Modularity

Modularity (Q) is commonly used for measuring the strength of community structures. The closer its value is to 1, the better the quality of community division (Yuan & Liu, 2021). Q can be calculated as follows in Eq. (6). Where m is the total number of edges in the network, $a_{ij}$ is an element in the adjacency matrix A of network G, and $d_i$ is the degree of node $v_i$. When $v_i$ and $v_j$ belong to the same community, ${\delta }_{i,j}=$1; otherwise ${\delta }_{i,j}=$0.

(6) $$Q={\displaystyle \frac{1}{2m}{\sum }_{i,j}\left(a_{ij}-{\displaystyle \frac{d_i{d}_j}{2m}}\right){\delta }_{i,j}}$$

Normalized mutual information

For networks with a known community structure, NMI is generally used to evaluate the community division effect. The higher the NMI, the more similar the result is to the realistic community structure. A value of 1 indicates that the partition result is completely consistent with the actual community structure. Assuming that $A=\left\{A_1,A_2,\dots ,A_k\right\}$ and $B=\left\{B_1,B_2,\dots ,B_{k^{\mathrm{\prime }}}\right\}$ are the realistic community structure and the division result of the network by an algorithm, respectively, $k$ and $k^{\prime }$ are the number of communities under the two divisions, NMI can be defined as follows in Eq. (7). Where n is the total number of nodes, $T$ is the confusion matrix, $T_{ij}$ is the number of common nodes included in the realistic communities.

$A_i$ and $B_j$, and $T_i$ is the sum of the elements in the i-th row of the confusion matrix.

(7) $$\mathrm{N}\mathrm{M}\mathrm{I}={\displaystyle \frac{2{\sum }_{i=1}^k{\sum }_{j=1}^{k^{\prime }}{T}_{ij}log{\displaystyle \frac{n{T}_{ij}}{T_i{T}_j}}}{-{\sum }_{i=1}^k{T}_{i}log{\displaystyle \frac{T_i}{n}-{\sum }_{j=1}^{k^{\prime }}{T}_{j}log{\displaystyle \frac{T_j}{n}}}}}$$

Stage 1: initial community detection

The first stage of the proposed LPA-ITSLR algorithm is shown in Table 1. First, each node is assigned a unique label, and the similarity between nodes is calculated according to Eq. (9). Then, the PC value of each node is calculated and sorted in ascending order. Next, the labels are updated according to the sorted nodes. In the label updating strategy, the similarity between the current node and other nodes is compared. If the node with the largest similarity is not unique, the LR is further compared; if not, one is randomly selected to obtain the rough initial community structure in the first stage.

Stage 2: community merge

The second stage of the LPA-ITSLR algorithm is shown in Table 2. In view of the problem that small communities may cause low modularity in the first stage, whether the initial community meets the weak community condition is judged first. If the condition is not met, the community with the largest number of connected edges is selected for merging; this process is repeated until the entire network meets the weak community condition. The research of LPA-TS (Wenping et al., 2018) shows that α is generally set as 2 in Eq. (3). In order to achieve more accurate results, α is set as 0.5, 1, 1.2, 1.5 and 2, respectively for the eight data sets used in this article. Through experiments, it can be known that when α is set as 1.5, good division results are achieved. Moreover, experiments are also carried out on 15 artificial data sets to verify the rationality of the value of α. Therefore, in our study, α is determined to be 1.5 to achieve better performance. Each community is regarded as a node, and its PC value is calculated using Eq. (1); then, the community with the most links is determined for merging. If the modularity increases after the merge, the merge will be selected; otherwise, it will not be merged, thus ensuring that the community structure after the second stage merge will have a higher modularity and be closer to the realistic community structure.

Realistic dataset

Six classic realistic datasets were used in the experiment; their attributes are presented in Table 3. Here, |V| represents the total number of nodes in the network, |E| represents the total number of edges, | $\mathrm{\Omega }$| represents the number of communities included in the network, max(k) represents the maximum node degree, <k> represents the average node degree, <L> represents the average path length, and <c> represents the clustering coefficient.

Community division results

The proposed LPA-ITSLR algorithm was used to divide communities in the six abovementioned real datasets. The results are illustrated in Fig. 4, where nodes in different communities are represented by different color.

Stability analysis of LPA-ITSLR

The proposed LPA-ITSLR algorithm and the LPA and LPA-TS algorithms were compared and analyzed in the six abovementioned real datasets. Each dataset was run independently for 10 times, and the average value of the three algorithms on the six datasets was obtained (denoted as <Q>), as shown in Table 4. The independent experimental results for each time are shown in Fig. 5. As can be seen from the experimental results presented in Table 4 and Fig. 5, LPA-ITSLR performed well in all datasets, with the exception that the average module degree on the NetScience was slightly lower than that obtained using the other two algorithms. Moreover, LPA-ITSLR yielded more stable community partitioning results and a higher modularity than the other two algorithms. NetScience is a weighted network; however, in the experiment, the weight was ignored, and it was transformed into a powerless network for community division. Therefore, the quality of community division on this network obtained using LPA-ITSLR was slightly lower than that obtained using the other two algorithms. However, in the 10 independent experiments, the results of the LPA and LPA-TS algorithms exhibited fluctuations, indicating that the two algorithms are unstable due to the randomness of node and label update. The module-degree value of the proposed LPA-ITSLR algorithm always remained stable for every network, indicating that LPA-ITSLR effectively solves the oscillation problem in the process of label propagation and has higher accuracy and stability.

To further verify the robustness of LPA-ITSLR, 100 independent experiments were conducted on the Karate, Dolphin, and Football networks; the results are presented in Fig. 6. The community division results obtained using the LPA algorithm exhibited the most serious fluctuations in the modularity value, followed by the LPA-TS algorithm. In contrast, LPA-ITSLR maintained the same community division results in 100 experiments, and the modularity was higher than that of LPA and LPA-TS.

To further evaluate the performance of the LPA-ITSLR algorithm, it was compared with four recent community detection algorithms based on label propagation. Among them, the COPRA algorithm (Gregory, 2010) realizes community division by assigning multiple labels with attribution coefficients to a node. The WLPA algorithm (Tong et al., 2015) first selects the label with a larger weight for propagation during the label propagation process. The LINSIA (Wu et al., 2016) algorithm is based on node importance and employs label importance to complete the community division. The LILPA (Zhang et al., 2020) algorithm uses a fixed label update sequence based on the ascending order of node importance. The modularity of the results obtained using the five algorithms on the four real datasets is presented in Table 5. From Table 5, it can be seen LPA-ITSLR yielded the highest modularity and most stable in community division results. Thus, instability caused by label oscillation is avoided effectively by using LPA-ITSLR.

Performance comparison of LPA-ITSLR with other algorithms

For further analysis of the effectiveness of the proposed algorithm for community partition and correctness, three classic datasets of Karate, Dolphins, and Football were used, and the LPA-ITSLR algorithm and seven classic community detection algorithms were employed for obtaining the division results for correlation analysis in terms of the number of communities $\left|\mathrm{\Omega }\right|$ and module Q as evaluation indicators, The results (Newman, 2006) are presented in Table 6 (Li et al., 2019; Xing et al., 2014).

From Table 6, it can be seen that the number of communities and modularity of the partition results of the eight algorithms on the three classical networks (Yin et al., 2018) were different, but LPA-ITSLR exhibited good performance on these datasets; moreover, the partition results and the number of communities were consistent with the realistic network structure, and the modularity was higher than that obtained using other algorithms.

Artificial datasets

Ten artificial networks were generated using the LFR benchmark (Yang, Perotti & Tessone, 2017); the basic information is presented in Table 7. The number of nodes |V| in the top eight artificial networks is 1,000, and the community size | $\mathrm{\Omega }$| is 10–50, that is, min $\left|\mathrm{\Omega }\right|$ = 10, max $\left|\mathrm{\Omega }\right|$ = 50. The average degree of nodes <k> is 20, and the maximum degree max(k) is 50. The values of 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, and 0.45 were employed as the mixing parameter µ, and the eight networks were denoted as LFR-1–LFR-8. The latter two artificial networks are more complicated. The number of nodes is 5,000, the community size is 50, the average degree of nodes is 10, and the maximum degree is 50. The mixing parameter µ was 0.1 and 0.3, respectively, and these two networks were denoted as LFR-9 and LFR-10.

Comparative analysis of algorithm performance

For the first eight artificial datasets, the proposed LPA-ITSLR algorithm was compared with the LPA and LPA-TS algorithms in terms of the community division results. The average modularity <Q> and NMI were used as evaluation indicators. The experimental results are shown in Fig. 7. As the value of µ increased, the network became more complex. The modularity of the community division results of the three algorithms on the corresponding network decreased by varying degrees, but LPA-ITSLR yielded higher modularity than the other algorithms. Moreover, the NMI value of LPA-ITSLR on the first seven networks was 1, and the NMI value of the network with a µ value of 0.45 was 0.9943, showing extremely strong stability and higher quality of community division.

For large-scale artificial networks LFR-9 and LFR-10 with high complexity, the proposed algorithm was compared with seven recent LPA algorithms. Q and NMI were considered as evaluation parameters. The results are presented in Table 8. It can be seen that the community division results (Liu et al., 2015) obtained using the seven algorithms (Xie, Szymanski & Liu, 2011) were not stable, and the algorithm proposed in this article maintains stable community division results on the two complex artificial data sets (Zhang et al., 2017). Although the NMI value was slightly lower than that for other algorithms, the modularity was far higher. In the community merge phase optimization strategy based on modularity, LPA-ITSLR is superior as it can yield stable and high-quality community division results.

For the above 10 artificial networks, the experimental results show that the proposed algorithm is superior to other algorithms in both Q and NMI. In order to further verify the superiority of the proposed algorithm, we compare the number of communities detected by LPA-ITSLR algorithm with the actual number of communities of the 10 networks, and the results are shown in Table 9. It can be seen from Table 9 that the number of communities detected by the algorithm proposed in this article is basically consistent with the actual number of communities. In general, good results are obtained except for small deviations in some networks.

Experiments on large scale artificial datasets

In order to further verify the effectiveness, the computational performance and utility of the proposed algorithm for large-scale networks, nine artificial data sets were used and the number of nodes of these networks was from 6,000 to 50,000, and these networks were denoted as LFR-11 to LFR-19, respectively. Table 10 shows the experimental results on these nine large-scale networks, including the actual number of communities, the number of communities detected by the LPA-ITSLR algorithm, Q and NMI.

It can be seen from Table 10 that, the algorithm performs well on these large data sets. With the increase of the number of nodes, the network scale and complexity continues to expand, and there is a discrepancy between the actual number of communities and the number of communities obtained by the algorithm. But the Q is always above 0.86, and the NMI is more than 0.96 by and large. Specially, on the dataset containing 20,000 to 50,000 nodes, the NMI basically reaches more than 0.98, which shows the utility of the proposed algorithm in community division for large-scale networks. In addition, the number of communities detected by the algorithm proposed is basically consistent with the actual number of communities, which further verifies the effectiveness and superiority of the LPA-ITSLR algorithm.

Experiments on large scale realistic datasets

Similarly, experiments were also carried out on large-scale realistic data sets to further verify the performance of the proposed algorithm. There are a variety of community discovery algorithms, mainly divided into split-based methods, such as GN (Girvan-Newman) algorithm (Poggiolini, 2012); methods based on modularity, such as the CNM (Clauset-Newman Modularity) algorithm (Clauset, Newman & Moore, 2004); methods based on spectral analysis, such as SC (Spectral Clustering) (Kumar et al., 2015), and methods based on label propagation, such as LPA. In addition to the six classic small realistic data sets mentioned above, three representative large-scale realistic networks were obtained, and comparative experiments were conducted with the four classic community discovery algorithms mentioned above. The three data sets are Email, Political Blogs (PB) and Power Grid (PG), and the topological properties are shown in Table 11.

The community discovery in realistic networks is more challenging than that in the simulation network, and the community structure cannot be predicted in advance, so the modularity can only be used for comparison. Table 12 shows the community division results of the proposed algorithm and the four classic algorithms on the above three large-scale realistic networks. In Table 12, the first column is a list of realistic networks, and the second to sixth columns are five classic community discovery algorithms. For each algorithm, the maximum modularity and the number of communities are computed. For example, in the Email network, the maximum modularity obtained by GN algorithm is 0.532, and the number of communities founded is 61, which is recorded as 0.532/61.

It can be seen from Table 12 that the algorithm proposed in this article has achieved better community division results on the three data sets in terms of modularity. On the Email network, the modularity is slightly lower than GN algorithm. On the PB network, the performance is better than the other four algorithms. On the PG network, the modularity of the proposed algorithm is only 0.003 lower than that of CNM algorithm. In general, the GN algorithm based on global, the CNM algorithm considering modularity increment and the LPA-ITSLR algorithm proposed in this article are not very different from each other in community division results for the three datasets. However, GN and CNM algorithms have higher computational complexity than other algorithms, while SC algorithm and LPA algorithm perform relatively poor. From the perspective of the number of communities, the GN algorithm tends to get more communities. For example, the GN algorithm divides the PB network into 205 communities, which is significantly higher than other algorithms. The LPA algorithm divides the PB network into three communities, which is closest to the realistic number of communities. For PB network, the algorithm proposed in this article achieves a result that is completely consistent with the number of realistic communities, while the number of communities given by other methods is more than 10. Obviously, the corresponding methods tend to over fit. Combining the two indicators of modularity and the number of communities, experimental results on six small-scale and three large-scale realistic data sets show that the LPA-ITSLR algorithm proposed can effectively realize good community division with a higher modularity, and the number of communities discovered is basic consistent with the realistic community structure.