In the perspective of directed complex networks, the modified gravity model states that the edge flow \(F_{ij}\) between two nodes *i* and *j* takes the form of

$$ F_{ij} = C {M_{i}^{\alpha_{1}}M_{j}^{\alpha_{2}}}/{d_{ij}^{\alpha_{3}}} $$

(16)

for \(i\neq{j}\), where *C* is a constant, \(d_{ij}\) stands for the distance between *i* and *j*, and \(M_{i}\) and \(M_{j}\) stand respectively for the economic dimensions of the two nodes *i* and *j* that are being measured. For undirected transportation networks, we have

$$ W_{ij} = C{M_{i}^{\alpha_{1}}M_{j}^{\alpha_{2}}}/{d_{ij}^{\alpha_{3}}} $$

(17)

for \(j>i\). In our analysis, we consider *M* to be GDP (*G*), population (*P*) and per capita GDP (\(G/P\)), respectively.

### 4.1 GDP in the modified gravity law

#### 4.1.1 Whole transportation networks

When *M* in Eq. (16) stands for GDP, the regression equation of the gravity law reads

$$ \lg F_{ij} = \alpha_{0}+\alpha_{1} \lg{G_{i}} + \alpha_{2}\lg{G_{j}}- \alpha_{3}\lg {d_{ij}} + \epsilon, $$

(18)

where *ϵ* is the error term. We obtain that \(\alpha_{0}=1.171\pm0.063\), \(\alpha_{1}=0.621\pm0.008\), \(\alpha_{2}=0.637\pm 0.009\), and \(\alpha_{3}=1.191\pm0.013\), where the adjusted \(R^{2}\) statistic is 0.477, the *F* statistic and *p*-value for the full model are respectively 28,447 and 0.000, and an estimate of the error variance is 0.352. Figure 13(a) shows the scatter plot of \(F_{ij}\) with respect to \({G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat{\alpha }_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) for the directed network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat{\alpha}_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) in Fig. 13(b). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ F_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat {\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(19)

A regression shows that \(\alpha_{0}=1.153\pm0.031\) and \(\alpha_{1}=1.578\pm0.028\) with their *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.995, and the *F* statistic and *p*-value for the full model are respectively 3258 and 0.0000.

When *M* in Eq. (17) stands for GDP, the regression equation of the gravity law reads

$$ \lg W_{ij} = \alpha_{0}+\alpha_{1} \lg{G_{i}} + \alpha_{2}\lg{G_{j}}- \alpha_{3}\lg {d_{ij}} + \epsilon. $$

(20)

We obtain that \(\alpha_{0}=1.273\pm0.081\), \(\alpha_{1}=0.743\pm0.011\), \(\alpha_{2}=0.649\pm 0.011\), and \(\alpha_{3}=1.255\pm0.016\), where the adjusted \(R^{2}\) statistic is 0.568, the *F* statistic and *p*-value for the full model are respectively 22,379 and 0.000, and an estimate of the error variance is 0.308. Figure 13(c) shows the scatter plot of \(W_{ij}\) with respect to \({G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat{\alpha }_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) for the undirected network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat{\alpha}_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) in Fig. 13(d). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ W_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{G_{i}^{\hat{\alpha}_{1}}G_{j}^{\hat {\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(21)

We bin the data with respect to *W* and illustrate the results in Fig. 13(d). We observe that there is a power-law dependence when *W* is greater than about 200. A regression shows that \(\alpha_{0}=1.314\pm0.030\) and \(\alpha_{1}=1.289\pm0.021\) with the *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.996, and the *F* statistic and *p*-value for the full model are respectively 3791 and 0.0000.

We find that, for the scatter plots, the adjusted \(R^{2}\) statistic for the undirected HFTN (0.568) is greater than that for the directed HFTN (0.477). This result is visible in Fig. 13, which shows that the scatter plot is thinner for \(W_{ij}\) when compared with the one for \(F_{ij}\).

#### 4.1.2 Daily transportation networks

We now test the modified gravity law with daily directed and undirected freight highway transportation networks. We find that most of the daily networks exhibit the modified gravity law. As an example, the results for the directed network on 15 January 2019 are illustrated in Fig. 14(a). Regression of Eq. (18) for the scatter data points gives that \(\alpha_{0}=-0.119\pm0.062\), \(\alpha_{1}=0.242\pm0.010\), \(\alpha_{2}=0.202\pm 0.010\), and \(\alpha_{3}=0.347\pm0.012\), where the adjusted \(R^{2}\) statistic is 0.195, the *F* statistic and *p*-value for the full model are respectively 2315 and 0.000, and an estimate of the error variance is 0.128. Regression of Eq. (19) for the binning data points shows that \(\alpha_{0}=-1.604\pm0.233\) and \(\alpha_{1}=4.240\pm0.370\), where the adjusted \(R^{2}\) statistic is 0.957, the *F* statistic and *p*-value for the full model are respectively 556 and 0.000, and an estimate of the error variance is 0.006. The results for the undirected network on 15 January 2019 are illustrated in Fig. 14(b). Regression of Eq. (20) for the scatter data points gives that \(\alpha_{0}=-0.139\pm0.078\), \(\alpha_{1}=0.333\pm0.013\), \(\alpha_{2}=0.274\pm 0.012\), and \(\alpha_{3}=0.482\pm0.016\), where the adjusted \(R^{2}\) statistic is 0.286, the *F* statistic and *p*-value for the full model are respectively 2815 and 0.000, and an estimate of the error variance is 0.141. Regression of Eq. (21) for the binning data points shows that \(\alpha_{0}=-1.241\pm0.180\) and \(\alpha_{1}=2.977\pm0.233\), where the adjusted \(R^{2}\) statistic is 0.965, the *F* statistic and *p*-value for the full model are respectively 694 and 0.000, and an estimate of the error variance is 0.005.

However, we find that the daily networks around the Chinese New Year (5 February 2019) do not exhibit the gravity law. The transportation flow decreased significantly during the Spring Festival because most of the truck drivers returned home to gather with their families and most companies were also closed. As an example, the results for the directed network on 4 February 2019 are illustrated in Fig. 14(c). Regression of Eq. (18) for the scatter data points gives that \(\alpha_{0}=-0.342\pm0.106\), \(\alpha_{1}=0.067\pm0.017\), \(\alpha_{2}=0.049\pm 0.017\), and \(\alpha_{3}=-0.041\pm0.021\), where the adjusted \(R^{2}\) statistic is 0.022, the *F* statistic and *p*-value for the full model are respectively 36.6 and 0.000, and an estimate of the error variance is 0.074. Regression of Eq. (19) for the binning data points shows that \(\alpha_{0}=-11.425\pm10.233\) and \(\alpha_{1}=22.681\pm19.102\), where the adjusted \(R^{2}\) statistic is 0.412, the *F* statistic and *p*-value for the full model are respectively 6.999 and 0.024, and an estimate of the error variance is 0.070. The results for the undirected network on 15 January 2019 are illustrated in Fig. 14(d). Regression of Eq. (20) for the scatter data points gives that \(\alpha_{0}=-0.369\pm0.118\), \(\alpha_{1}=0.085\pm0.019\), \(\alpha_{2}=0.077\pm 0.019\), and \(\alpha_{3}=-0.006\pm0.023\), where the adjusted \(R^{2}\) statistic is 0.032, the *F* statistic and *p*-value for the full model are respectively 48.8 and 0.000, and an estimate of the error variance is 0.084. Regression of Eq. (21) for the binning data points shows that \(\alpha_{0}=-9.981\pm3.990\) and \(\alpha_{1}=17.909\pm6.637\), where the adjusted \(R^{2}\) statistic is 0.742, the *F* statistic and *p*-value for the full model are respectively 34.563 and 0.000, and an estimate of the error variance is 0.032. The two adjusted \(R^{2}\) statistics for the scatter data are close to zero, implying that the term \(G_{i}^{\alpha_{1}}G_{j}^{\alpha _{2}}/d_{ij}^{\alpha_{3}}\) does not have explanatory power for the transportation flow \(F_{ij}\) or \(W_{ij}\) and the modified gravity law is absent.

In Fig. 15, we present the evolution of the exponents \(\alpha_{1}\) of daily directed and undirected freight highway transportation networks. In each case, the exponent fluctuates roughly around a constant. The cone peak or valley corresponds to the dates around the Spring Festival during which the gravity law does not hold.

### 4.2 Population in the modified gravity law

#### 4.2.1 Whole transportation networks

When *M* in Eq. (16) stands for population, the regression equation of the gravity law reads

$$ \lg F_{ij} = \alpha_{0}+\alpha_{1} \lg{P_{i}} + \alpha_{2}\lg{P_{j}}- \alpha_{3}\lg {d_{ij}} + \epsilon, $$

(22)

where *ϵ* is the error term. We obtain that \(\alpha_{0}=1.409\pm0.066\), \(\alpha_{1}=0.789\pm0.011\), \(\alpha_{2}=0.721\pm 0.012\), and \(\alpha_{3}=1.192\pm0.013\), where the adjusted \(R^{2}\) statistic is 0.446, the *F* statistic and *p*-value for the full model are respectively 25,128 and 0.000, and an estimate of the error variance is 0.372. Figure 16(a) shows the scatter plot of \(F_{ij}\) with respect to \({P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat{\alpha }_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) for the directed network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat{\alpha}_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) in Fig. 16(b). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ F_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat {\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(23)

A regression shows that \(\alpha_{0}=1.539\pm0.024\) and \(\alpha_{1}=1.639\pm0.028\) with their *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.995, and the *F* statistic and *p*-value for the full model are respectively 3453.6 and 0.0000.

When *M* in Eq. (17) stands for population, the regression equation of the gravity law reads

$$ \lg W_{ij} = \alpha_{0}+\alpha_{1} \lg{P_{i}} + \alpha_{2}\lg{P_{j}}- \alpha_{3}\lg {d_{ij}} + \epsilon. $$

(24)

We obtain that \(\alpha_{0}=1.593\pm0.085\), \(\alpha_{1}=0.868\pm0.014\), \(\alpha_{2}=0.797\pm 0.015\), and \(\alpha_{3}=1.271\pm0.017\), where the adjusted \(R^{2}\) statistic is 0.531, the *F* statistic and *p*-value for the full model are respectively 19,295 and 0.000, and an estimate of the error variance is 0.334. Figure 16(c) shows the scatter plot of \(W_{ij}\) with respect to \({P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat{\alpha }_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) for the undirected network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat{\alpha}_{2}}}{d_{ij}^{\hat{\alpha}_{3}}}\) in Fig. 16(d). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ W_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{P_{i}^{\hat{\alpha}_{1}}P_{j}^{\hat {\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(25)

We bin the data with respect to *W* and illustrate the results in Fig. 16(d). We observe that there is a power-law dependence when *W* is greater than about 200. A regression shows that \(\alpha_{0}=1.741\pm0.024\) and \(\alpha_{1}=1.333\pm0.022\) with the *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.995, and the *F* statistic and *p*-value for the full model are respectively 3631 and 0.0000.

We find that, for the scatter plots, the adjusted \(R^{2}\) statistic for the undirected HFTN (0.531) is greater than that for the directed HFTN (0.446). This result is visible in Fig. 16, which shows that the scatter plot is thinner for \(W_{ij}\) when compared with the one for \(F_{ij}\).

#### 4.2.2 Daily transportation networks

We now test the modified gravity law with daily directed and undirected freight highway transportation networks. We find that most of the daily networks exhibit the modified gravity law. As an example, the results for the directed network on 15 January 2019 are illustrated in Fig. 17(a). Regression of Eq. (22) for the scatter data points gives that \(\alpha_{0}=-0.244\pm0.066\), \(\alpha_{1}=0.353\pm0.014\), \(\alpha_{2}=0.251\pm 0.014\), and \(\alpha_{3}=0.336\pm0.013\), where the adjusted \(R^{2}\) statistic is 0.192, the *F* statistic and *p*-value for the full model are respectively 2278 and 0.000, and an estimate of the error variance is 0.129. Regression of Eq. (23) for the binning data points shows that \(\alpha_{0}=-2.188\pm0.319\) and \(\alpha_{1}=4.334\pm0.426\), where the adjusted \(R^{2}\) statistic is 0.946, the *F* statistic and *p*-value for the full model are respectively 439 and 0.000, and an estimate of the error variance is 0.007. The results for the undirected network on 15 January 2019 are illustrated in Fig. 17(b). Regression of Eq. (24) for the scatter data points gives that \(\alpha_{0}=-0.189\pm0.083\), \(\alpha_{1}=0.420\pm0.017\), \(\alpha_{2}=0.361\pm 0.017\), and \(\alpha_{3}=0.470\pm0.016\), where the adjusted \(R^{2}\) statistic is 0.271, the *F* statistic and *p*-value for the full model are respectively 2616 and 0.000, and an estimate of the error variance is 0.144. Regression of Eq. (25) for the binning data points shows that \(\alpha_{0}=-1.427\pm0.210\) and \(\alpha_{1}=3.046\pm0.257\), where the adjusted \(R^{2}\) statistic is 0.960, the *F* statistic and *p*-value for the full model are respectively 596 and 0.000, and an estimate of the error variance is 0.006.

However, we find that the daily networks around the Chinese New Year (5 February 2019) do not exhibit the gravity law. The transportation flow decreased significantly during the Spring Festival because most of the truck drivers returned home to gather with their families and most companies were also closed. As an example, the results for the directed network on 4 February 2019 are illustrated in Fig. 17(c). Regression of Eq. (22) for the scatter data points gives that \(\alpha_{0}=-0.223\pm0.114\), \(\alpha_{1}=0.082\pm0.025\), \(\alpha_{2}=0.026\pm 0.024\), and \(\alpha_{3}=-0.037\pm0.021\), where the adjusted \(R^{2}\) statistic is 0.012, the *F* statistic and *p*-value for the full model are respectively 20.3 and 0.000, and an estimate of the error variance is 0.075. Regression of Eq. (23) for the binning data points shows that \(\alpha_{0}= -9.526\pm13.747\) and \(\alpha_{1}=24.918\pm33.416\), where the adjusted \(R^{2}\) statistic is 0.216, the *F* statistic and *p*-value for the full model are respectively 2.76 and 0.128, and an estimate of the error variance is 0.093. The results for the undirected network on 15 January 2019 are illustrated in Fig. 17(d). Regression of Eq. (24) for the scatter data points gives that \(\alpha_{0}=-0.242\pm0.128\), \(\alpha_{1}=0.097\pm0.027\), \(\alpha_{2}=0.066\pm 0.027\), and \(\alpha_{3}=-0.005\pm0.024\), where the adjusted \(R^{2}\) statistic is 0.017, the *F* statistic and *p*-value for the full model are respectively 24.7 and 0.000, and an estimate of the error variance is 0.085. Regression of Eq. (25) for the binning data points shows that \(\alpha_{0}=-12.333\pm5.064\) and \(\alpha_{1}=28.166\pm10.873\), where the adjusted \(R^{2}\) statistic is 0.726, the *F* statistic and *p*-value for the full model are respectively 31.854 and 0.000, and an estimate of the error variance is 0.034. The two adjusted \(R^{2}\) statistics for the scatter data are close to zero, implying that the term \(P_{i}^{\alpha_{1}}P_{j}^{\alpha _{2}}/d_{ij}^{\alpha_{3}}\) does not have explanatory power for the transportation flow \(F_{ij}\) or \(W_{ij}\) and the modified gravity law is absent.

In Fig. 18, we present the evolution of the exponents \(\alpha_{1}\) of daily directed and undirected freight highway transportation networks. In each case, the exponent fluctuates roughly around a constant. The cone peak or valley corresponds to the dates around the Spring Festival during which the gravity law does not hold.

### 4.3 Per capita GDP in the modified gravity law

#### 4.3.1 Whole transportation networks

When *M* in Eq. (16) stands for per capita GDP, the regression equation of the modified gravity law reads

$$ \lg F_{ij} = \alpha_{0}+\alpha_{1} \lg(G_{i}/P_{i}) + \alpha_{2}\lg (G_{j}/P_{j})-\alpha_{3}\lg{d_{ij}} + \epsilon, $$

(26)

where *ϵ* is the error term. We obtain that \(\alpha_{0}=4.984\pm0.050\), \(\alpha_{1}=0.544\pm0.018\), \(\alpha_{2}=0.681\pm 0.019\), and \(\alpha_{3}=1.385\pm0.014\), where the adjusted \(R^{2}\) statistic is 0.330, the *F* statistic and *p*-value for the full model are respectively 15,365 and 0.000, and an estimate of the error variance is 0.451. Figure 19(a) shows the scatter plot of \(F_{ij}\) with respect to \({(G_{i}/P_{i})^{\hat{\alpha}_{1}}(G_{j}/P_{j})^{\hat {\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}}\) for the directed network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({(G_{i}/P_{i})^{\hat{\alpha}_{1}}(G_{j}/P_{j})^{\hat{\alpha}_{2}}}/{d_{ij}^{\hat {\alpha}_{3}}}\) in Fig. 19(b). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ F_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{(G_{i}/P_{i})^{\hat{\alpha }_{1}}(G_{j}/P_{j})^{\hat{\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(27)

A regression shows that \(\alpha_{0}=9.317\pm0.158\) and \(\alpha_{1}=2.191\pm0.053\) with their *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.990, and the *F* statistic and *p*-value for the full model are respectively 1736 and 0.0000.

When *M* in Eq. (17) stands for population, the regression equation of the gravity law reads

$$ \lg W_{ij} = \alpha_{0}+\alpha_{1}\lg (G_{i}/P_{i} ) + \alpha_{2}\lg (G_{j}/P_{j} )-\alpha_{3}\lg{d_{ij}} + \epsilon. $$

(28)

We obtain that \(\alpha_{0}=5.621\pm0.067\), \(\alpha_{1}=0.660\pm0.026\), \(\alpha_{2}=0.701\pm 0.023\), and \(\alpha_{3}=1.520\pm0.019\), where the adjusted \(R^{2}\) statistic is 0.388, the *F* statistic and *p*-value for the full model are respectively 10,769 and 0.000, and an estimate of the error variance is 0.436. Figure 19(c) shows the scatter plot of \(W_{ij}\) with respect to \((G_{i}/P_{i})^{\alpha_{1}}(G_{j}/P_{j})^{\alpha _{2}}/d_{ij}^{\alpha_{3}}\) for the undirected network. We bin the data with respect to *F* and illustrate \(F_{ij}\) against \({(G_{i}/P_{i})^{\hat{\alpha}_{1}}(G_{j}/P_{j})^{\hat{\alpha}_{2}}}/{d_{ij}^{\hat {\alpha}_{3}}}\) in Fig. 19(d). We observe that there is a power-law dependence when *F* is greater than about 30:

$$ W_{ij} = \alpha_{0} + \alpha_{1} \lg \bigl[{(G_{i}/P_{i})^{\hat{\alpha }_{1}}(G_{j}/P_{j})^{\hat{\alpha}_{2}}}/{d_{ij}^{\hat{\alpha}_{3}}} \bigr] + \epsilon. $$

(29)

We bin the data with respect to *W* and illustrate the results in Fig. 19(d). We observe that there is a power-law dependence when *W* is greater than about 200. A regression shows that \(\alpha_{0}=8.641\pm0.140\) and \(\alpha_{1}=1.726\pm0.043\) with the *p*-values being 0.0000 and 0.0000. The adjusted \(R^{2}\) statistic is 0.990, and the *F* statistic and *p*-value for the full model are respectively 1642 and 0.0000.

We find that, for the scatter plots, the adjusted \(R^{2}\) statistic for the undirected HFTN (0.388) is greater than that for the directed HFTN (0.330). This result is visible in Fig. 19, which shows that the scatter plot is thinner for \(W_{ij}\) when compared with the one for \(F_{ij}\).

#### 4.3.2 Daily transportation networks

We now test the modified gravity law with daily directed and undirected freight highway transportation networks. We find that most of the daily networks exhibit the modified gravity law. As an example, the results for the directed network on 15 January 2019 are illustrated in Fig. 20(a). Regression of Eq. (26) for the scatter data points gives that \(\alpha_{0}= 1.147\pm0.043\), \(\alpha_{1}=0.199\pm0.019\), \(\alpha_{2}=0.197\pm 0.019\), and \(\alpha_{3}=0.364\pm0.013\), where the adjusted \(R^{2}\) statistic is 0.117, the *F* statistic and *p*-value for the full model are respectively 1264 and 0.000, and an estimate of the error variance is 0.141. Regression of Eq. (27) for the binning data points shows that \(\alpha_{0}=5.689\pm0.583\) and \(\alpha_{1}=6.644\pm0.830\), where the adjusted \(R^{2}\) statistic is 0.916, the *F* statistic and *p*-value for the full model are respectively 272 and 0.000, and an estimate of the error variance is 0.012. The results for the undirected network on 15 January 2019 are illustrated in Fig. 20(b). Regression of Eq. (28) for the scatter data points gives that \(\alpha_{0}= 1.582\pm0.056\), \(\alpha_{1}=0.271\pm0.024\), \(\alpha_{2}=0.272\pm 0.023\), and \(\alpha_{3}=0.509\pm0.017\), where the adjusted \(R^{2}\) statistic is 0.175, the *F* statistic and *p*-value for the full model are respectively 1489 and 0.000, and an estimate of the error variance is 0.163. Regression of Eq. (29) for the binning data points shows that \(\alpha_{0}= 5.773\pm0.584\) and \(\alpha_{1}=4.622\pm0.569\), where the adjusted \(R^{2}\) statistic is 0.918, the *F* statistic and *p*-value for the full model are respectively 280 and 0.000, and an estimate of the error variance is 0.011.

However, we find that the daily networks around the Chinese New Year (5 February 2019) do not exhibit the gravity law. The transportation flow decreased significantly during the Spring Festival because most of the truck drivers returned home to gather with their families and most companies were also closed. As an example, the results for the directed network on 4 February 2019 are illustrated in Fig. 20(c). Regression of Eq. (26) for the scatter data points gives that \(\alpha_{0}=-0.103\pm0.071\), \(\alpha_{1}=0.097\pm0.032\), \(\alpha_{2}=0.124\pm 0.032\), and \(\alpha_{3}=-0.039\pm0.021\), where the adjusted \(R^{2}\) statistic is 0.023, the *F* statistic and *p*-value for the full model are respectively 37.4 and 0.000, and an estimate of the error variance is 0.074. Regression of Eq. (27) for the binning data points shows that \(\alpha_{0}=-8.942\pm4.624\) and \(\alpha_{1}=32.579\pm15.580\), where the adjusted \(R^{2}\) statistic is 0.685, the *F* statistic and *p*-value for the full model are respectively 21.7 and 0.001, and an estimate of the error variance is 0.037. The results for the undirected network on 15 January 2019 are illustrated in Fig. 20(d). Regression of Eq. (28) for the scatter data points gives that \(\alpha_{0}= 0.007\pm0.079\), \(\alpha_{1}=0.122\pm0.037\), \(\alpha_{2}=0.156\pm 0.034\), and \(\alpha_{3}= 0.004\pm0.023\), where the adjusted \(R^{2}\) statistic is 0.028, the *F* statistic and *p*-value for the full model are respectively 42.6 and 0.000, and an estimate of the error variance is 0.084. Regression of Eq. (29) for the binning data points shows that \(\alpha_{0}= -2.627\pm2.288\) and \(\alpha_{1}=15.534\pm10.407\), where the adjusted \(R^{2}\) statistic is 0.468, the *F* statistic and *p*-value for the full model are respectively 10.6 and 0.007, and an estimate of the error variance is 0.065. The two adjusted \(R^{2}\) statistics for the scatter data are close to zero, implying that the term \((G_{i}/P_{i})^{\alpha_{1}}(G_{j}/P_{j})^{\alpha _{2}}/d_{ij}^{\alpha_{3}}\) does not have explanatory power for the transportation flow \(F_{ij}\) or \(W_{ij}\) and the modified gravity law is absent.

In Fig. 21, we present the evolution of the exponents \(\alpha_{1}\) of daily directed and undirected freight highway transportation networks. In each case, the exponent fluctuates roughly around a constant. The cone peak or valley corresponds to the dates around the Spring Festival during which the gravity law does not hold.