## 3 Methodology
### Indices Deriving: Calculating Crowd Dynamics Indicators  
Crowd dynamics indicators describe the characteristics of attractions as affected by human movements in tourism, and provide the source data for route design and adjustment. Based on previous research and the impact of crowd mobility on the tourism experience, this paper derives three crowd dynamics indicators of each attraction from the tourism activity chains: (1) crowd flow indicator (relative crowding at the attraction); (2) crowd interaction indicators (degree of centrality in the crowd interaction network of attractions); (3) crowd structure indicators (similarity of crowd structure between attractions). 1. Crowd Flow Indicator. To some extent, the crowding of attractions represents a higher popularity and a more entertaining atmosphere; however, \"over-tourism\" poses more significant security risks and reduces the quality of the tourism experience. Since the congestion at each attraction has time-period regularity, the crowding experienced on the tour can be reduced by avoiding the peak hours at each attraction. The crowd flow indicator (\(\overline{F}_{i,w_{k}}\)) represents the average relative crowding between different days in the same time window, which is calculated as follows: \[\overline{F}_{i,w_{k}}=\frac{1}{N_{D}}\sum_{D=1}^{N_{D}}\frac{flow_{i,D,w_{k}}} {MAX(flow_{i})}\] (1) where \(flow_{i,D,w_{k}}\) denotes the number of tourists at \(A_{i}\) within time window \(w_{k}\) of day D; \(MAX(flow_{i})\) denotes the maximum number of tourists at \(A_{i}\) in all time windows over several days; \(N_{D}\) is the total number of days. 2. Crowd Interaction Indicator. Crowd interaction refers to the movement of crowds between attractions resulting from the transformation of tourists. The directed crowd interactive network of attractions in different time windows is constructed by calculating the number of transfers between attractions with the tourism activity chains of all tourists, which are used as the weight of edges. In the interaction network, the nodes with a higher degree centrality have higher significance and status. The in-degree and out-degree of centrality of the node in different time windows can be calculated as follows: \[D_{i,in,W_{k}}=\sum_{j}^{N_{A}}r_{ij,in,W_{k}}\] (2) \[D_{i,out,W_{k}}=\sum_{j}^{N_{A}}r_{ij,out,W_{k}}\] (3) where \(r_{ij,in,W_{k}}\) is the number of tourists from attraction \(A_{i}\) to \(A_{j}\) in time window \(W_{k}\); \(r_{ij,out,w_{k}}\) is the number of tourists from \(A_{j}\) to \(A_{i}\) in \(W_{k}\). 3. Crowd Structure Indicator. The tourism behavioral characteristics of tourists can reflect the characteristics of the target audience of the attraction, which can be obtained from the tourism activity chains, listed as follows. (1) duration of stay at an attraction (\(D_{A}\)); (2) average ticket expense per day (\(E_{Avg}\)); (3) average number of attractions per day (\(N_{Avg}\)); (4) average duration of stay per attraction (\(D_{Avg}\)); (5) number of days to play (\(D_{T}\)); (6) activity entropy (\(A_{e}\)); (7) radius of gyration (\(R_{g}\)). Activity entropy measures the diversity of an individual's mobility behavior [57]: \[A_{e}=-\sum_{i=1}^{N_{A}}p_{i}\text{log}(p_{i})\] (4)where \(p_{i}\) is the proportion of stay time at attraction \(A_{i}\). A larger value for \(A_{e}\) implies the tourist has a higher diversity of activities. Radius of gyration is commonly used to evaluate the spatial dispersion of an individual's movement [58]:  
\[R_{g}=\sqrt{\frac{\sum_{i=1}^{N_{A}}{(\overrightarrow{l_{i}}-\overrightarrow{l _{c}})}^{2}}{N_{A}}} \tag{5}\]  
where \(\overrightarrow{l_{i}}\) denotes the \(x\), \(y\) coordinates vector of \(A_{i}\); \(l_{c}=\sum_{i=1}^{N_{A}}{\overrightarrow{l_{i}}/N_{A}}\), which refers to the center of all the location of tourism attractions in the tourist's tourism activity chain. A large value for \(R_{g}\) reveals that the tourist is willing to travel in a large space.