# C(RD-SH) - Relative Degree Structural Hole Centrality

#### Definition

This method ranked nodes based on structural hole in combination with the degree ratio of a node and its neighbors.The structural hole gives a comprehensive attention of the information about the node topology in relation to its neighbors, whereas the degree ratio of nodes reflects its significance against the neighbors. Combination of these two measures summarized in the structural hole leverage matrix demonstrates the importance of a node according to its position in the network structure.

The network constraint index used to measure the closeness centrality and the structural holes of network. The network constraint index of node $i$ is expressed as:

$$C_{cst}(i)={\underset{j\in \sigma (i)}{\sum}}(p_{ij}+{\underset{k,k\ne i,k\ne j}{\sum}} p_{ik} p_{kj})^2$$

where, node $k$ is an indirect node that connects node $i$ to node $j$, $p_{ik}$ and $p_{kj}$ are respectively the proportion of total attempt that node $i$ and node $j$ do for maintaining its neighborhood with its common neighbor $k$ and expressed as:

$$p_{ij}=a_{ij}/{\underset{j\in \sigma (i)}{\sum}} a_{ij}$$

where, $p_{ij}$ is the proportion of efforts which node $i$ attempts to maintain the neighborhood relationship with node $j$, $\sigma (i)$ is the set of nodes which are neighbors of the node $i$. $a_{ij} = 1$ if node $i$ is connected to node $j$. Otherwise, $a_{ij} = 0$.

Combination of leverage matrix and network constraint index used to create Structural Hole Leverage Matrix.

This measure includes both local information of nodes (the correlations between degree of a node and those of its direct neighbors) and their positional information in the network (provided by structural holes concept). It is expected that this measure can identify which nodes act as influential diffusers especially among the bridge nodes.

The network constraint index used to measure the closeness centrality and the structural holes of network. The network constraint index of node $i$ is expressed as:

$$C_{cst}(i)={\underset{j\in \sigma (i)}{\sum}}(p_{ij}+{\underset{k,k\ne i,k\ne j}{\sum}} p_{ik} p_{kj})^2$$

where, node $k$ is an indirect node that connects node $i$ to node $j$, $p_{ik}$ and $p_{kj}$ are respectively the proportion of total attempt that node $i$ and node $j$ do for maintaining its neighborhood with its common neighbor $k$ and expressed as:

$$p_{ij}=a_{ij}/{\underset{j\in \sigma (i)}{\sum}} a_{ij}$$

where, $p_{ij}$ is the proportion of efforts which node $i$ attempts to maintain the neighborhood relationship with node $j$, $\sigma (i)$ is the set of nodes which are neighbors of the node $i$. $a_{ij} = 1$ if node $i$ is connected to node $j$. Otherwise, $a_{ij} = 0$.

Combination of leverage matrix and network constraint index used to create Structural Hole Leverage Matrix.

This measure includes both local information of nodes (the correlations between degree of a node and those of its direct neighbors) and their positional information in the network (provided by structural holes concept). It is expected that this measure can identify which nodes act as influential diffusers especially among the bridge nodes.