# Network Structure and Synergy Characteristics in the Guangdong-Hong Kong-Macao Greater Bay Area
## 3 Research Data and Methods
### Research Methods
#### 3.3.2 Empirical Quantitative Metrics  
(1) Degree (\(k\)): The most intuitive indicator characterizing the importance of cities, measured from three perspectives: inflow, outflow, and aggregation [36]. Among them, for outflow direction, outgoing degree indicates a city's ability to disseminate or export resources and flows to the external entities. For inflow direction, the incoming degree indicates a city's ability to receive resources and flows. For aggregation, the total degree combines both inflow and outflow metrics, reflecting the overall influence and activity of the city within the network. Specifically, it is as follows:  
\[k_{out}(i)=\sum_{j}w_{ij},k_{in}(i)=\sum_{j}w_{ij},k_{w}(i)=k_{out}(i)+k_{in}(i) \tag{2}\]  
where \(k_{out}(i)\), \(k_{in}(i)\), and \(k_{w}(i)\) are the outgoing, incoming, and total degrees of the cities, respectively. (2) Clustering coefficient (C): the average of the clustering coefficients of all cities in the network [37]. \[C(i)=\frac{2E(i)}{K(i)(K(i)-1)},C=\frac{1}{N}{\sum_{i=1}^{N}C(i)} \tag{3}\]  
where \(E(i)\) is the actual number of edges among the cities connected by the city \(i\). (3) Efficiency (\(E\)): The distribution efficiency of the capital and transportation flow elements is represented by the average of the shortest path lengths between any two cities in the network [38]. \[E=\frac{1}{N(N-1)}{\sum_{i\
eq j}\frac{1}{d_{ij}}} \tag{4}\]  
where \(d_{ij}\) is the shortest path length between city \(i\) and city \(j\). \(N\) is the number of nodes in the network, \(N=11\). (4) Density (\(D\)): the ratio of the number of edges actually present in the network to the maximum number of possible edges, which is used to measure the connectivity of the network [38]. \[D=\frac{e}{N(N-1)} \tag{5}\]where \(e\) is the number of edges actually present in the network. (5) Centrality: quantifies the heterogeneity and hierarchy of the roles, positions, and functions of cities in networks of capital flows and networks of transport flows by the extent to which they are central to the network. Based on the network perspective, centrality is known as central potential and characterizes the degree of closeness of the entire network [36]. Degree characterizes the nature of individual nodes, and potential characterizes the nature of the entire graph. \[DC_{net}=\frac{\sum_{i=1}^{N}(DC(i)_{max}-DC(i))}{N-2}\]  
\[BC_{net}=\frac{\sum_{i=1}^{N}(BC(i)_{max}-BC(i))}{N-1}\]  
\[CC_{net}=\frac{2N-3}{(N-1)(N-2)}{\sum_{i=1}^{N}(CC(i)_{max}-CC(i))} \tag{6}\]  
\[DC(i)=\frac{1}{N-1}{\sum_{j\in N}w_{ij}}\]  
\[BC(i)=\frac{2}{(N-1)(N-2)}{\sum_{ \begin{subarray}{c}1\leq j<l\leq N\\ j\
eq i\
eq l\end{subarray} }}\]  
\[CC(i)=\frac{N-1}{\sum_{j=1,j\
eq l}^{N}d_{ij}} \tag{7}\]  
where \(DC_{net}\), \(BC_{net}\), and \(CC_{net}\) are the degree, proximity, and mediating centrality of the network. \(DC(i)\), \(BC(i)\), and \(CC(i)\) are the degree, proximity, and mediating centrality of the cities. \(n_{jl}\) is the number of shortest paths between city \(i\) and city \(j\), and \(n_{jl}(i)\) is the number of cities that pass through city \(i\) in the shortest path between city \(j\) and city \(l\). The above indicators make it possible to quantify the directionality of capital and transport flows while measuring the overall degree of interaction between cities. (6) Degree of coupling and coordination: describes the degree to which two or more systems interact and influence each other [39]. The degree of coupling and coordination determines the evolution of the system. Logistics and capital flow networks are independent and interacting systems. The coupling degree coordination model is used to quantify the degree of development coordination of logistics and capital flow networks in the Greater Bay Area metropolitan cluster. At the same time, the relative development coordination degree is introduced to measure the relative development level of logistics and capital flow in each city (as shown in Table 1).