more direct and intuitive measures of the intensity by quantifying movement in and out of an area; on the other hand, it may oversimplify complex urban dynamics without considering the urban function of the origins and destinations of flows. Therefore, it is more suitable for quantifying the usage intensity of transportation hubs, commercial centers, or any urban functions where the number of movements is a critical aspect of its function. The second set comprises the indices that consider the importance of flows (Wu et al., 2022; Zhong et al., 2014), such as PageRank value (Jia et al., 2019). This set of indices accounts not only for the volume of flows but also their interconnectedness; however, it cannot directly indicate the urban function of the geographical locations as it relies on the network structure. Thus, it is more suited for quantifying the centrality of areas in urban networks, such as key nodes in transportation or utility networks, and for urban planning where understanding the network dynamics is crucial. As to a specific type of urban function, its intensity can be measured by the inflow it attracts and the outflow it discharges. In a network, the inflow/outflow of a node usually refers to the degree centrality. Wei et al. (2020) constructed the spatial interaction network with taxi OD flows in Shanghai and applied the degree centrality to quantify the centrality of places. The differences in centrality reveal varying degrees of the intensity of urban function in Shanghai. Lai et al. (2022) identified the residence-employment functional flows from mobile phone signaling data to generate the social network and adopted relative degree centrality and commuting density of the nodes in the network to measure functional centrality, which can depict the degree of the intensity of urban function and functional influence. In addition to the value of inflow or outflow, the indices of quantifying the intensity relative to resources were also developed. Gao et al. (2013); Su et al. (2020); Yan et al. (2022) estimated the traffic capacity for a given road or region through evaluating the balance between the share of traffic flows with that of road resources. Other than the relative intensity, the different patterns of intensity can also be revealed through the variation of the ratio between inflow and outflow. Liu et al. (2020) detected traffic black holes or volcanoes by finding a subgraph of a road network with inflow significantly greater or lower than the outflow from taxi OD flow data. Although the inflows and outflows are efficient in quantifying the intensity, the neighborhood information aside from the place is neglected. Differing from the first set of indices, the one proposed by Jia et al. (2019) highlighted the differences in intensities of urban functions generated by different types of flows. This research employed an improved PageRank algorithm, which assumes that a flow passing through a region is equivalent to a flow voting for the region, and the intensity of the region is determined by the cumulative votes cast by all flows. Quantifying the intensity of urban function can help get insight into the urban structure and thereby the essence of urban function (Zhong et al., 2017).

4 Evaluation of the influence of urban function

In addition to the intensity, as discussed above, another aspect to quantify the urban function is the influence. Generally, an urban function with stronger influence tends to have a larger service area. As a result, the influence can be measured by the service area. Since the shapes of objects, including subregions or public facilities in a city, can be roughly grouped into two types, that is, planar and linear (the former, such as urban parks, hospitals and transportation hubs, and the latter such as the subway and bus lines), we then summarize the progress regarding the evaluation of these two different types of service areas.

4.1 Delineation of the service area for planar objects

Regarding a planar object (either a region or a facility), its service area is defined as the influence zone surrounding it and can be determined by the distribution of flows that relate to it (Hendon, 1974; Lefever, 1926). Because the origins or destinations of flows may not be evenly distributed, the key is to estimate the service areas from those points (Guo et al., 2019; Yang, Song, Shu, et al., 2016a, 2016b). Previous studies have developed two methods for estimating service areas: the topological method and the statistical method (Boone et al., 2009; Oh & Jeong, 2007; Sister et al., 2010).

The topological method for the service area of a facility assumes that residents tend to visit the nearest facilities most frequently. Therefore, the estimation of the service area is distance-based, such as buffer areas and Thiessen polygons (Boone et al., 2009; Oh & Jeong, 2007; Sister et al., 2010). However, these methods can only adapt to the facilities with the same function, such as bank, post office, and petrol station. For facilities with different functions, like park and hospital, the nearest neighbor assumption is not always true. Increasing evidence shows that residents may not visit the nearest parks most frequently but instead visit others for their different attractive features (Schipperijn et al., 2010; Zhang & Zhou, 2018). Besides parks, the service areas of hospitals are not limited to neighborhoods either because patients tend to bypass nearby hospitals and go to more reputable ones, even if they are farther away (Yang et al., 2016a, 2016b). In addition to those common facilities, the service area of some unique facilities (e.g., airports, train stations, and embassies) has less to do with nearby areas, and more to do with the distribution of origins (inflows) and destinations (outflows) of its visitors (Yang et al., 2016a, 2016b).

According to the analysis above, it is necessary to develop statistical methods for estimating influence scopes that do not adhere to the nearest neighbor assumption. To delineate the service areas of urban parks, Guo et al. (2019) proposed to use the standard deviational ellipse (SDE) of the park visitors’ homes. The SDE of visiting flows can be expressed as:

$${\sigma }_{x}=\sqrt{2}*\sqrt{\frac{\sum_{i=1}^{n}{\left[\left({x}_{i}^{O}-\overline{{X }^{O}}\right){\text{cos}}\theta -\left({y}_{i}^{O}-\overline{{Y }^{O}}\right){\text{sin}}\theta \right]}^{2}}{n}}$$
(5)
$${\sigma }_{y}=\sqrt{2}*\sqrt{\frac{\sum_{i=1}^{n}{\left[\left({x}_{i}^{O}-\overline{{X }^{O}}\right){\text{sin}}\theta +\left({y}_{i}^{O}-\overline{{Y }^{O}}\right){\text{cos}}\theta \right]}^{2}}{n}}$$
(6)

where \({\sigma }_{x}\) and \({\sigma }_{y}\) are the standard deviations for the \(x\)-axis and \(y\)-axis; \(\theta\) is the angle between the \(x\)-axis and the north-south axis; \({x}_{i}^{O}\) and \({y}_{i}^{O}\) are the coordinates for the O point of the visiting flow \({f}_{i}\); \(n\) is the number of visiting flows;\(\left(\overline{{X }^{O}},\overline{{Y }^{O}}\right)\) is the coordinate of the center of the SDE, which is calculated as:

$$\overline{{X }^{O}}=\frac{\sum_{i}^{n}{x}_{i}^{O}}{n}$$
(7)
$$\overline{{Y }^{O}}=\frac{\sum_{i}^{n}{y}_{i}^{O}}{n}$$
(8)

based on above equations, the SDE can be determined and plotted on a map by equating the derivative of the standard deviation functions (\({\sigma }_{x}\) or \({\sigma }_{y}\)) with regard to \(\theta\), to zero (Lefever, 1926).

Based on the mobile phone data, they first identified the travel flows to each park using location analyses; then, calculated the geometric areas within 1-SDE (covering approximately 68% of data values which fall within one standard deviation of the mean) of the flows’ origins as the park service areas. In doing so, the estimated park service area is more realistic because it includes not only the zone of influence on nearby residents but also the zone of influence on distant visitors. Nevertheless, the SDE method can only provide the boundary of the service area and treat each location within the area equally. In fact, the probability of visiting a facility varies across different locations. To depict the distribution of the probability of visiting a facility like hospital, Jia et al. (2017) first extracted patient-to-hospital travel flows from inpatient discharge records and used these flows to estimate distance decay functions for hospitalization. Then, the distance decay function was incorporated into the Huff model to estimate the probabilities of patients from different areas visiting each hospital. Finally, the areas with the highest probabilities of visiting a certain hospital formed the service area of this hospital.

4.2 Evaluation of the service scope of linear objects

We have summarized the indices used for identifying the influence scope of planar facilities. Regarding the linear facilities, one of the most important is the public transportation system, typically consisting of buses and subways. As the urbanization continues to expand rapidly, the public transportation system has emerged as the primary mode of transportation (Ambarwati et al., 2016; Çolak et al., 2016). Therefore, evaluating its service area is crucial to ensure adequate accessibility (Ferguson, 1990; Giuliano, 1991). Among the various indicators used to assess the transportation service area, the coverage stands out as one of the most important. It determines whether the public transportation system offers convenient services to citizens and has been employed to evaluate the design of the public transportation system (Murray, 2001). Traditionally, the service coverage of public transportation is defined as the population/area coverage of a public transportation station within a certain walking distance threshold (Biba et al., 2010; Gutiérrez et al., 2008). Previous studies paid more attention to the coverage of the origin station (Murray, 2001; Murray et al., 1998) and the most common method for determining coverage is to analyze census data using a buffering approach (O'Neill et al., 1992; Zhao et al., 2003). However, the coverage of the destination station was overlooked. That is, even though the origin of a trip is covered by a subway station, travelers may not choose it if there is no corresponding station near the destination. In other words, current methods only assessed the service coverage of public transportation from a static perspective, while coverage analysis should be implemented from a dynamic perspective. As a result, we should focus on both origins and destinations of the travel demands instead of only one endpoint.

To solve this problem, Chen et al. (2022) proposed a flow buffer approach to analyze the coverage of public transportation services by considering the supply of public transportation at both the origin and destination. This method constructs the public transport station flow from the origin station to the destination station, and evaluates its coverage based on the buffer zone of the flow. Specifically, the buffer zone of a flow, which connects two subway stations, is measured by the volume of the flow sphere around it and can be computed with Eq. (4). The study also found that the traditional buffering method overestimates the coverage ratio which is based on the comfortable walking distance threshold. In other words, the regions, where the origin of the trip is within the threshold distance of the starting station, but the additive distances between the trip flow and the station flow (as defined in Eq. 2) exceeds twice the threshold distance, should be excluded. Therefore, the traditional methods are not suitable for the actual travels of residents. The flow buffer approach from the flow space perspective may improve the understanding of the urban transportation and aid in designing the public transportation network.

5 Quantifying importance or irreplaceability concerning number and length of flows

Each region or facility has its urban function, some of which are similar while others are different. For the similar urban functions, we need to determine which is more important, while for the different urban functions, we need to identify which is irreplaceable. Although the intensity and influence of urban functions can be quantified by indices based on the number of flows or their coverage, neither index alone can fully reflect the importance or irreplaceability of a given urban function. Specifically, regarding the indices for intensity, since they treat each flow equally and do not take into account their lengths and directions, which may indicate the varying contributions of inflows/outflows to the overall intensity, the importance or the irreplaceability cannot be determined. For instance, a place or public facility that attracts distant flows may be more important than one that only attracts nearby flows (Yang et al., 2016a, 2016b). As for the indices for coverage, they only consider the service area and neglect the intensity indicated by the number of flows. For example, a hospital with a large service area but low outpatient volume may be due to the designated hospital system (where patients are only allowed to seek medical treatment at a certain hospital) rather than its actual importance.

To couple the intensity with the coverage, Wang et al. (2021) proposed a flow-based locational measure, I-index, to quantify the urban function based on both the flow volume and flow length. The I-index inherits the merit of the famous H-index (Hirsch, 2005) in bibliometric analysis by considering a public facility or a place as a scientist, flows to this facility as papers and flow length as citation count, and can be expressed as:

$$I={\text{max}}\{k|{d}_{k}\ge \alpha *k\}$$
(9)

where \(k\) is the ranking of flows to this facility based on their lengths in descending order; \({d}_{k}\) is the length of the \(k\)-th longest flow; \(\alpha\) is a conversion factor that can be determined adaptively from the flow dataset. In a nutshell, the I-index of a facility is the maximum value of \(k\) such that the facility has received at least \(k\) flows with a length of at least \(\alpha *k\) meters each. Due to its simplicity and flow-based perspective, the I-index is a more effective measure for quantifying urban function. They extracted flows to tertiary hospitals in Beijing based on taxi trajectory data and calculated the I-index for each hospital. The results show that the Peking Union Medical College Hospital is the most important hospital in Beijing. Besides, several well-known general hospitals (#301 Hospital, Peking University Third Hospital, and Peking University People's Hospital) and specialized hospitals (Beijing Children's Hospital, Cancer Hospital) also rank among the top. Because the information of flow direction may also have close relationship with the service area (for instance, if the flows comes only from limited directions, then the service area will not be too large. As a result, the direction information should be considered for the quantification of urban function). Due to this reason, Wang et al. (2023), based on the idea of I-index, developed an enhanced measurement called X-index to quantify the importance of a place by considering both the direction and volume. A place receiving more flows from more directions tends to have a larger X-index. The application of the X-index to Beijing based on taxi OD flows successfully identifies several well-recognized important centers, such as the Central Business District (CBD) and the Financial Street, and echoes the classic central place theory at the intra-urban level.

6 Conclusion

While the identification of urban function has been extensively studied, the research on the quantification of urban function remains insufficient. As the technology of location-based developed, the flow data, which record the origin and destination of mobility, have become an important resource in urban research. This paper highlights the importance of the quantification of urban function and presents the research framework for using the flow data to quantify the urban function (Fig. 2).

Fig. 2
figure 2

Framework for using the flow data to quantify the urban function

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The quantification of urban function is divided into two aspects: intensity and influence. As for the intensity, the indices based on the inflow/outflow and their ratio have been developed. Specifically, the volume of inflow/outflow can quantify the intensity in a place or a facility, while their ratio can identify whether the pattern of the intensity is a volcano or blackhole. As for the influence, two categories of indices are applied to facilities of different shapes. One is for the planar facilities, such as parks and hospitals, which is subdivided into two types: topological indices and statistical indices. The other is for the linear facilities like the subway and bus lines. For the sake of estimating the service areas for linear facilities, the indices based on the flow buffer zone are realized in the flow space. As an application of quantifying urban functions, measuring the importance or irreplaceability between urban functions can be achieved by integrating its intensity and influence; specifically, indices containing the volume and the length (even the direction) were established and showed efficiency in quantifying the importance of places or facilities.

In conclusion, this paper underscores the importance of flow data in quantifying urban function. Compared with the traditional data and methods, flow data along with the flow space provide a new perspective for understanding the urban function and thereby facilitate the proper (re)allocation of urban resources.