## 3 Methodology and Data
### Network Mixing Patterns
#### 3.1.2 Measuring Assortativity Coefficient  
This paper applies theoretical ideas to examine the mixing patterns within complex networks to reveal the mixing patterns of Chinese cities. It also analyzes the general patterns exhibited by the structural features or socioeconomic attributes of cities in their mixing patterns. The mixing patterns of networks, which exhibit assortativity, is described as the similarity of node attributes at both ends of the network links. The assortativity coefficient is a key quantitative measure of assortative mixing in networks, as shown in Equation (1). \[r=\frac{\sum_{xy}xy(e_{xy}-a_{x}a_{y})}{\sigma_{a}\sigma_{b}} \tag{1}\]  
where \(x\) and \(y\) represent certain attribute values of the nodes at both ends of a link in the network. represents the proportion of links in the network where the attribute values at  
Figure 2: Mixing patterns of urban networks: (**a**) centripetal mixing pattern; (**b**) balanced mixing pattern. both ends of the link are x and y, respectively; \(a_{x}\) and \(b_{y}\) represent the proportion of links where the attribute values at one end are x or y, respectively. \(\sigma_{a}\) (\(\sigma_{b}\)) denote the standard deviations of the distributions of \(a_{x}\) (\(b_{y}\)). The value of r ranges between (\(-1\),\(1\)), and when \(\mid\) r \(\mid>0.1\), the assortative (or disassortative) mixing features of the network are significant. In this paper, the assortativity coefficients of urban agglomerations are calculated from two dimensions: structural features and socioeconomic attributes, as follows:  
(1) Structural assortativity coefficient: The micro-structural features of network nodes are used as the node attribute, including degree centrality and betweenness centrality, which represent the city's ability to occupy and control resources, respectively. The information and investment networks are directed networks, the out-degree and in-degree of nodes will be discussed as two independent node structural features. Additionally, the assortativity coefficient of betweenness centrality will also be calculated as a node attribute. Degree centrality and betweenness centrality are defined as follows: Degree centrality is a fundamental measure of node importance in networks. Based on the adjacency matrix representation, degree centrality is defined as shown in Equation (2). \[C_{d}(i)=\frac{\sum_{j=1}^{n}a_{ij}}{n-1} \tag{2}\]  
where \(a_{ij}\) represents elements in the adjacency matrix; \(\sum_{j=1}a_{ij}\) represents the number of nodes directly connected to node \(i\); \(n\) is the network size. The normalization represents the ratio of the node's actual degree to its theoretical maximum degree, thus also known as relative degree centrality. In this study, a higher degree of centrality indicates that a city occupies a more central position within the urban agglomeration, while a lower value suggests the city is more isolated. Betweenness centrality measures a node's ability to control network resources. A node achieves higher betweenness centrality when it lies on more geodesic paths between other node pairs. According to the definition, network betweenness centrality is defined as shown in Equation (3). \[C_{b}(i)=\frac{2\sum_{j}^{n}\sum_{k}^{n}b_{jk}}{n^{2}-3n+2} \tag{3}\]  
where \(b_{jk}\) represents the node's betweenness, describing the proportion of geodesic paths between nodes \(j\) and \(k\) that pass through node \(i\). Similar to the normalization of degree centrality, it standardizes the measure to eliminate the effect of network size, making betweenness centrality a measure of how close a node is to achieving maximum betweenness centrality. Specifically, in urban agglomeration networks, nodes with higher betweenness centrality occupy crucial positions within the entire urban network, effectively controlling and coordinating resource flows and personnel movement between cities.