## 3 Solution for DRLP
### Construction of Similarity Adjacency Matrix
However, as the mixing parameter \(\mu\) exceeds 0.5, the performance of the algorithms begins to diverge: the NMI values of the LPA_CL algorithm exhibit a cliff-like decline, completely failing at \(\mu=0.65\); except for DRLP and CNLLP, the detection accuracy of all other algorithms shows significant degradation; and when \(\mu\) rises to 0.7, the network community structure becomes highly blurred, yet the DRLP algorithm still maintains stable detection performance. This further verifies that our DRLP algorithm can detect effective community structures in networks regardless of the community size, and it is further shown that the DRLP algorithm can detect effective community structures for both large-scale and small-scale networks, regardless of having a clear or fuzzy community structure. Figure 4: Community detection results on the LFR2 network dataset. Figure 5: Community detection results on the LFR3 network dataset. ## 5 Conclusions

Addressing complex and diverse community detection tasks, this paper proposes a Degree and PageRank-based Label Propagation (DRLP) algorithm. First, we calculate the weight matrix for each node, then compute the node importance based on the weight matrix, and finally sort the nodes in descending order of importance. During the label propagation process, a novel neighbor closeness strategy is adopted, which avoids the random selection of the traditional LPA and effectively improves the accuracy of community detection. Using different evaluation indicators, DRLP was compared and analyzed with several label improvement algorithms. Simulation results on real and synthetic network datasets show that our algorithm achieves better modular density with maintaining high modularity, and it demonstrates high \(NMI\) accuracy for both small-scale and medium-to-large-scale synthetic networks. However, the algorithm's \(O(|V|^{2})\) complexity might present scalability limitations for very large networks (e.g., \(|V|>10^{6}\)), necessitating additional optimization. While DRLP demonstrates good performance across medium-to-large-scale networks, we recognize that its \(O(|V|^{2})\) complexity may impose scalability limitations for extremely large networks. Therefore, future work could explore some optimization strategies such as parallel computing or similar techniques to reduce the algorithm's time complexity. Furthermore, based on the existing framework, we plan to extend DRLP's applicability to more complex network scenarios in the future, including weighted networks, dynamic networks, and optimized implementations for overlapping community detection. Additionally, we will explore potential synergies between DRLP and GNN architectures to further enhance its performance. These comprehensive optimizations will collectively expand the algorithm's practical utility while preserving its core advantage in community detection accuracy. Conceptualization, M.L. ; data curation, M.L., A.W. and B.L. ; formal analysis, M.L. ; funding acquisition, A.W. and B.L. ; methodology, M.L. and A.W. ; project administration, M.L. ; resources, A.W. and B.L. ; software, X.G. ; supervision, A.W. ; validation, M.L., X.G. and B.L. ; visualization, M.L. and X.G. ; writing--original draft, M.L. ; writing--review and editing, A.W. and X.G. All authors have read and agreed to the published version of the manuscript. This work was funded by the Collaborative Research Projects of \"Chunhui Program\", Ministry of Education of China, grant number RZ2300003745, and in part by Shanxi Province Graduate Student Practice and Innovation Program. Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:

Complex network. Set of nodes in the G.

Set of edges in the G.

nodes in the G.

Similarity adjacency matrix. Set of neighbors of node u. The label of node u. Set of neighbors of node with same label.