Overlapping community discovery based on graph embedding and label propagation algorithm

Node importance evaluation integrating multidimensional indicators

When evaluating the importance of a node in overlapping community discovery algorithms, existing research mainly uses a single indicator, such as degree centrality, to measure it (Liu et al., 2020). Given this, we propose a node importance evaluation indicator integrating multidimensional features, comprehensively considering three dimensions: degree centrality, PageRank value, and the clustering coefficient of nodes. First, degree centrality measures the direct connection relationship between nodes and other nodes. Second, the PageRank value evaluates the influence of nodes on the entire network. Finally, the clustering coefficient measures the degree of connectivity between neighbors of a node. After normalization, we fuse these three indicators to avoid the potential one-sidedness of a single indicator. In this article, the importance of node $v$ is denoted as $NI\left(v\right)$, as shown in Formula (5). Here, $DC\mathrm{\_}norm\left(v\right)$, $PR\mathrm{\_}norm\left(v\right)$, and $CC\mathrm{\_}norm\left(v\right)$ represent the normalized degree centrality, normalized PageRank value, and normalized clustering coefficient of node $v$, respectively. Their calculation methods are shown in Formulas (6), (7), and (8), respectively.

(5) $$NI\left(v\right)=DC\mathrm{\_}norm\left(v\right)\times PR\mathrm{\_}norm\left(v\right)\times \left(1+CC\mathrm{\_}norm\left(v\right)\right).$$

(6) $$DC\mathrm{\_}norm\left(v\right)={\displaystyle \frac{DC\left(v\right)}{{max}_{u\in V}DC\left(u\right)}.}$$

(7) $$PR\mathrm{\_}norm\left(v\right)={\displaystyle \frac{PR\left(v\right)}{{max}_{u\in V}PR\left(u\right)}.}$$

(8) $$CC\mathrm{\_}norm\left(v\right)={\displaystyle \frac{CC\left(v\right)}{{max}_{u\in V}CC\left(u\right)}.}$$

Here, $DC\left(v\right)$, $PR\left(v\right)$, and $CC\left(v\right)$ are the degree centrality, PageRank value, and clustering coefficient of node $v$ defined in Formulas (1), (2), and (3), respectively. $Max()$ represents taking the maximum value of the corresponding indicator.

The strategy of label initialization

Based on the above node importance evaluation method, this article adopts the following strategy for label initialization. Firstly, we calculate the importance values of all nodes in the network. Secondly, we sort them in descending order to form an updated sequence of node labels. Finally, for each node, we assign a unique initial label and set the attribution coefficient of its initial label to 1. The pseudocode for the node importance-based label initialization strategy is presented as follows.

Label update strategy based on node attribution coefficient

When nodes receive multiple candidate labels with identical attribution coefficients during filtering, random label selection compromises algorithm stability (Li et al., 2022b). To reduce the instability and low accuracy of community discovery results caused by the randomness of label selection, the algorithm proposed in this article introduces the Node2vec graph embedding technique. Our method uses cosine similarity to construct a node similarity matrix and introduces a novel Label Membership Coefficient $\left(LMC\right)$. This coefficient not only considers the label weights of neighboring nodes but also introduces node similarity based on Node2vec as a weight factor, which enables the label propagation process to reflect the strength of relationships between nodes more accurately.

The label membership coefficient of a node reflects the degree to which that node belongs to a community. The membership coefficient of node $v$ to label $l$ is denoted as $LMC\left(v,l\right)$, which is defined as Formula (9), where $L\left(v\right)$ represents the label set of node $v$. Here, $L\left(v\right)$ represents the set of neighboring labels of node $v$, defined as shown in Formula (10).

(9) $$LMC\left(v,l\right)={\displaystyle \frac{{\sum }_{u\in \mathrm{\Gamma }\left(v\right)}sim\left(v,u\right)\times LMC\left(u,l\right)}{{\sum }_{u\in \mathrm{\Gamma }\left(v\right)}{\sum }_{l^{\prime }\in L\left(u\right)}sim\left(v,u\right)\times LMC\left(u,l^{\prime }\right)}.}$$

(10) $$L\left(v\right)=\{\left(l,LMC\left(u,l\right)\right)|u\in \mathrm{\Gamma }\left(v\right)\}.$$

Rules of adaptive label filtering

This article introduces an adaptive label filtering mechanism to ensure the effectiveness of label propagation. For node v, we set a label filtering threshold $\theta ={\displaystyle \frac{1}{\left|L\left(v\right)\right|}}$. Here, $\left|L\left(v\right)\right|$ represents the number of labels received by node $v$. If the attribution coefficient $LMC\left(v,l\right)$ of a particular label $l$ to node $v$ is lower than the threshold $\theta$, we delete the label. If a node retains no neighbor labels after filtering, it keeps the neighbor label with the highest attribution coefficient.

Label propagation process

The specific rules for the label propagation process are as follows.

• (1)

  Data preprocessing. Using the Node2vec model to generate embedding vectors for nodes, calculate the cosine similarity between nodes based on Formula (4) and construct the inter-node similarity matrix S based on this.

• (2)

  Community label initialization. For each node v in network graph G, calculate each node’s importance index NI(v) according to Formula (5), sort all nodes in descending order according to their NI values, and form an updated sequence of node labels, denoted as update_sequence. Setting each node’s initial label attribution coefficient LMC(v, l) is 1.

• (3)

  Label collection. Collect the label information of neighboring nodes for node v to be updated in updates_sequence and form a neighbor label set L(v) according to Formula (10).

• (4)

  Calculation of attribution coefficient. Based on the similarity matrix S and the label set L(v) of nodes, calculate the new attribution coefficient LMC(v, l) of node v to label l according to Formula (9).

• (5)

  Label filtering. Filter the label attribution coefficient of node v according to the set threshold $\theta$, and delete labels below the threshold. If no labels remain after filtering, the algorithm retains the label with the highest attribution coefficient.

• (6)

  Normalization processing. Normalize the attribution coefficients in the retained label set so that the sum of the attribution coefficients for all labels is 1, resulting in a new label set L’(v) for node v.

• (7)

  Iteration until termination. Repeat steps (3) to (6), with one iteration per round. When the algorithm reaches the maximum number of iterations or the label attribution coefficients of all nodes remain unchanged between two consecutive iterations, the algorithm ends. Ultimately, all labels retained by each node represent the community to which it belongs.

Real datasets

We used six real datasets in the experiment. These datasets come from the Stanford University datasets website. Table 1 shows the features of the datasets, where $\left|V\right|$ represents the number of nodes, $\left|E\right|$ denotes the number of edges, |Ω| is the number of real communities in the network, $max\left(k\right)$ is the maximum value of node degree, <k> represents the average degree of nodes, <d> denotes the average path length of the network, and <c> signifies the average clustering coefficient of nodes.

Comparison with classic overlapping community discovery algorithms

We compared and analyzed the algorithm proposed in this study with five classic overlapping community discovery algorithms on the six real datasets mentioned above. We ran each algorithm independently 100 times on the corresponding datasets and obtained the average $EQ$ of the six algorithms across the six datasets, as shown in Fig. 1.

Experimental results demonstrate that GELPA-OCD consistently outperforms all five baseline algorithms across all datasets regarding $EQ$ values. Especially on large-scale networks such as Power and the Internet, the performance of the proposed algorithm significantly improved, with an average improvement of 94.97% and 88.56%, respectively, demonstrating its superiority in handling large-scale networks. In the medium-scaled network, compared with other algorithms, the proposed algorithm improved $EQ$ values by 4.77~38.31% (average improvement of 27.07%) on the Dolphin dataset, 2.66~18.55% (average improvement of 10.33%) on the Football dataset, and 3.02~46.05% (average improvement of 16.79%) on the Polbooks dataset. On the small-scale dataset Karate, the proposed algorithm achieved an average improvement of 39.68% in $EQ$ values compared to the other five algorithms. The experimental results fully validated the proposed algorithm’s effectiveness, correctness, and accuracy in discovering overlapping communities.

Performance comparison with newly appearing algorithms

To further verify the correctness and effectiveness of the proposed algorithm for overlapping community detection, we conducted comparative analyses with seven recent community detection algorithms: CLEM (Xue & Tang, 2020), NIEM (Yang et al., 2023), EMOFM (Tian, Yang & Zhang, 2019), OCLN (Yang et al., 2023), CFCD (Zhang, Ding & Yang, 2019), CDSAT (Jabbour et al., 2020), and PCDSAT (Jabbour et al., 2020). We selected four classic datasets, Karate, Dolphin, Football, and Polbooks, and used the $EQ$ value as the evaluation index. Table 2 shows the results.

Table 2 showed that the $EQ$ values computed for the community discovery results of the eight algorithms on the four classic datasets were not completely the same. However, the GELPA-OCD algorithm proposed in this article performed well on all four classic networks, with $EQ$ values higher than those of the other seven algorithms. The above results verified the proposed algorithm’s effectiveness, correctness, and higher quality for overlapping community discovery.

Algorithm stability analysis

We conducted 100 repeated experiments on the Karate and Polbooks datasets to evaluate algorithm stability, comparing GELPA-OCD with two classic algorithms, COPRA and LFM. Figure 2 presents the $EQ$ values across all experiments.

As shown in Fig. 2, the proposed algorithm showed significant advantages in stabilizing community discovery results. On the Karate and Polbooks datasets, the GELPA-OCD algorithm’s $EQ$ values remained unchanged with minimal fluctuations, while the COPRA and LFM algorithms showed significant fluctuations. The consistency of EQ values across multiple runs is crucial as it indicates algorithmic stability and reliability. In practical applications, stable algorithms ensure that users obtain consistent community detection results regardless of random initialization, which is essential for reproducible research and reliable decision-making processes. The experimental results showed that the proposed algorithm effectively improved the randomness of node update order by introducing a multidimensional feature-based node importance evaluation index. The improved label propagation process used similarity weights and adaptive label filtering thresholds, effectively improving algorithm stability and enhancing the quality of community discovery.

Analysis of algorithm convergence

To verify the convergence of the proposed algorithm, we recorded label changes of network nodes with increasing iteration numbers for both the proposed GELPA-OCD algorithm and comparative algorithms. The maximum number of iterations in the experiments was set to 30 for small and 300 for large datasets. Taking the classic Karate and Football datasets as examples, Fig. 3 illustrates the convergence process of three algorithms, with the horizontal axis representing the number of iterations and the vertical axis showing the proportion of nodes whose labels changed in the current iteration relative to the total number of nodes in the network, i.e., the label change rate.

As shown in Fig. 3, all algorithms converged within 10 iterations, but GELPA-OCD converged faster. On the Karate network, convergence reached just four iterations, compared to 6 and 10 for BMLPA and COPRA, respectively. On the more complex Football network, GELPA-OCD achieved near-zero label change by the 3rd iteration, while BMLPA and COPRA required 7 to 10 iterations. These results confirmed the proposed algorithm’s faster convergence and improved efficiency.

Artificial datasets

Due to the unclear community structure of many real network datasets, it is not easy to effectively measure the accuracy and effectiveness of algorithms. Therefore, we used an LFR benchmark (Lancichinetti, Fortunato & Radicchi, 2008) to generate four groups of artificial networks: A, B, C, and D. The total number of artificial datasets is 25. Table 3 shows the basic information on artificial networks. Among them, $\left|V\right|$ is the number of network nodes, and the algorithm in this article takes values from 1,000 to 5,000; The mixing parameter $\mu$ is defined as $\mu ={\displaystyle \frac{d_{out\left(v\right)}}{d_v}}$, which measures the ratio of external to total connections for each node, with values ranging from 0.1 to 0.5 in this article; $k$ is the average degree of nodes, with a uniform value of 10; $max\left(k\right)$ is the maximum degree of the network, with a uniform value of 50; min|Ω| represents the minimum size of the community, with a uniform value of 10; max|Ω| represents the maximum size of the community, with a uniform value of 50; $\left|on\right|$ is the number of overlapping nodes in the network, with values ranging from 100 to 500 in this article; $\left|om\right|$represents the number of communities to which overlapping nodes belong, with values ranging from 2 to 8 in this article.

Comparison of NMI values for community discovery of different algorithms

We compared the proposed algorithm with five others on four groups of synthetic datasets, using $NMI$ as the evaluation metric to assess performance across different network structures. Figure 4 shows the results.

Figure 4 shows that the proposed algorithm performed best on the four artificial networks, followed by the LPANNI algorithm. In Group A, as the mixing parameter increased, all algorithms showed declining $NMI$ values, but our algorithm maintained the highest scores throughout, demonstrating strong robustness in fuzzy community structures. In Group B, increasing the number of overlapping nodes from 200 to 500 made the networks more complex. Although all methods showed reduced performance, our algorithm consistently achieved the highest $NMI$. In Group C, as the number of communities per overlapping node increased, all algorithms experienced performance drops, with BMLPA and CPM declining the most. Our algorithm showed the slowest decline, indicating superior performance in highly overlapping scenarios. In Group D, as the network grew from 2,000 to 5,000 nodes, the proposed algorithm’s $NMI$ values remained stable and slightly increased, confirming its scalability and strong performance on large-scale networks.

Quality assessment of overlapping node detection

To more comprehensively evaluate the effectiveness and accuracy of the proposed algorithm in identifying overlapping nodes, precision, recall, and F1-score were used as evaluation metrics to compare the overlapping node identification performance of different algorithms on seven artificially synthesized networks of group D. This group of networks maintained the same mixing parameter ( $\mu =0.1$) and number of overlapping nodes ( $on=100$), while varying the network scale. Table 4 compares precision, recall, and F1-score for overlapping node identification by six algorithms on seven artificial networks of different scales.

As shown in Table 4, GELPA-OCD consistently achieved excellent performance. It maintained high precision across all networks, while other algorithms showed fluctuating results. For instance, BMLPA reached 100% precision on the LFR19 network but dropped sharply on larger networks. CPM and LFM performed poorly, with precision falling below 50% and even down to 0 on some networks. Regarding recall, GELPA-OCD stayed above 86% across all networks, with an average of 89%, indicating strong coverage. COPRA followed with an average recall of 84.9%. In contrast, LPANNI showed inconsistent results, and CPM and LFM performed poorly, especially with CPM achieving 0% recall on LFR24 and LFM falling under 30% on several large networks. For the F1-score, which balances precision and recall, GELPA-OCD again led, averaging 93% and showing strong stability across different scales. COPRA ranked second, while LPANNI was stable but underperformed compared to GELPA-OCD. CPM and LFM showed significant instability, with the F1-score dropping to 0 on some networks. Overall, GELPA-OCD demonstrated superior performance in community detection and overlapping node identification, particularly in complex and large-scale network environments.

Ablation experiments on node importance metric combination strategies

To verify the effectiveness of this article’s node importance metric combination method, we compared the community detection performance of four combination strategies: multiplicative combination, additive combination, weighted combination, and nonlinear combination of the proposed node importance metrics, under identical conditions. Formulas (5), (11), (12), and (13) show the multiplicative combination (the combination method used in this article), additive combination, weighted combination, and nonlinear combination, respectively.

(11) $$NI\left(v\right)=DC\mathrm{\_}norm\left(v\right)+PR\mathrm{\_}norm\left(v\right)+\left(1+CC\mathrm{\_}norm\left(v\right)\right)$$

(12) $$NI\left(v\right)={\omega }_1\ast DC\mathrm{\_}norm\left(v\right)+{\omega }_2\ast PR\mathrm{\_}norm\left(v\right)+{\omega }_3\left(1+CC\mathrm{\_}norm\left(v\right)\right)$$

(13) $$NI\left(v\right)=DC\mathrm{\_}norm\left(v\right)+PR\mathrm{\_}norm{\left(v\right)}^{\alpha }+{\left(1+CC\mathrm{\_}norm\left(v\right)\right)}^{\beta }$$where $NI\left(v\right)$ represents the importance of node v. The calculation methods for $DC\mathrm{\_}norm\left(v\right)$, $PR\mathrm{\_}norm\left(v\right)$, and $CC\mathrm{\_}norm\left(v\right)$ are shown in Formulas (6), (7), and (8) above. ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ are weight coefficients, where ${\omega }_1+{\omega }_2+{\omega }_3=1$. $\alpha$ and $\beta$ are weight coefficients for the nonlinear combination, where $\alpha +\beta =1$, facilitating normalized comparison. To ensure the effectiveness of the node importance metric combination method proposed in this article, we conducted a parameter sensitivity analysis for weighted and nonlinear combinations in the experiments, using the Karate and Dolphin networks as examples. We ran each set of weights independently 10 times and recorded the $EQ$ values. Table 5 shows the $EQ$ values for different weights.

As shown in Table 5, different weighting strategies slightly affected algorithm performance. We selected the parameter combination with the highest $EQ$ value, (0.3, 0.3, 0.4) for the weighted method and (0.3, 0.7) for the nonlinear method.

To verify the effectiveness and superiority of the multiplicative combination method of node importance metrics proposed in this algorithm, we conducted experiments on two classic datasets, Karate and Dolphin. We maintained other parameters of the proposed algorithm unchanged while only altering the calculation method for node importance. Each combination strategy was run independently 10 times, with averages taken to reduce the impact of randomness.

The nonlinear, additive, weighted, and multiplicative combination strategies for the karate dataset achieved $EQ$ values of 0.4071, 0.4242, 0.4032, and 0.4498, respectively. The $EQ$ values of the four combination strategies on the dolphin dataset were 0.4065, 0.3999, 0.4461, and 0.4962, respectively. The multiplication combination strategy achieved the highest $EQ$ value on the two classical datasets. The multiplication combination strategy selected in this article avoids the subjectivity of artificial parameter adjustment. Although the multiplication combination is sensitive to a single indicator near zero, the value range of the three basic indicators is controlled between [0, 1] after normalization. There are a few extreme cases near zero. In addition, the form of $\left(1+CC\mathrm{\_}norm\left(v\right)\right)$ is adopted to ensure that even if the clustering coefficient is zero, the overall importance of the node will not be zero. The above experiments proved the superiority of the node importance index under the multiplication combination strategy proposed in this article.

Parameter sensitivity analysis of the graph embedding algorithm

To investigate the impact of key parameters in Node2vec on the performance of the proposed algorithm, we conducted experiments on a small-scale dataset, Karate, a medium-scale dataset, Football, and large-scale datasets, Power and Internet, using $EQ$ as the evaluation metric, with a focus on analyzing the effects of embedding dimension and random walk parameters $p$ and $q$ on algorithm performance.

(1) Impact of embedding dimension on algorithm performance. To investigate the impact of embedding dimension on algorithm performance, we fixed random walk parameters at $p=1.0$ and $q=1.5$ and tested five different embedding dimensions: 32, 64, 128, 256, and 512. As shown in Fig. 5, the horizontal axis D represents the embedding dimension, while the vertical axis represents the $EQ$ values corresponding to the algorithm’s community detection results under different embedding dimensions.

As shown in Fig. 5, increasing the embedding dimension generally improved community detection quality up to a point, after which performance stabilized or declined. On the Karate network, the $EQ$ value increased from 0.417 to 0.45 as the dimension rose from 32 to 64, but declined beyond 128, suggesting that excessively high dimensions may cause overfitting. On the Internet network, $EQ$ values kept improving but with diminishing gains beyond 256. The Power and Football networks showed more stable trends, with optimal performance typically at 128 and 256 dimensions. Based on the experiments in this experiment of this article, for small networks ( $\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s}<100$), we can achieve relatively good results by selecting a dimension of 64; for medium-scale networks (nodes between 100–1,000), we typically choose a dimension of 128, while for large-scale networks ( $\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s}>1\text{,}000$), selecting a dimension of 256 achieves a better balance between performance and computational efficiency.

(2) Impact of random walk parameters on algorithm performance. We conducted experiments on three representative networks, Karate, Football, and Power, to evaluate how Node2vec’s random walk parameters $p$ and $q$ affected community detection. Parameter $p$ controls the probability of returning to the previous node, while $q$ influences the exploration of new nodes. In our experiment, we fixed the embedding dimension at 128 and varied $p$ and $q$ to observe their impact. Table 6 shows the experimental results.

From the experimental results shown in Table 6, we observed that random walk parameters had a certain impact on community detection quality. On the Football network, when $p=1.0$ and $q=1.5$, the algorithm achieved the optimal $EQ$ value of 0.616. This parameter combination leveraged the ability to capture structural equivalence when $q<1$, which was beneficial for identifying relatively distinct community boundaries in the network. For the Karate network, the algorithm performed best $\left(EQ=0.449\right)$ when $p=1.0$ and $q=1.5$, indicating that small-scale networks required appropriately enhanced stability in label propagation to avoid excessive focus on local structures. For the medium-scale Power network, the parameter combination of $p=1.0$ and $q=1.0$ showed optimal performance $\left(EQ=0.762\right)$. This balanced random walk strategy was more suitable for handling larger and structurally complex networks. When the $p$ and $q$ parameters deviated from the optimal combination, algorithm performance varied across all three networks.

Overall, GELPA-OCD performed consistently well when p = 1.0 and q ranged between 0.5 and 1.5. These findings confirmed that Node2vec’s biased walks significantly influenced overlapping community detection. Accordingly, our main experiments used p = 1.0 and q = 1.5.

Performance comparison of different graph embedding algorithms

To validate the choice of Node2vec as the graph embedding method, we compared its performance with the classic Deepwalk algorithm on four benchmark datasets. We kept embedding dimensions, walk length, and number of walks consistent across both methods and varied only the $p$ and $q$ parameters for Node2vec. Table 7 presents the $EQ$ values and the number of detected communities.

The results showed that Node2vec consistently outperformed Deepwalk, achieving higher $EQ$ values across all datasets. On the Karate and Dolphin networks, our algorithm detected three communities instead of the actual two, but achieved higher modularity while maintaining stability and efficiency. On the Football network, the Node2vec-based method produced results closest to the actual number of communities. These findings validate Node2vec as an appropriate embedding technique, with its parameterized random walks effectively balancing node homogeneity and structural equivalence, providing more accurate node similarity information for label propagation.

Theoretical analysis of adaptive threshold filtering mechanism

The adaptive threshold $\theta =\frac{1}{\left|L\left(v\right)\right|}$ proposed in this article improved the label propagation mechanism. From the perspective of algorithm convergence, this threshold ensured monotonicity in the label propagation process; in each iteration, labels with affiliation coefficients below the “average level” were filtered out, causing the retained label affiliation coefficients to become more concentrated on dominant communities after normalization, thereby accelerating the algorithm’s convergence speed. In this article’s experimental section, “Analysis of algorithm convergence,” convergence analysis was conducted on the classic Karate and Football datasets, comparing the proposed algorithm with three classic overlapping community detection algorithms. Compared to COPRA and BMLPA algorithms, the proposed algorithm demonstrated faster label distribution change rate convergence, typically reaching a stable state after 3 or 4 iterations.

Furthermore, this threshold mechanism could automatically balance the precision and recall of label retention in networks with different topological structural properties. In densely connected areas (within communities), nodes typically received multiple labels (larger $\left|L\left(v\right)\right|$), resulting in a lower threshold that allowed nodes to retain multiple labels with higher affiliation coefficients, ensuring high recall. In sparsely connected areas (community boundaries), nodes received fewer labels (smaller $\left|L\left(v\right)\right|$), resulting in a higher threshold that filtered out most labels with low affiliation coefficients through this mechanism, improving the precision of label selection. In this article’s experimental section, “Assessment of overlapping node detection,” experimental results on artificial datasets of different scales showed that the proposed algorithm demonstrated good recall and precision in identifying overlapping nodes.