## 3 Solution for DRLP
### Construction of Similarity Adjacency Matrix  
In this section, we present a weighted adjacency matrix that can reflect the correlation between network nodes. Many network analysis techniques, such as Shortest Path Distance [36], Average Path Length [4], Diameter [4], Clustering Coefficient [4], and Jaccard Similarity [37], can be used to measure the distance or similarity between nodes in a graph or complex network. Among these metrics, the Shortest Path Distance and Diameter are usually necessary for calculating the shortest path lengths between nodes, resulting in high time complexity, which gives poor performance in large-scale networks. Although the density of connections between nodes and their neighbors is provided by the Clustering Coefficient, the distance or similarity between individual nodes is not directly reflected by the coefficient. Jaccard Similarity partially denotes the intersection of neighbor sets between two nodes, but in real-world social networks, the similarity or distance between two nodes is not always consistent. For example, if node A has only node B as its neighbor while node B has other neighbors besides A, the influence of B on A is much greater than that of A on B. Therefore, we propose a new metric to measure the distance or similarity between nodes. **Definition 2** (Similarity adjacency matrix, \(S\)). : _The model of a social network is an undirected complex graph \(G=(V,E)\), where each node \(u\) in \(V\) is a user or entity in the social network, and each edge \(e=(u,v)\) in \(E\) is the social relationship between entities \(u\) and \(v\). For any two nodes \(u\) and \(v\), the similarity index of node \(u\) for node \(v\) in graph \(G\) is defined as follows:_  
\[S(u,v)=\frac{|N_{u}\cap N_{v}|+1}{|N_{u}|} \tag{2}\]  
The method for calculating the similarity adjacency matrix \(S\) is given in Algorithm 1. ```
Input : undirected complex network \(G\)(\(V\),\(E\)), adjacency matrix \(A\) Output : Similarity adjacency matrix \(S\in\mathbb{R}^{|V|\times|V|}\) \(1:S\leftarrow\mathbf{0.0}_{|V|\times|V|}\) \(2:\textbf{for}\textit{each}\textit{node}\ u\ \textit{in}\ V\ \textbf{do}\) \(3:\quad\textbf{for}\textit{each}\ \textit{node}\ v\ \textit{in}\ V\ \textbf{do}\) \(4:\quad\textbf{if}\ \textit{



**Definition 3** (Node importance, NI). : _For any node \(u\) in \(V\), the node importance index of node \(u\) in graph \(G\) is defined as follows:_

\[NI(u)=(1-\alpha)+\alpha*\sum_{v\in N_{u}}\frac{DC(u)}{d_{v}}*S(u,v)\quad\beta=1 \tag{3}\]

\[NI(u)=(1-\alpha)+\alpha*\sum_{v\in N_{u}}\frac{NI(v)}{d_{v}}\qquad\qquad\beta>1 \tag{4}\]

_where \(NI(u)\) is the node importance index of node \(u\), \(d_{v}\) is the degree of node \(v\), and \(\alpha\) is the damping factor. The importance of each node is initialized first based on the degree centrality and neighborhood relevance among the given importance metrics, and then the importance of each node is iteratively updated following the PageRank strategy, with the number of iterations denoted as \(\beta\)._

The method for calculating the node importance (NI) is given in Algorithm 2. ```
Input : undirected complex network \(G\)(\(V\),\(E\)), Similarity adjacency matrix \(S\) Output : Node Importance List NI\(\in\mathbb{R}^{|V|}\)\(1:NI\leftarrow\mathbf{0}_{|V|}\)\(2:\)for each node \(u\) in \(V\)do\(3:NI(u)\leftarrow\) calculate the initial importance of nodes according to Equation (2)\(4:\)endfor \(5:\)repeat: \(6:\)for each node \(u\) in \(V\)do\(7:NI(u)\leftarrow\) calculate the final importance of nodes according to Equation (3)\(8:\)endfor \(9:\)until (Reach the number of iterations \(\beta\))\(10:\)returnNI
```
**Algorithm 2** Solving node importance index

### Modified Label Selection Process

In this section, we propose a modified label update rule and introduce a preference label selection strategy. Differently from the existing algorithms, we not only analyze the frequency of labels among the neighbors of the node, but also refer to the intimacy between nodes and their neighbors and the influence of neighbor nodes.