## 3 Solution for DRLP
### Construction of Similarity Adjacency Matrix
And higher Q-values mean better community division results, stronger connections within communities, and sparser connections between communities. \(Q\) is defined as follows:

\[Q_{c}=\frac{1}{2|E|}\sum\limits_{i,j\in V}[A_{ij}-\frac{d_{i}d_{j}}{2|E|}] \delta(c_{i},c_{j}) \tag{7}\]

where \(Q_{c}\) is the modularity, \(i\) and \(j\) are two nodes in the graph, \(d_{i}\) is the degree of node \(i\), and \(\delta(c_{i},c_{j})\) is the Dirac function. \(\delta(c_{i},c_{j})=1\) if \(c_{i}=c_{j}\), which indicates that node i and node j are in the same community; otherwise, \(\delta(c_{i},c_{j})=0\). #### 4.1.3 Modularity Density D

Modularity density is another widely used evaluation metric for measuring the effectiveness of community partitioning. In general, if nodes in the network have more connections within the same community and fewer connections between different communities, the \(D\) will be higher; this means that the network is more densely structured and can be more easily divided into independent functional modules or communities. \(D\) is defined as follows:

\[D_{c}=\sum\limits_{i=1}^{N}\frac{L(c_{i},c_{i})-L(c_{i},\overline{c_{i}})}{|c _{i}|} \tag{8}\]

where \(D_{c}\) is the modularity density of the community partitioning result. \(L(c_{i},c_{i})\) is the number of connections between nodes within community \(c_{i}\), \(L(c_{i},\overline{c_{i}})\) is the number of connections between nodes within community \(c_{i}\) and nodes in other communities, and \(|c_{i}|\) is the number of nodes in community \(c_{i}\). ### Comparative Algorithms and Test Set

To assess the effectiveness of DRLP, we compare it with various community detection algorithms on three synthetic networks and seven real-world network datasets. Since DRLP is an optimization algorithm based on label propagation, the compared algorithms include the traditional LPA [19], spectral clustering (SC), and the Louvain algorithm [40], as well as existing optimized algorithms based on label propagation, which include CNLLP [28], LPA_CL [41], and CDEP [42]. Additionally, all experiments are implemented in Python 3.9.18 and executed on a Windows 10 environment with an Intel(r) Core(tm) [EMAIL_ADDRESS] GHz CPU and 32.00 GB of memory. Seven real-world network datasets are selected from the datasets applied in most community detection algorithms. Their sizes range from 34 nodes to 23,133 nodes, covering both small-scale and large-scale networks. They are the Zachary karate club dataset [43], the dolphin dataset [44], the polbooks dataset [45], the football dataset [6], the PGP dataset [46], the CA_AstroPh dataset, and the CA_CondMat dataset [47]. By comparing the DRLP algorithm on datasets of different sizes, we comprehensively validate its effectiveness. Table 1 provides the number of node (_Nodes_) in the network and the node relationships (_edges_). Additionally, the actual number of communities (\(C_{n}\)) of the real-world network datasets is given if it exits. For synthetic networks, we consider both the Stochastic Block Model (SBM) [48] and the LFR benchmark network [49], ultimately adopting the latter due to its richer parameterization for modeling realistic community structures. The LFR network includes various adjustable parameters: the number of nodes (_Nodes_), the node degree distribution (the average node degree (\(k_{\textit{race}}\)), the max node degree (\(k_{\textit{max}}\))), community size (the min community size \(C_{\textit{min}}\) and the max community size \(C_{\textit{max}}\)), and the power distribution exponent for node degree and power distribution exponent for community size (\(\tau_{1}\), \(\tau_{2}\)). Furthermore, the fuzziness of community boundaries in the LFR network can be influenced by adjusting the mixing parameter \(\mu\). As \(\mu\) increases, the community structure becomes more fuzzy, making community detection more difficult accordingly. We perform experiments on three different sizes of LFR networks (LFR1, LFR2, and LFR3).