*3.1. Formulae*

According to the radiation models (2) we adopt, the transportation probability *pij* from city *i* to city *j* is

$$p\_{i\bar{j}} = \frac{M\_i M\_{\bar{j}}}{(M\_{\bar{i}} + S\_{i\bar{j}})(M\_{\bar{i}} + M\_{\bar{j}} + S\_{i\bar{j}})} \,. \tag{8}$$

When we choose population *P* for *M*, the transportation probability becomes

$$p\_{i\bar{j}} = \frac{P\_{\bar{i}} P\_{\bar{j}}}{(P\_{\bar{i}} + S\_{i\bar{j}})(P\_{\bar{i}} + P\_{\bar{j}} + S\_{i\bar{j}})} \tag{9}$$

where *Sij* is the total population in the circle of radius *dij* centered at *i* but excluding the source and destination population. Alternatively, when we use GDP as the proxy, we have

$$p\_{ij} = \frac{G\_i G\_j}{(G\_i + S\_{ij})(G\_i + G\_j + S\_{ij})} \,\prime \tag{10}$$

where *Sij* is the total GDP in the circle of radius *dij* centered at *i* but excluding the source and destination population.

The transportation probabilities *pij* of the raw radiation model using geographic distance and the cost-based radiation model using driving distance are calculated with respect to population *P* in Equation (9) and gross domestic product *G* in Equation (10).

#### *3.2. Power-Law Distribution of pij*

Figure 1 illustrates the four empirical distributions of the transportation probability *pij* between two cities for the two radiation models with *M* = *P* and *M* = *G*, respectively. We observe a nice power-law tail in each case and the exponents are the same for the four cases:

$$f(p\_{ij}) \sim p\_{ij}^{-a-1},\tag{11}$$

**Figure 1.** Power-law tailed distribution of the transportation probability between two cities. The solid lines are power laws with the same exponent of −1.5. (**a**) Population *P* is used in the raw radiation model with the geographic distance. (**b**) Population *P* is used in the cost-based radiation model with the driving distance. (**c**) Gross domestic product (GDP) *G* is used in the raw radiation model with the geographic distance. (**d**) Gross domestic product *G* is used in the cost-based radiation model with the driving distance.

where the tail exponents *α* ≈ 0.5 and the intercepts are almost the same. The power-law relationship holds over three orders of magnitude. The smallest transportation probabilities deviate from the power-law distributions with higher probability density. Theoretically, we know that two cities with longer distance usually have a smaller transportation probability. Indeed, it we plot *pij* with respect to *dij*, we find that the points fluctuate around a powerlaw scaling with an exponent of −4:

$$p\_{ij} \sim d\_{ij}^{-4}\,. \tag{12}$$

which corresponds to the case of uniform population (or GDP) density [14]. The standard deviation of the data points from this reference power law quantifies the strength of heterogeneity of the spatial distribution of population and GDP in mainland China.

#### *3.3. Asymmetric Relationship between pij and pji*

We illustrate in Figure 2 the asymmetric relationship between *pij* and *pji* for the two radiation models using population. The results for GDP is very similar for each model. It is striking that the predicted values of transportation probability span nine orders of magnitude. We also find that the scatter points lies close to the diagonal *pij* = *pji*. The points from the cost-based model in Figure 2b concentrate more to the diagonal than the points in Figure 2a and thus the transportation probability matrix {*pij*} is less asymmetric. The two dashed lines impose a restriction on the transportation probability values, requiring that

$$p\_{ij} + p\_{ji} = 1,\tag{13}$$

which is more visible if we use linear coordinates. This restriction can be derived as follows. According to Equation (9), the probability of transportation from city *j* to city *i* is

$$\mathcal{P}\_{\vec{\mu}} = \frac{P\_{\vec{i}} P\_{\vec{j}}}{(P\_{\vec{j}} + S\_{\vec{j}\vec{i}})(P\_{\vec{i}} + P\_{\vec{j}} + S\_{\vec{j}\vec{i}})}.\tag{14}$$

For two given cities *i* and *j*, it is easy to notice that *pij* and *pji* reach their maxima when the two cities are adjacent, that is

$$S\_{\bar{i}\bar{j}} = S\_{\bar{j}\bar{i}} = 0.\tag{15}$$

In this case, we have

$$p\_{ij} = \frac{P\_j}{P\_i + P\_j} \tag{16}$$

and

$$p\_{ji} = \frac{P\_i}{P\_i + P\_j}.\tag{17}$$

The restriction shown in Equation (13) is thus obtained. This argumen<sup>t</sup> holds for both of the radiation models, because the derivation is independent of the definition of the distance between two cities. It also applies to the two models based on GDP, as expressed in Equation (10).

#### *3.4. Comparison between p*geo *ijand p*cost *ij*

We compare the predicted transportation probabilities from the two models. The results are shown in Figure 3. We find that the points fluctuate around the diagonal line

$$p\_{ij}^{\text{cost}} = p\_{ij}^{\text{sco}}.\tag{18}$$

The insets show that there are many points that fall exactly on the diagonal. These points correspond to the situations when

$$S\_{ij}^{\text{geo}} = S\_{ij}^{\text{opt}}.\tag{19}$$

Usually, this condition (19) is more likely to be fulfilled when the two cities *i* and *j* are close. As a special case, when city *j* is the closest city of city *i*, we have *S*geo *ij* = *S*cost *ij* = 0. In this case, the two transportation probabilities *p*geo *ij*and *p*cost *ij*are identical.

**Figure 2.** Asymmetric relationship between *pij* and *pji*. (**a**) Population *P* is used in the raw radiation model with the geographic distance. (**b**) Population *P* is used in the cost-based radiation model with the driving distance. (**c**) Gross domestic product *G* is used in the raw radiation model with the geographic distance. (**d**) Gross domestic product *G* is used in the cost-based radiation model with the driving distance.

**Figure 3.** Comparison of the transportation probabilities *pij* from the two models based on geographic distance and driving distance. The insets are the same data in linear coordinates. (**a**) The radiation models are based on population. (**b**) The radiation models are based on GDP.

#### **4. Transportation Diversity**

We now define the transportation diversity of a city *i* based on its transportation probability *pij* as follows

$$D\_i = -\sum\_{i \neq j} p\_{i\dot{j}} \ln p\_{i\dot{j}\dot{\nu}} \tag{20}$$

where *pij* can be calculated from the two radiation models using either population *P* or gross domestic product *G*. We calculate four sets of diversity *<sup>D</sup>M*,*<sup>d</sup> i* , where *M* = *P* or *M* = *G* and *d* = *d*geo or *d* = *d*cost. Indeed, human mobility or communication diversity has been proposed and studied [25–27].

#### *4.1. Comparison of Diversity Based on Population and Gross Domestic Product*

In Figure 4, we compare six pairs of any two diversity sets obtained. The two plots in the top row show the influence of distance on diversity for fixed choice of *M*, while the two plots in the bottom row illustrate the influence of the choice of *M* on diversity in a given model. We find that, in each plot, there is a nice linear relationship:

$$D\_i^{M^{(1)},d^{(1)}} = D\_i^{M^{(2)},d^{(2)}}.\tag{21}$$

It is found that the influence is weaker for the choice of model than for the choice of *M*.

**Figure 4.** Comparison of the two transportation diversity measures *<sup>D</sup>M*(1),*d*(1) *i* and *<sup>D</sup>M*(2),*d*(2) *i* calculated using population *P* and gross domestic product *G* for the raw radiation model and the cost-based radiation model. (**a**) *M*(1) = *M*(2) = *P*, *d*(1) = *d*geo and *d*(2) = *d*cost. (**b**) *M*(1) = *M*(2) = *G*, *d*(1) = *d*geo and *d*(2) = *d*cost. (**c**) *d*(1) = *d*(2) = *d*geo, *M*(1) = *G*, and *M*(2) = *P*. (**d**) *d*(1) = *d*(2) = *d*cost, *M*(1) = *G*, and *M*(2) = *P*. The solid lines are the diagonal lines.

#### *4.2. Dependence of City Traits on Diversity*

We further check the dependence of city traits (*P*, *G*, *F*out, or *F*in) on the truck transportation diversity *Di*, where *<sup>F</sup>*in*i*is total in-flux arriving at city *i*

$$F\_i^{\rm in} = \sum\_{j \neq i} F\_{ji}. \tag{22}$$

The results are depicted in Figure 5. In the four plots of Figure 5e–h for *DP*,cost *i* , we observe two outliers that seem isolated from other points. These outliers correspond to two same cities, Shennongjia Forestry District and Ali District. The diversities of these two cities are respectively 0.1496 and 0.1529.

**Figure 5.** Dependence of city traits (*P*, *G*, *F*out, and *F*in) on truck transportation diversity (*D<sup>P</sup>*,geo). The diversity is calculated from the raw radiation model based on population and geographic distance. The solid lines are power-law fits.

We observe power-law dependence in each plot. We can write that

$$Y\_i \sim (D\_i^{M,d})^{\otimes (Y,M,d)},\tag{23}$$

where *Y* represents *P*, *G*, *F*out or *F*in, *M* stands for population *P* or gross domestic product *G* in the radiation model, and *d* determines the geographic or driving distance. The powerlaw exponents *β*(*<sup>Y</sup>*, *M*, *d*) are estimated with the ordinary least-squares regression, which are presented in Table 1. For a given city trait and the chosen *M*, the two power-law exponents are similar in the raw radiation model and the cost-based radiation model.

In contrast, the power-law exponent is larger when we use population *P* as *M* in the radiation models.


**Table 1.** Power-law exponents *β*(*<sup>Y</sup>*, *M*, *d*) for the cost-based radiation model.

#### **5. Discussion and Conclusions**

In this work, we investigated the highway freight transportation diversity of 338 Chinese cities based on the transportation probability *pij* from one city to the other. The transportation probabilities are calculated from the raw radiation model based on geographic distance and the cost-based radiation model based on driving distance as the proxy of cost.

We found that, in either the raw radiation model or the cost-based radiation model, the results obtained with the population and the gross domestic product are quantitatively similar. It is mainly due to the nice power-law scaling between population and GDP of Chinese cities, where the power-law scaling exponent is estimated to be 1.15 ± 0.08 [6,28].

We investigated several important properties of the truck transportation probability *pij*. It is found that the transportation probabilities are distributed broadly with a nice power-law tail and the tail exponents are close to 0.5 for the four models. It is also found that the transportation probability matrix in each model is asymmetric such that *pij* does not necessary equal to *pji*, which is consistent with our intuition.

We also found 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 radiation model and the cost-based radiation model. It is intuitive that a city with higher GDP (often with larger population) usually has higher diversity in its industrial structure. These cities usually have higher diversity in highway freight transportation.

The strong correlation between transportation diversity and economic development implies a strong association between industry diversity and economic development. Although a causal direction of this relationship cannot be established through our analysis, transportation diversity at least provides a structural signal for the economic development of a city, highlighting the potential benefit of industry-targeted policies for economic development. Further research is required to obtain reliable policy implications. In particular, longitudinal data sets for transportation networks and economic development are required to establish a possible causal relationship.

**Author Contributions:** Funding acquisition, W.-X.Z.; Investigation, L.W., J.-C.M., Z.-Q.J. and W.Y.; Methodology, L.W. and W.-X.Z.; Supervision, W.-X.Z.; Writing—original draft, L.W. and W.-X.Z.; Writing—review & editing, L.W., J.-C.M., Z.-Q.J., W.Y. and W.-X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partly supported by the Fundamental Research Funds for the Central Universities.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** We signed a confidentiality agreemen<sup>t</sup> with the transportation company who provided us the data used in this work. Hence the data will not be shared.

**Conflicts of Interest:** The authors declare no conflict of interest.
