An interactive city choice model and its application for measuring the intercity interaction

The rapid development of cities worldwide and the acceleration of urbanization have led to more than half of the world’s population living in citiesRitchieRoser-482 ,and thus cities have become the main location for human activities in today’s societyBarthelemy-550 . The connections between cities through the transportation network promotes the flow of people, goods, information, money and skills among cities; such flow between cities is called intercity interactionOrd-552 ; Besag-549 . Understanding and predicting intercity interaction patterns has long been an important research topic in sociology, geography, economics, transportation science and many other disciplinesWilson-560 ; najem2018debye . It also has great significance in the rational formulation of urban development strategiesBatty-486 ; paldino2015urban , the promotion of regional sustainable developmentGatelyHutyra-556 , communicable disease controlHufnagelBrockmann-490 ; EubankGuclu-491 ; BalcanColizza-492 and other fields. As the intercity interaction intensity increases, cities are no longer regarded as isolated individuals but as interdependent urban systemsBatten-553 . Therefore, understanding the intercity interaction and establishing a model that can accurately measure the interaction between cities are of great value for optimizing the spatial structure of urban agglomerationsBarthelemy-554 ; LeNEChet-555 .

The gravity model was the first model proposed to measure intercity interactionStewart-557 . The model assumes that the intensity of interaction between two cities is proportional to the product of their sizes (e.g., population, GDP) and inversely proportional to a power law function of their distance. The gravity model is simple in form and is widely used to predict intercity interactions, such as intercity travelJungWang-499 , commuting tripsViboudBjORnstad-501 ; lenormand2016systematic , population migrationTobler-487 and international tradeFagiolo-567 . However, this model is based on analogy with Newton’s law of universal gravitation and does not involve individual spatial choice behaviorWilson-560 ; YanZhou-508 . Furthermore, the parameter of the gravity model’s power law distance function is artificially defined. For example, some researchers set the parameter to 1GoldenbergLevy-564 ; backstrom2010find ; rahman2009australia , while others set it to 2KringsCalabrese-534 ; Keum-577 ; AlamUddin-578 . Therefore, it would be a valuable contribution to establish a parameter-free model to measure intercity interactions from the perspective of individual spatial choice behavior.

Simini et al. took an important step forward in spatial interaction modeling by establishing a parameter-free model named the radiation modelSiminiGonzALez-509 to predict commuting trips between counties in the U.S. This model assumes that the individual will consider the employment opportunities provided by the work location and the benefits that the opportunities may bring to him/her when choosing a work location. He/she will choose the work location nearest to his/her home that offers a benefit greater than the best offer available in his/her home county. Some researchers improve the radiation model and propose various commuting prediction models, such as the radiation model with selectionsimini2013human and the flow and jump modelvarga2018commuting . Recently, many researchers have applied the radiation model or improved radiation models to measure intercity interaction intensityLiFeng-561 ; Tian-562 ; ZhengKuang-563 . However, the radiation model assumes that the individual will only choose the nearest location with a higher benefit than his/her home, which reflects a cautious tendency of individual choice behavior. It can predict commuting trips but is not suitable for predicting general travelLiuYan-566 because travelers may choose not only the closest location with a higher benefit than the origin but also other locations with higher benefits than the origin and intervening destinations . To solve this problem, Liu and Yan proposed another parameter-free model, named the opportunity priority selection (OPS) modelLiuYan-519 , that adopts the perspective of individual destination choice behavior. The OPS model assumes that when the individual chooses a destination, he/she will choose a location with a higher benefit than the benefit of the origin and the benefits of the intervening opportunitiesStouffer-498 . This reflects an exploratory tendency in individual choice behavior and can accurately predict human mobility within and between cities. Compared with the radiation model, the OPS model can better describe individual destination choice behavior between cities, which implies that the OPS model is more suitable for measuring the intercity interaction intensity. However, applications of the OPS model to measure intercity interactions is still lacking.

In this paper, we establish an intercity interaction measurement model named the interactive city choice (ICC) model by improving the OPS model. We apply this model to measure the intercity interaction intensity in China and analyze the impact of the change in China’s land transportation network from 2005 to 2018 on the city interaction intensity. Compared with the earlier gravity model and radiation model, the results obtained by our model are more consistent with reality, implying that it captures the underlying mechanism of individual behavior in choosing to interact with cities.

The OPS modelLiuYan-519 assumes that when an individual chooses a destination, similar to the classic radiation modelSiminiGonzALez-509 and the population-weighted opportunities modelYanZhao-512 ; YanWang-515 , he/she first evaluates the benefit of the opportunities in each location, in which the number of opportunities in a location is proportional to the location’s population, and the benefit of opportunities is a random variable with a distribution of $p(z)$ ($p(z)$ can be any continuous distribution). After evaluating the benefit, the individual will select a location that presents higher benefits than the origin and any intervening opportunities. According to the above assumption, when an individual at location $i$ makes a choice for location $j$, the probability of location $j$ being selected is

$$Q_{ij}=\int_{0}^{\infty}{\mathrm{Pr}_{m_{i}+s_{ij}}(z)}\mathrm{Pr}_{m_{j}}(>z)\mathrm{d}z,$$

(1)

where $m_{i}$ is the number of opportunities at location $i$, and $s_{ij}$ is the number of intervening opportunities (i.e., the sum of the number of opportunities at all locations at a shorter distance to $i$ than $j$Stouffer-498 ; see Fig. 1(a)). ${\mathrm{Pr}_{m_{i}+s_{ij}}(z)}$ is the probability that the maximum benefit obtained after $m_{i}+s_{ij}$ sampling is exactly $z$, and ${\mathrm{Pr}_{m_{j}}(>z)}$ is the probability that the maximum benefit obtained after $m_{j}$ samplings is greater than z.

Since $\mathrm{Pr}_{x}(<z)=p(<z)^{x}$, we can obtain

$$\mathrm{Pr}_{x}(z)=\frac{\mathrm{d}\mathrm{Pr}_{x}(<z)}{\mathrm{d}z}=xp(<z)^{x-1}\frac{\mathrm{d}p(<z)}{\mathrm{d}z}.$$

(2)

Substituting Eq. (2) in Eq.(1), we obtain

	$\displaystyle Q_{ij}$	$\displaystyle=\int_{0}^{\infty}(m_{i}+s_{ij})p(<z)^{m_{i}+s_{ij}-1}\frac{\mathrm{d}p(<z)}{\mathrm{d}z}[1-p(<z)^{m_{j}}]\mathrm{d}z$		(3)
		$\displaystyle=(m_{i}+s_{ij})\int_{0}^{1}{[p(<z)^{s_{ij}+m_{i}-1}-p(<z)^{m_{j}+s_{ij}+m_{i}-1}]}\mathrm{d}p(<z)$
		$\displaystyle=(m_{i}+s_{ij})(\frac{p(<z)^{s_{ij}+m_{i}}}{s_{ij}+m_{i}}\Big{|}_{0}^{1}-\frac{p(<z)^{m_{j}+s_{ij}+m_{i}}}{m_{j}+s_{ij}+m_{i}}\Big{|}_{0}^{1})$
		$\displaystyle=(m_{i}+s_{ij})(\frac{1}{s_{ij}+m_{i}}-\frac{1}{m_{j}+s_{ij}+m_{i}})$
		$\displaystyle=\frac{m_{j}}{m_{i}+s_{ij}+m_{j}}.$

From Eq. (3), we can see that the OPS model can calculate the probability of an individual choosing a destination without any adjustable parameters. However, there are two problems in applying the OPS model to measure intercity interactions. One is that the intervening opportunities $s_{ij}$ in the OPS model are calculated by the geographic distance between two locations, as shown in Fig. 1(a). However, in reality, locations are connected by a transportation network. Most individuals compare which locations are easier to reach by taking travel time, instead of geographic distance, as the main factor. Therefore, the intervening opportunities $s_{ij}$ should be calculated by the travel time between two locationsRenErcsey-Ravasz-516 , as shown in Fig. 1 (b). The other is that the OPS model assumes that the number of location opportunities is proportional to its population. However, the number of opportunities provided by each city is not directly proportional to the population but is more related to the city’s industrial scale, GDP or other economic indicatorsReillyOthers-568 , among which the most commonly used indicator is GDPFagiolo-567 . Therefore, it is more reasonable to use GDP to reflect the number of opportunities. To solve these two problems, we establish a probability model for individuals to choose to interact with a city. We assume that the intervening opportunities $s_{ij}$ are calculated by the travel time and that the number of location opportunities is proportional to the city’s GDP. Given the GDP of each city, we can calculate the probability that an individual located in city $i$ selects city $j$ as

$$Q_{ij}=\frac{m_{j}}{m_{i}+s_{ij}+m_{j}},$$

(4)

where $m_{i}$ is the GDP of city $i$, and $s_{ij}$ is the sum of the GDP of all cities whose travel time from city $i$ is shorter than that of city $j$ (see the orange area in Fig. 1 (b)). Furthermore, if we know the total population $n_{i}$ of city $i$, we can calculate the interaction intensity from city $i$ to city $j$ as

$$T_{ij}={n_{i}}{Q_{ij}}=\frac{{n_{i}}{m_{j}}}{m_{i}+s_{ij}+m_{j}}.$$

(5)

We name Eq. (5) the interactive city choice (ICC) model. It should be noted that the spatial interaction intensity $T_{ij}$ is not an actual flow volume but a dimensionless value. Furthermore, according to the spatial interaction intensity $T_{ij}$, we can calculate the interaction intensity of the city as

$\displaystyle T_{i}=\frac{\sum\limits_{{j}\neq{i}}T_{ij}+\sum\limits_{{k}\neq{i}}T_{ki}}{2},$

(6)

where $\sum\limits_{{j}\neq{i}}T_{ij}$ is the sum of the outgoing interaction intensity and $\sum\limits_{{k}\neq{i}}T_{ki}$ is that of the incoming interaction intensitySimYaliraki-513 .

In this section, we apply the ICC model to measure the interaction intensity between cities in China and analyze the impact of the change in the Chinese land transportation network from 2005 to 2018 on city interaction intensity. It should be noted that China started to build high-speed railways in 2005; thus, we select 2005 as the starting year. Because we can only download Chinese economic and demographic data up to 2018, when we started this work, we select 2018 as the ending year.

The measurement of intercity interaction has important research significance. In this paper, we develop the ICC model from the perspective of individual choice behavior, which assumes that the probability of an individual choosing to interact with a city is proportional to the number of opportunities, as expressed by the GDP of the destination city, and inversely proportional to the number of intervening opportunities, calculated by the shortest travel time in the land transportation network. Multiplying this probability by the origin city’s population, one can obtain the intercity interaction intensity. To demonstrate the advantage of the ICC model, we apply the ICC model to measure the interaction intensity among 339 cities in China. After collecting and processing the big data related to intercity interaction, we analyze the impact of the change in the land transportation network from 2005 to 2018 on the intercity and city interaction intensity. We find that the travel time between cities has decreased and the interaction intensity between large cities has increased due to the development of land transportation. In particular, the interaction intensity of cities along high-speed railways has greatly increased. Compared with the previously widely used spatial interaction models (i.e., the gravity model and radiation model), the results obtained by the ICC model are more consistent with the actual situation.

The proposed ICC model not only helps us better measure the intercity interaction intensity but also offers potential additional applications. For example, the ICC model provides a new perspective for identifying suburbs, which is a hot topic in geographical research. The traditional suburban identification method usually refers to population density and the nature of residential landTian-562 . The ICC model introduces the spatial interaction intensity between city districts, which can improve the method of suburban identification. In addition, the ICC model can calculate the interaction intensity within and between urban agglomerations, providing valuable indicators for a comprehensive evaluation of the degree of urban agglomerationLiFeng-561 ; ShiYan-572 , which is of great significance for urban agglomeration sustainable development.

Although the ICC model can obtain more realistic results when measuring intercity interaction intensity, it still has room for expansion in practical applications. In this paper, we use GDP, which is a key factor affecting the number of opportunities, to reflect the number of opportunities. In reality, there are many other factors, e.g., urban population, industrial size and industrial structure, that also affect a city’s opportunities. Therefore, we can use multiple factors to calculate the number of opportunities in future applications. In addition, we only use the travel time calculated by the shortest time path algorithm in the land transportation networks, including roads and railways, to measure the interaction intensity among 339 Chinese cities. However, the importance of various transportation modes is different in different countries or regions. For example, airways are an important mode of passenger transportation between the U.S. citieszhang2017energy , and waterways are the main mode of freight transport between European citiesbenga2019assesment . Therefore, future research can consider extending the land transportation network to a more comprehensive three-dimensional transportation network including roads, railways, airways and waterways to make more reasonable measurements of intercity interactions.

We thank Professor Kai Liu and her postgraduate student Wei-Hua Zhu at Beijing Normal University for their help with our early research.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 71822102, 71621001, 71871010).