3.1. Analysis of the Correlation for the CGSPFN
Many scholars have conducted a systematic study and believe that there is a close correlation about the gasoline prices in different areas [
16,
18]. In this section, we will explore the correlation based on the nodes of the two networks. According to the above results, the number of the nodes of the CGSPFN is 1592 and 1634 in NYH and GC, respectively. The number of the same nodes is 1288 among the network nodes. The same node strength is counted in the whole period. According to the above results, the following Formula (7) is applied to calculate the similarity of the CGSPFN in NYH and GC.
where
and
denote the number of the node of the CGSPFN in NYH and GC, respectively.
denotes the number of the same nodes of the CGSPFN in NYH and GC. Meanwhile,
, when
, is shown that the similarity is the lowest between the two networks; when
, it is shown that the similarity is the highest between the two networks. According to the calculation, we can obtain that the correlation of the node strength is 0.7405 between the two CGSPFNs, which shows there are close relationships of the strength of the same nodes of the CGSPFN and there is a higher similarity in NYH and GC.
In terms of the SFP and FFP, the correlation is also explored about the nodes of the CGSPFN based on
Table 4. The number of the same nodes is 1222 in the SFP, but it is 966 in the FFP. According to Formula (7), the network similarity in the SFP is 0.4986, but only 0.3069 in the FFP, which illustrates that the CGSPFNs have a higher interdependence between the NYH and GC in the SSP than in the FFP.
By means of the comprehensive analysis, the interdependence between the CGSP fluctuations of the NYH and GC can be described by means of the network similarity, which shows that there is a higher interdependence between the two networks in the whole period, but the interdependence gradually decreases in SFP and FFP.
3.2. Analysis of the Regularity for the New Nodes
Based on the CGSPFN of different harbors in different periods (i.e., the whole period, the SFP and the FFP), we explore the regularity of the cumulative time interval for the new nodes of the network. As for the new nodes, the same number of the nodes is chosen from the first appeared node, which means the same number of the nodes is decided by the least number of the nodes in the network. Then, we obtain the regularity of the cumulative time interval for the new nodes in NYH and GC, as shown in
Figure 5.
In
Figure 5a, the green line represents the equal interval curve, and the new nodes of the CGSPFNs appear at unequal intervals as the time goes on, but gradually increase, and it shows a trend of the linear growth. The least square method is utilized to regress the cumulative time interval for the new nodes of the CGSPFN in NYH and GC, and the corresponding regression equations are
and
respectively. Meanwhile, the corresponding coefficients of determination (
) of the trend line are 0.9611 and 0.9674. Obviously, the result has a higher credibility and the cumulative time interval for the new nodes of the CGSPFN has a linear growth trend and shows a good regularity in NYH and GC, which reflects that the two networks have the similarity and the foresight of the CGSP under the temporal distribution.
In
Figure 5b, the image of the cumulative time interval for the new nodes is presented. It can be seen that the cumulative time intervals for the new nodes of the CGSPFN also have a good regularity in the SFP and FFP, i.e., a trend of the linear growth. The corresponding regression equations and coefficients of determination are shown in
Table 5.
According to
Table 5, the result has a higher credibility, which illustrates that the cumulative time interval for the new nodes of the CGSPFN has a trend of the linear growth, very high similarity and a good regularity in the SFP and FFP. In addition, the cumulative time interval for the new nodes in the FFP is larger than that in the SFP, as shown in
Figure 5b.
To sum up, the new nodes of the CGSPFN mean that the price of the abnormal fluctuations has appeared and the new nodes are different from the previous fluctuations; and the CGSP has complex nonlinear features in NYH and GC, but the cumulative time interval for the price of abnormal fluctuations shows a trend of the linear growth. The time can be identified effectively by the regularity when the price of the abnormal fluctuation appears, which can help the decision makers to respond to the fluctuations of the new price nodes in time, so as to improve the accuracy of the judgment and reduce the risk for the CGSP in NYH and GC.
3.3. Analysis of the Node Strength and its Distribution
In order to further reveal the temporal distribution characteristics and power law distribution of the important nodes for the CGSPFN in NYH and GC, the strength [
41] and its distribution [
32] about the nodes are explored in the whole period, as shown in
Figure 6.
According to
Figure 6, most of the node strengths are small and only the minorities are large in the whole period, which shows a feature of the typical scale-free network. In addition, the important nodes mean that the node strengths are greater than or equal to 45 and they are different nodes. Meanwhile, some statistical indicators of the important nodes of the CGSPFN are given in
Table 6.
Based on the above analysis, it can be seen that the gasoline price fluctuation states in NYH and GC convert frequently and are complex, but the core fluctuation states are in the top 1.6% of the nodes, which can be used to reflect the state and conversion relationship between the fluctuation states, and attempt approximate description of the essential characteristics of the gasoline price fluctuations in NYH and GC.
To some extent, the importance and influence of the nodes can be reflected by the node strengths in the whole network. Therefore, the important nodes are explored and we obtain the name, the strength and the temporal distribution feature about the nodes, as shown in
Figure 7. In
Figure 7, the symbol indicates the name of the node, the number on the line represents the weights between the two nodes, the red line donates the positive direction and the blue line donates the positive direction.
In the terms of the temporal distribution feature of the important nodes in
Figure 7, we can obtain the name, the strength, and the temporal distribution feature of the important nodes. As for the CGSPFN in NYH, the 23 important nodes are located in the first 112 nodes, the largest node strengths first appeared on 25 and 26 August, 1986, and they are located in the 30th and 31st node. Meanwhile, the value of the strength is 65, and the names of the nodes are ‘idiii‘ and ‘diiii‘, respectively. However, there are only 8 nodes and their strengths are 1 in the first 112 nodes; the earliest node is ‘diIie‘, it is located in the 18th node and appeared on 7 August, 1986. As for the CGSPFN in GC, the 25 important nodes are located in the first 107 nodes; the largest node strengths first appeared on 3 October, 1986, and it is located in the 53rd node. Meanwhile, the value of the strength is 62, and the name of the node is ‘diddd‘. However, there are only 9 nodes and their strengths are 1 in the first 107 nodes; the earliest node is ‘deide‘, it appeared on 23 February, 1987 and is located in 83rd node. The results show that the nodes with a large strength must be the nodes appearing in the earlier time, but the nodes appearing in the earlier time are not necessarily nodes with a large strength. Therefore, it can be seen that the links between the important nodes with a large strength are very close based on the weights of the connected edges among the nodes of the CGSPFN in NYH and GC.
In the terms of the average contribution rate of the connections among the important nodes of the CGSPFN in NYH and GC, it can be found that different CGSPFNs have obvious positive correlation features, i.e., the larger node strengths tend to be connected with a larger node strength. Although the conversion among the state of the CGSP is very frequent and the progress is complicated, the first 2.6% nodes (including the same nodes) can reflect their key fluctuations, which mean that the states of the CGSP fluctuations in the future may have already appeared in the early stage. Therefore, the fluctuation states and transformation relationships about the first 2.6% nodes are researched, and then the essential features of the CGSP can be found.
As is known to all, the network is a scale-free network if the strength distribution
can be fitted by the power-law distribution described by
where the
means the strength distribution,
refers to the proportionality constant,
is the strength, and
is the power law index. Meanwhile, the larger the power law index is, the stronger the power law distribution of the network is.
According to Formula (8), the strength distribution of the node is fitted by using the least squares method in different periods, then coefficient of determination (
), power law index (
), and P-Value of the double logarithmic can be obtained, as shown in
Table 7.
According to
Table 6, the fitting results have a high credibility and the CGSPFNs obey the power law distribution on the whole, which indicates that they are the scale-free network. In the terms of the scale-free network with higher levels of the power law distribution, the corresponding power law indexes are larger. Therefore, the level of the power law distribution about the CGSPFN in GC is higher than that in NYH in the whole period, which also illustrates the higher complexity of the CGSPFN in GC from another angle. As for the SFP and FFP, the power law indexes of the corresponding period in GC are larger than that in NYH, which illustrate that the strength of the power law distribution about the CGSPFN in GC is higher in the corresponding period and the power law distribution also shows a part of complexity and regularity. These results fully indicate that there are some differences in the two CGSPFNs.
3.4. Analysis of the Fluctuation Mode about the CGSP in NYH and GC
According to the relevant definition of the distance and average path length [
42,
43], the distance between any of two nodes in the two CGSPFNs is calculated by the Floyd algorithm [
44], and the proportion of different lengths about the path is shown in
Figure 8.
In
Figure 8, we counted the distance between any of two nodes and the proportion of the same distance, then we obtained the proportion of the length of the path. The distance between any of two nodes means the time required from one mode to be converted to another, and the average path length indicates the conversion cycle among the fluctuation modes. Therefore, the conversion cycle of the fluctuation mode of the CGSPFN can be obtained by calculating the distance and the average path length among the nodes. According to
Figure 8, the relevant statistical indicators of the fluctuation modes of the CGSPNs are given in the whole period, as shown in
Table 8.
The relevant statistical indicators show that most of the fluctuation modes display short-range correlation and their conversions are more frequent with an average of 7–8. Comparing the diameter of the CGSPFN in NYH and GC, the diameter of the CGSPFN in GC is smaller than that in NYH, but they are basically the same conversion cycle. These results provide a basis for predicting the regularity of the periodic conversion of the CGSP.
In the SFP and FFP, we give the distribution of the length path among the fluctuation modes of the CGSPFNs, as shown in
Figure 9.
In order to display more clearly and intuitively, we give the relevant statistical indicators of the fluctuation modes of the CGSPFN in the SFP and FFP, as shown in
Table 9.
According to
Table 9, the diameter of the CGSPFNs in NYH in the SFP is larger than that in GC, but it is the opposite in the FFP, which means that the path length had changed, i.e., the change of the conversion cycle. In the SFP, the conversion cycle of the fluctuation modes in GC is shorter than that in NYH, which means that the conversion is more frequent. However, it is opposite in the FFP. Moreover, there is a little difference in the value of the statistical indicators, which illustrates that the fluctuation modes of the two CGSPFNs have an approximate conversion cycle in the corresponding period. Therefore, we can predict the time of the conversion among the nodes by identifying the conversion cycle, which is beneficial to decrease the risk of the CGSP fluctuation for the decision makers.
3.5. Analysis of the Betweenness about the CGSPFNs in NYH and GC
According to the definition and calculation formula of the betweenness of the node about the network [
32,
34,
45], we explore the fluctuation mode about the CGSP in the whole, and the evolutionary relationships between the betweenness and the strengths about the CGSPFN in the whole period are analyzed, as shown in
Figure 10.
According to
Figure 10, there are less nodes with a larger betweenness and the larger betweenness of the nodes have less strength, and it appeared in an earlier time. Meanwhile, the larger node strengths have less betweenness and it appeared in an earlier time. In addition, the largest betweenness and strength in NYH are larger than that in GC, which illustrates that the nodes of the CGSPFN in NYH have stronger ability of the connectivity than that in GC.
As shown in
Figure 10, we count the betweenness of the nodes of the CGSPFN in descending order of its value in NYH and GC, then we list the first six and the corresponding strengths, the name and the first time that the node appears, as shown in
Table 10.
In terms of the relationship between the betweenness and the strength in the whole period, it can be seen that the betweenness of the nodes is larger but the strength is smaller in
Figure 8 and
Table 10, which illustrates that the nodes with less strength have important connectivity and influence in the CGSPFN.
In order to analyze the relationships between the betweenness and the strength under the temporal distribution in the SFP and FFP, their evolutionary relationships are researched, as shown in
Figure 11.
As for the CGSPFN in NYH in
Figure 11a,b, the first six betweenness of the nodes in the SFP are 0.03258, 0.0313, 0.03119, 0.03038, 0.02991 and 0.02943, respectively, and the corresponding strengths are 5, 4, 4, 3, 5 and 3. In the FFP, the first six betweenness of the nodes are 0.03367, 0.03341, 0.03309, 0.03133, 0.02939 and 0.02865, respectively, and the corresponding strengths are 5, 4, 4, 3, 5 and 4. As for the CGSPFN in GC in
Figure 11c,d, the first six betweenness of the nodes in the SFP are 0.03258, 0.0313, 0.03119, 0.03038, 0.02991 and 0.02943, respectively, and the corresponding strengths are 5, 4, 4, 3, 5 and 3. In the FFP, the first six betweenness of the nodes are 0.03367, 0.03341, 0.03309, 0.03133, 0.02939 and 0.02865, respectively, and the corresponding strengths are 5, 4, 6, 7, 5 and 5. By the comparative analysis, the betweenness of the nodes in the FFP is larger than that in SFP, which illustrates that the impact of some unexpected emergencies enhances the intermediary betweenness of the nodes and indicates a higher influence. Meanwhile, the betweenness of the nodes shows a downward trend in the SFP, i.e., the impact of the nodes decreases. Comparing the relationships between the betweenness and the strength, the nodes with the smaller strength play a major role of the connectivity in the CGSPFN, but the nodes connect with other nodes by the nodes with the larger strength. When the betweenness is higher and there is a tendency with the continual increase, which illustrates that other nodes must pass through this node so as to connect with the other nodes, then this period is an important period. According to the results, we can give an effective prejudgment of the CGSP in the next period.
3.6. Analysis of the Clustering Features between the Fluctuation Modes
According to the definition and calculation formula of the clustering coefficient [
34,
46] and average clustering coefficient [
47] of the nodes about the network, the evolutionary relationships between the clustering coefficient and the strength, the first appeared time of the nodes of the CGSPFN, are analyzed in NYH and GC, as shown in
Figure 12.
According to
Figure 12a, there are only 15 nodes of the CGSPFN whose clustering coefficients are not 0 in NYH. According to the descending order of the clustering coefficient s of the nodes, they are 0.25, 0.1875, 0.17857, 0.16667, 0.10714, 0.10606, 0.10417, 0.09488, 0.09091, 0.07778, 0.05049, 0.04545, 0.04545, 0.02778 and 0.01450, respectively. The corresponding names are ‘iDDDD’, ‘iIIII’, ‘Ddddd’, ‘DDDDD’, ‘dIIII’, ‘dDDDD’, ‘IDDDD’, ‘diiii’, ‘Iiiii’, ‘DIIII’, ‘idddd’, ‘Idddd’, ‘IIIII’, ‘iiiii’ and ‘ddddd’. Meanwhile, the corresponding strengths are 3, 8, 7, 18, 7, 11, 24, 65, 11, 15, 52, 11, 11, 45 and 23. As shown in
Figure 12b, there are 18 nodes of the CGSPFN whose clustering coefficients are not 0 in GC. According to the descending order of the clustering coefficients of the nodes, they are 0.675, 0.13636, 0.125, 0.125, 0.11667, 0.10606, 0.09876, 0.08333, 0.07692, 0.07407, 0.06667, 0.06026, 0.05324, 0.05208, 0.04167, 0.039474, 0.02273 and 0.01075, respectively. The corresponding names are ‘iiiiD’, ‘edddd’, ‘iDDDD’, ‘dDDDD’, ‘DIIII’, ‘Idddd’, ‘IDDDD’, ‘Iiiii’, ‘IIIII’, ‘iIIII’, ‘eiiii’, ‘idddd’, ‘diiii’, ‘dIIII’, ‘Diiii’, ‘DDDDD’, ‘ddddd’ and ‘iiiii’. Meanwhile, the corresponding strengths are 10, 11, 8, 6, 10, 11, 27, 18, 13, 9, 10, 56, 54, 12, 9, 19, 33 and 31. On the basis of the above results, the clustering coefficients and their average are all smaller in NYH and GC, where the average clustering coefficients are 0.00097, 0.00120, respectively; i.e., the probability that two nodes connected to the same nodes are also connected to each other is small. However, the number of the clustering coefficients that are not 0 is more than that in NYH, and the average clustering coefficients of the CGSPFN in GC are larger than that in NYH, which all illustrate that the CGSPFN in GC is more closely tied than that in NYH. As for the nodes where the clustering coefficients are not 0, most of the strength of which are smaller, but there are a few nodes with a large strength, which show that the CGSPFN is not completely random in NYH and GC, and has the features of the community. The conclusion can be found in
Figure 4, where the nodes with the same color present a community. According to these results, we can find that the obvious clustering features in the CGSPFN may occur not only in small communities, but also in large ones. In addition, we explore the temporal distribution characteristics of the nodes where the clustering coefficients are not 0, as shown in
Table 11.
According to
Table 11, we can find that the first appeared time in NYH is the same as in GC, which illustrates that the two CGSPFNs have similar time distribution characteristics in a small-time scale; but their differences increase when the time scale is expanded, which shows that the clustering of the CGSP fluctuations may be reflected not only on a large time scale, but also on a small-time scale.
In the SFP and FFP, we give the evolutionary image of the clustering coefficients of the nodes, and the clustering coefficients are not 0, as shown in
Figure 13.
As for the CGSPFN in NYH, there are only 8 nodes whose clustering coefficients are not 0 in the SFP. According to the descending order of the clustering coefficients of the nodes in the SFP, they are 0.11111, 0.11111, 0.1, 0.1, 0.1, 0.09091, 0.07778 and 0.06667, respectively. The corresponding names are ‘Diiii’, ‘IDDDD’, ‘IIIII’, ‘DDDDD’, ‘iIIII’, ‘iiiii’, ‘diiii’ and ‘DIIII’, then the corresponding strengths are 9, 9, 5, 5, 5, 11, 15 and 10. Meanwhile, there are only 9 nodes whose clustering coefficients are not 0 in the FFP. According to the descending order of the clustering coefficients of the nodes in the FFP, they are 0.25, 0.25, 0.15, 0.142857, 0.14286, 0.125, 0.1, 0.09091 and 0.05, respectively. The corresponding names are ‘ddddd’, ‘DIIII’, ‘dDDDD’, ‘diiii’, ‘Diiii’, ‘Idddd’, ‘IDDDD’, ‘iiiii’ and ‘DDDDD’, then the corresponding strengths are 4, 4, 5, 7, 7, 4, 5, 11 and 5. As for the CGSPFN in GC, there are 15 nodes whose clustering coefficients are not 0 in the SFP. According to the descending order of the CCs of the nodes in the SFP, they are 0.5, 0.33333, 0.25, 0.2, 0.1875, 0.16667, 0.11905, 0.11905, 0.10417, 0.08333, 0.07143, 0.05556, 0.05 and 0.03704, respectively. The corresponding names are ‘dDDDD’, ‘IIIII’, ‘iIIII’, ‘DDDDD’, ‘Idddd’, ‘IDDDD’, ‘Iiiii’, ‘Diiii’, ‘DIIII’, ‘edddd’, ‘idddd’, ‘Ddddd’, ‘iiiii’ and ‘ddddd’, then the corresponding strengths are 9, 9, 5, 5, 5, 11, 15 and 10. Meanwhile, there are 5 nodes whose clustering coefficients are not 0 in the FFP. According to the descending order of the clustering coefficients of the nodes in the FFP, they are 0.3, 0.2, 0.12963, 0.09615 and 0.06410, respectively. The corresponding names are ‘Diiii’, ‘diiii’, ‘idddd’, ‘Iiiii’ and ‘iiiii’, then the corresponding strengths are 5, 5, 9, 13 and 13, with average clustering coefficients shown in
Table 12.
By analyzing the clustering coefficients in the SFP and FFP, we can find that the number of the nodes whose clustering coefficients are not 0 in GC is more than that in NYH in the SFP, and it is opposite in the FFP, which illustrate that the CGSPFN structure of the GC is closer in the SFP, but the CGSPFN structure of the NYH is closer than that in GC in the FFP. These results all show that the CGSP of different harbors have more complex characteristics in different periods.