**1. Introduction**

The growing volumes of passenger and freight transport around regionally and globally witness their important role for economic development of different countries [1–5]. Aviation, railway, highway and shipping are four main transportation methods in modern societies. Unlike other three ones, information about highway transportation is less publicly available. In mainland China, the highway system has experienced a very rapid development since the Reform and Opening-up of China, forming a rapidly expanding multiplex network which contains national highways, provincial highways, county highways and countryside highways [6]. China has the longest expressway network in the world, which includes about 0.143 million kilometers expressways.

In the past decades, the gravity law is the most adopted in understanding transportation networks and predicting transportation fluxes [7–11], which reads

$$\mathcal{W}\_{ij} \sim \frac{\mathcal{M}\_i^a \mathcal{M}\_j^{\mathcal{G}}}{d\_{ij}^{\gamma}},\tag{1}$$

**Citation:** Wang, L.; Ma, J.-C.; Jiang, Z.-Q.; Yan, W.; Zhou, W.-X. Highway Freight Transportation Diversity of Cities Based on Radiation Models. *Entropy* **2021**, *23*, 637. https://doi.org/10.3390/e23050637

Academic Editor: Ryszard Kutner

Received: 19 April 2021 Accepted: 13 May 2021 Published: 20 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where *Wij* is the flow between locations *i* and *j*, *Mi* (or *Mj*) is usually the population or gross domestic product (GDP) of location *i* (or *j*), *dij* is the distance between *i* and *j*, and *α*, *β* and *γ* are the model parameters. Very relevantly, the gravity law has been investigated and confirmed in the Korean highway network between 30 largest cities [7], the express bus flow in Korea consisting of 74 cities and 170 bus routes with 6692 operating buses per day [12], and the urban bus networks of Korean cities [13], and the highway freight transportation networks of 338 Chinese cities [6].

However, the gravity model has several limitations, especially the requirement of previous traffic data to fit the parameters [14]. To overcome those limitations, the radiation model has been proposed [14], in which the predicted flux *F* ˜ *ij* from city *i* to city *j* is obtained as follows

$$F\_{ij} = F\_i^{\text{out}} \frac{M\_i M\_j}{(M\_i + S\_{ij})(M\_i + M\_j + S\_{ij})} \,\text{}\tag{2}$$

where *Sij* is the total "mass" (population or GDP) in the circle of radius *dij* centered at *i* but excluding the source and destination population, and *F*out *i* is total out-flux departing from city *i*

$$F\_i^{\text{out}} = \sum\_{j \neq i} F\_{ij\prime} \tag{3}$$

where *Fij* is the real flux from *i* to *j*. Obviously, the data of *F*out *i* are much easier to collect than *Fij*.

In the raw radiation model, *dij* is the geographic distance between *i* and *j*. The costbased radiation model has been soon proposed based on the intuition that an individual will choose the site that has the lowest travel cost on the network, where the travel cost can be measured by the path length or travel time from *i* to *j* [15]. In this work, *dij* is measure by the path length or driving distance from *i* to *j*. Later, to better estimate the fluxes at different spatial scales, a scaling parameter is introduced into the radiation model [16]. By combining memory effect and population-induced competition, a general model has been developed to enable accurate prediction of human mobility based on population distribution only, which also has a parameter qualifying the memory effect [17].

Although the radiation model has been adopted in the study of trip distributions [9,18–21], applications to freight transportation are rare. In this work, using a unique data set about the highway freight transportation by trucks between 338 cities in mainland China, we investigate the transportation probability *pij* between two cities *i* and *j* and the transportation diversity of a city calculated from *pij*. Although most studies dealt with undirected transportation networks [6,22,23], radiation models enable us to consider directed transportation networks due to the availability of data [24]. The raw radiation model and the cost-based radiation model are adopted because they are parameter free.

It has been reported that higher social network diversity provides greater access to social and economic opportunities and has a strong correlation with the economic development [25]. With the highway freight transportation data between Chinese cities available, we aim to investigate the relationship between highway freight transportation network diversity and economic development of cities. Such an analysis has not been conducted due to the difficulty in obtaining the highway freight transportation data. Our analysis shows that the population, the gross domestic product, the in-flux, and the out-flux scale as power laws with respect to the transportation diversity in the raw and cost-based radiation models, which implies that a more developed city usually has higher diversity in highway truck transportation. This finding reflects the fact that a more developed city usually has a more diverse economic structure.

The remainder of this work is organized as follows. Section 2 describes the data sets we analyze. Section 3 studies the basic properties of transportation probability. Section 4 deals with the transportation diversity of cities and their relationship with population and GDP. We discuss and summarize in Section 5.

#### **2. Data Sets**

The data set we analyze was provided by a leading truck logistics company in China, which records the highway truck freight transportation between 338 cities in mainland China over the period from 1 January 2019 to 31 May 2019 [6]. The data cleaning was done by the company, who used the data set in their truck scheduling and route planning. There are about 15.06 million truck freight transportation records in total, each entry containing the origin and destination cities and the starting date of the transportation. We can construct the flux matrix **F** = *Fij*<sup>338</sup>×338, where *Fij* stands for the number of trucks with freights driven from city *i* to city *j*. Unloaded trucks are not counted in. Because radiation models do not consider intra-city transportation, we set that

$$
\mathbf{0} = \mathbf{0}.\tag{4}
$$

It is obvious that *Fij* is not necessary to be equal to *Fji* for *i* = *j*.

The GDP and population data for the 338 Chinese cities in 2017 were retrieved online from the Complete Collection of World Population (http://www.chamiji.com, accessed on 18 May 2021), which are publicly available except for a few cities. We supplemented the missing data by searching Baidu Encyclopedias (https://baike.baidu.com, accessed on 18 May 2021).

*Fii*

The geographic distance *d*geo *ij* is the shortest surface distance between two cities located by the longitude and latitude, which is the length of the grea<sup>t</sup> circle arc connecting two points on the surface of the earth. The longitude and latitude of each city can be easily obtained online for free. The data set of the driving distances *d*cost *ij* between pairs of cities was provided by the same truck logistics company, which were collected by their truck drivers. The driving distance between two cities are usually "optimized" by the truck drivers because they always have the motivation to find a path connecting the two cities with the least cost (time and money). Such an optimization is achieved either by their own experience or by information from buddy truck drivers they trust. It is obvious that

$$d\_{ij}^{\text{QCD}} \prec^{\text{\text{\textquotedblleft}}} d\_{ij}^{\text{\textquotedblright}} \tag{5}$$

for all pairs of cities. The difference between these two distances increases when the two cities are farther away to each other. By definition, the geographic distance matrix is symmetric, that is,

$$d\_{ij}^{\text{geo}} = d\_{ji}^{\text{geo}}.\tag{6}$$

In contrast, the driving distance matrix is asymmetric, i.e.,

$$d\_{ij}^{\text{cost}} \neq d\_{ji}^{\text{cost}},\tag{7}$$

which is mainly due to the fact that, besides highways, there are often local roads that a truck driver has to take from one city to the other.

#### **3. Transportation Probability**
