## 3 Results
### 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.  
Figure 4: Spatial distributions of three weighted centrality indices of the total flow network: (**a**) degree; (**b**) betweenness; (**c**) closeness.  
Figure 3: Density distributions of different types of retail stores determined by KDE: (**a**) building material stores; (**b**) specialty stores; (**c**) supermarkets; (**d**) electronics stores; (**e**) shopping malls; (**f**) convenience stores.  
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.  
Figure 5: Temporal variations of the three weighted centrality indices on weekdays and weekends: (**a**) degree; (**b**) betweenness; (**c**) closeness.