3.2. Distribution Characteristics of Weighted Centrality
Figure 4 shows the spatial distributions of three weighted centrality indices: namely, weighted degree, weighted betweenness, and weighted closeness, based on the network constructed by the total public transport flows in the week. The lighter the color, the lower the centrality value. The degree values gradually decrease from the core to peripheral areas, and high values are mainly distributed within the fourth ring road. Betweenness presents a pattern with a high-value core and multiple secondary centers. The high-value core of the closeness is mainly distributed between the second east ring road and fourth east ring road. The closeness also exhibits a decreasing trend from the core to the peripheral areas, and the area with a high value covers a wider range.
Figure 5 shows the temporal changes in the average values of the three weighted centrality indices on weekends and on weekdays. The horizontal axis represents time, and the points on the graph correspond to the median values of the different time periods.
The vertical axis represents the centrality values. Overall, three indices show quite different temporal patterns. For weekdays, the degree curve shows two peaks, which indicate a morning peak from 7:00 to 9:00 and an evening peak from 17:00 to 19:00, and the value of the early peak is greater than that of the late peak. The betweenness and closeness curve also show two peaks, but the late peak is greater than the early peak. The low point of three indices appeared at 11:00–13:00, and an extra low point appeared at 13:00–15:00 for betweenness.
Compared with weekdays, the weighted degree curve for weekends fluctuates mildly before 17:00. The peak appeared at 17:00–19:00 and then the low point appeared at 19:00–21:00. For betweenness, the curve of weighted betweenness for weekends shows a trend of high in the middle and low on both sides. The peak appeared at 15:00–17:00, which is earlier than the time of the evening peak for weekdays (17:00–19:00). Compared with weekdays, the range of fluctuation for the weighted closeness curve for weekends is smaller. The peak appeared at 17:00–19:00.
The spatial distributions of weighted centrality indices in seven periods of a day from 7:00 to 21:00 are calculated, and here, we present three of them, including the morning period from 7:00 to 9:00, noon period from 11:00 to 13:00, and after-work period from 19:00 to 21:00.
Figure 6 shows the spatial distributions on weekdays. In general, the core area of Beijing maintains an advantageous position in the networks with public transit flow. Although the distributions exhibit certain similarities for different time periods for the same index, there are some differences. The degree centrality values between the west second ring and west third ring road change with time, with a trend of increasing first and then decreasing. For betweenness, the two secondary centers between the west second ring road and west third ring road and the south second ring road and south third ring road change over time, with a trend of increasing first and then decreasing. The other sub-centers also exhibit minor changes with time. For closeness, peripheral areas change slightly with time, and the central area also exhibits minor changes with time.
Figure 7 shows the spatial distributions on weekends. Compared with weekdays, the weighted centrality values change relatively smoothly over the weekends.
3.3. Relationships between Retail Store Locations and Weighted Centrality from a Temporal Perspective
This section examines how the density distribution of retail stores may correlate with the weighted centrality indices. First, the flow network without temporal division is examined, which is constructed by the total public transit flows of the whole week.
Table 3 shows the highest correlation coefficients between various retail stores and weighted centrality indices. Pearson’s correlation analysis was conducted between the density of retail stores and weighted centrality indices.
First, most store types have rather high correlation coefficients with weighted centrality indices. Convenience stores, shopping malls, supermarkets, and electronics stores have strong correlations with all weighted centrality indices, with coefficients above 0.6. The highest correlation coefficients for each type of store are more than 0.7, and the highest coefficient is achieved by convenience stores (with values above 0.8). Specialty stores have the highest coefficient, exceeding 0.5. Only building material stores exhibit weak correlations with weighted centrality indices (the highest coefficient is less than 0.4), which is consistent with the previous findings by using street centrality indices [
22,
23,
24], which implies that building material stores may be relatively less correlated to the public transport flow. These results indicate that most of the six types of retail stores are highly correlated to weighted centralities in the public transport flow network.
Second, four types, namely, supermarkets, convenience stores, electronics stores, and specialty stores, show the highest correlations with weighted degree. Only shopping malls show the highest correlation coefficients with weighted betweenness, with the highest correlation coefficient value reaching 0.785. This finding indicates that high-grade retail stores prefer nodes that are included in paths with relatively large flows. In comparison, it has been reported that betweenness performs well in previous physical street network studies [
18,
22,
23,
24]. This is also consistent with our findings, as betweenness in street networks reflects the frequency of the shortest paths passing through, while the weighted degree in this study directly reflects public traffic volume. The results indicate that transport volume has a significant impact on the location patterns of retail stores.
Then, the flow networks for different periods of a day with a distinction between weekdays and weekends are examined.
Table 4,
Table 5 and
Table 6 show the temporal analysis results.
Table 4 and
Table 5 show the correlation coefficients to the three weighted centrality indices across store types at different periods, and
Table 6 shows the highest correlation coefficients for each store type across the periods of a day and the relative centrality indices.
Table 4 and
Table 5 indicate that correlation coefficients vary across the day. For the relationship between weighted closeness and most retail stores on weekends, there is a continuous slight upward trend in the correlation coefficients with time. The relationships between the weighted degree and building material stores on weekdays are high in the morning and evening and low at noon. However, for the weighted degrees among specialty stores on weekdays, this pattern is reversed.
Table 6 shows that most of the highest correlation coefficients are rather large both on weekends and on weekdays. Compared with the results for the total flow network (
Table 3), the values of the highest correlation coefficients here are larger, which means that analyses without time divisions may underestimate correlations. For the same store types, most types, except for shopping malls, have higher correlations on weekends. Most types show consistency in a preference for the highest centrality index from weekdays to weekends. Only the index type of specialty stores changes in degree on weekdays to betweenness on weekends, but its correlations are less than 0.7.
It is noteworthy that the three types of shopping malls, supermarkets, and convenience stores sell general commodities but differ in store size and diversity in their commodity types. For these three types, they all nearly achieve the highest correlations during the period of 19:00–21:00 for the whole week, while the only outlier is that the shopping mall type correlates more strongly to a different period of 11:00–13:00 on weekends. The same period implies that most consumers go shopping after work, but shopping behavior for malls on weekends may differ, as people may like to spend time in malls.
Another interesting result for the three types is that the highest centrality indices are different: convenience stores correlate best with degree, supermarkets correlate best with closeness, and shopping malls correlate best with betweenness. Recall that for the total flow network without periods in
Table 3, the highest centrality index for the supermarket changed here from degree to closeness. In this case, the results of the total flow network may be misleading. Moreover, recall that the degree reflects the total traffic flow, the closeness reflects the closeness to all nodes in the flow network, and the betweenness reflects the traffic corridor. Thus, it can be inferred that the higher levels of store types are associated with higher correlations to the key structure of the flow network.
For the two types of specialty stores and electronics stores, both correlate best to degree centrality on weekdays, and neither correlate best with the period of 19:00–21:00 on weekends. These results imply that people may visit these types of stores after work on weekdays and may visit them at various periods in the daytime on weekends.