#### 3.1.3 Structure Hole  
In social networks, individuals occupying structural hole positions hold a significant advantage by acting as bridges between otherwise disconnected communities. These nodes facilitate the flow of information across different groups, granting them access to more resources and opportunities. The concept of structural holes provides a valuable framework for evaluating the significance of a node's position within the network structure. In networks with multiple communities, nodes occupying structural holes play a critical role in enabling the efficient circulation of information across the entire network.  
As shown in Figure 3, node B occupies a structurally pivotal position. If node B were removed, the network would fragment into three disconnected components. This highlights the importance of structural hole nodes like B, which serve as the only connection between different parts of the network. Node B not only has access to information from multiple communities but also acts as a shortcut for information flow, enhancing its influence. The strategic position of structural hole users can even grant them a certain level of \"power\" over neighboring nodes, as they control the flow of information between otherwise isolated groups.  
To quantify the importance of structural hole positions in social networks, we used various metrics, including effective network size [40], network constraint, and betweenness centrality, to measure the presence of structural holes. The effective size of the network for node \(i\) is calculated using Equation (5).  
\[ES_{i}=\sum_{j\in\tau(i)}\left(1-\sum_{q}p_{iq}p_{jq}\right)=n-\frac{1}{n}\sum _{q}P_{iq}P_{jq} \tag{5}\]  
where \(n\) represents the degree of node \(i\), \(\tau(i)\) is the set of all neighbors of node \(i\), and node \(q\) is a common neighbor of nodes \(i\) and \(j\). The terms \(p_{iq}\) and \(p_{jq}\) represent the proportion of node \(q\) among the neighbors of nodes \(i\) and \(j\), respectively.  
Figure 1: Schematic diagram of local clustering coefficient.  
Figure 2: Clustering coefficient. The red node labeled \(i\) represents the central node in the clustering coefficient calculation, while the blue nodes represent its neighboring nodes.  
The network constraint coefficient refers to the degree of dependency between nodes. The larger the value, the stronger the constraint effect, meaning the node's capacity is weaker and the likelihood of forming structural holes is smaller. The constraint is measured based on dependency relationships, calculating the strength of ties across structural holes. The formula for constraint is shown in Equation (6).  
\[C_{i}=\sum_{j\in\tau(i)}\left(p_{ij}+\sum_{q\
eq i\
eq j}p_{iq}p_{qj}\right)^{2} \tag{6}\]  
where \(\tau(i)\) is the set of neighboring nodes of node \(i\) and \(p_{ij}\) represents the proportion of the neighbor relationship between nodes \(i\) and \(j\) relative to all neighbors of node \(i\). The terms \(p_{iq}\) and \(p_{qj}\) are calculated similarly to \(p_{ij}\). The relationships among nodes \(i\), \(j\), and \(q\) are illustrated in Figure 4.  
Betweenness centrality refers to the extent to which a node occupies a structural hole position within the network. The larger the value, the more central the node is within the structural hole. The calculation for the betweenness centrality of node \(i\) is shown in Equation (7).  
\[C_{B}(i)=\sum_{u}\sum_{v}\frac{g_{uv}(i)}{g_{uv}} \tag{7}\]  
In this formula, \(g_{uv}\) represents the total number of shortest paths between nodes \(u\) and \(v\), and \(g_{uv}(i)\) represents the number of shortest paths between nodes \(u\) and \(v\) that pass through node \(i\). The more frequently node \(i\) appears in these shortest paths, the more central it is within the network, particularly in bridging otherwise disconnected communities.