## 3 Solution for DRLP
### Construction of Similarity Adjacency Matrix
According to our strategy, the greater the proportion of shared neighbors in all the neighbors of the target node, the greater the influence over the target node. Similarly, the node with greater importance will have a greater influence on the target node. Therefore, we define the label influence as the impact of label groups among neighbors on the node. **Definition 4** (Label influence, \(Lif\)). : _For any node \(u\) in \(V\) and \(v\) in \(NSL(u,l_{c})\), the influence of the label \(l_{c}\) on node \(u\) is defined as follows:_

\[LIF(u,l_{c})=\sum_{v\in NSL(u,l_{c})}S(u,v)*NI(v) \tag{5}\]

_where \(LIF(u,l_{c})\) is the influence of label \(l_{c}\) on node \(u\), and \(NSL(u,l_{c})\) is the set of neighbors of node \(u\) with label \(l_{c}\)._

The process of improved label selection is given in Algorithm 3. ```
Input : undirected complex network G(V,E), \(S\), NI, LC node u: Node that are ready to update Output : \(l_{o}\):The label update result of node u \(1:NSL\leftarrow\phi,l_{max}\leftarrow-\infty\) \(2:t\gets 0,C\leftarrow\phi\) \(3:\)for each node v in \(N_{u}\)do \(4:\)\(NSL\leftarrow\) (v/\(l_{v}\)) // Store node v and \(l_{v}\) into NSL \(5:\)endfor \(6:\)for each label \(l_{c}\) in \(NSL\)do \(7:\)\(LIF(u,l_{c})\leftarrow\) calculate the influence of the label \(l_{c}\) according to Equation (4) \(8:\)if\(LIF(\textbf{u},l_{c})>l_{max}\)then \(9:\)\(l_{max}\gets LIF(u,l_{c}),\)\(l_{o}\gets l_{c}\) \(10:\)endif \(11:\)endfor \(12:\)return\(l_{o}\)
```
**Algorithm 3** The modified label selection process

### DegreeRank Label Propagation (DRLP) Algorithm

After giving some core definitions of the algorithm, this section demonstrates the detailed process of the DegreeRank Label Propagation Algorithm, as shown in Algorithm 4. ```
Input : undirected complex network G(V,E), The max number of iterations \(t_{max}\) Output : Community set \(C\) \(1:LC\leftarrow\) give each node a unique label \(2:t\gets 0,C\leftarrow\phi\) \(3:N\Gamma\leftarrow\) Invoke Algorithm \(2\) \(4:\)\(SNO\leftarrow\) Sort the nodes by NI value \(5:\) repeat \(6:\)\(L\gets LC\) \(7:\)for node u in \(SNO\)do \(8:\)\(L(u)\gets l_{o}\) Invoke Algorithm \(3\) \(9:\) endfor \(10:\)until(\(LC\) = \(L\) or t = \(t_{max}\)) \(11:\)return\(C\)
```

**Algorithm 4** DRLP algorithm

## 4 Experimental Results and Analysis

### Evaluation Metrics

#### 4.1.1 Normalized Mutual Information (NMI)

\(NMI\)[39] is commonly used to evaluate the degree of similarity between the results of two clustered partitions. In this article, it is used to measure the effectiveness of the network partition results. The definition of \(NMI\) is as follows:

\[NMI(C_{x},C_{y})=\frac{2\sum\limits_{c_{i}\in C_{x}}^{C_{x}}\sum\limits_{c_{j} \in C_{y}}^{C_{y}}M_{ij}log(\frac{M_{i}M_{j}}{M_{ij}M})}{\sum\limits_{c_{i}\in C _{x}}^{C_{x}}M_{i}log(\frac{M_{i}}{M})+\sum\limits_{c_{j}\in C_{y}}^{C_{y}}M_{ j}log(\frac{M_{j}}{M})} \tag{6}\]

where \(C_{x}\) is the set of partitioning results of the real community; \(C_{y}\) is the set of partitioning results measured by the experiment. \(c_{i}\) is the result of a community division in \(C_{x}\). \(M\) is the partition information matrix, where the rows are actual communities and the columns are communities detected by experiments; \(M_{ij}\) is the number of nodes that are contained by both community \(C_{i}\) and community \(C_{j}\). \(M_{i}\) is the sum of the row of the partition information matrix representing the community \(c_{i}\), and \(M_{j}\) is the sum of the column of the community \(c_{j}\). #### 4.1.2 Modularity Q

Modularity [9] is a globally relevant metric of the network that could quantify the topological characteristics of communities. It is usually used to evaluate the quality of community partitioning results. The value of modularity is generally distributed in the range of [0, 1]; when \(Q=0\), it means that there is no community structure, and \(Q=1\) means that all the nodes in the same community.