# Detect Overlapping Community Based on the Combination of Local Expansion and Label Propagation
## 4 Experimental Result
### Synthetic Networks  
It is necessary to have benchmarks to observe the performance of our algorithm and compare it with others. Due to the lack of ground truth, real-world networks do not show clear contrast results but the synthetic networks are adaptable. We adopt \(LFR\) benchmarks proposed in [30] and set the parameters of the \(LFR\) generator referring to [14]: the size of networks are between 1000 and 5000. The average degree is set to 10, which is the same as most real-world social networks. The node degree and community size are governed by power law distributions with exponents \(\tau_{1}=2\) and \(\tau_{2}=1\), respectively. The maximum degree is \(k_{max}=50\), and community sizes vary within a small range \(s=(10,50)\) or large range \(l=(20,100)\). The mixing parameter \(\mu\) is set to 0.1 or 0.3. \(\mathcal{O}_{n}\) and \(\mathcal{O}_{m}\) together control the overlap, where the former is the number of overlapping nodes, while the latter is the number of communities to which each overlapping node belongs. \(\mathcal{O}_{n}\) is set to 10% or 50% of the total nodes so as to indicate the low or high density of overlapping nodes, respectively. \(\mathcal{O}_{m}\) varies from 2 to 8, indicating the diverse membership of overlapping nodes. By changing the combination of \(\mathcal{O}_{n}\) and \(\mathcal{O}_{m}\), we create detection tasks with different difficulties so that we can compare the performance of the algorithms in greater detail. The specific parameter settings of \(LFR\) benchmarks are shown in Table 1. For each set of parameters, we generate 100 instantiations and adopted the average performance of each algorithm. For each set of parameters, we generated 100 instantiations and 100 times experiments were carried out on each instantiation. We adopted the average of all these experiments as the algorithm performance. We compare \(LELP\) with some representative algorithms. For algorithms with tunable parameters, the setting that can obtain the best result is selected. For \(CFinder\), \(k\) varies from 3 to 8. For \(LFM\), \(\alpha\) ranges from 0.8 to 1.6 with an interval of 0.1. For \(SLPA\), parameter \(r\) varies from 0.05 to 0.5 with an interval of 0.05. For \(LELP\), we measure its performance with different \(d\), which controls the range of node neighbors in _closeness_. In general, the local structure information around the nodes is not fully reflected if \(d\) is small, while the local details may be ignored if \(d\) is too large. Therefore, we set \(d\) from 1 to 4 to observe \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Name** & \(\mathbf{N}\) & \(\overline{\mathbf{k}}\) & \(max_{\mathbf{k}}\) & \(\mu\) & \(min_{\mathbf{c}}\) & \(max_{\mathbf{c}}\) & \(\mathcal{O}_{n}\) & \(\mathcal{O}_{m}\) \\ \hline LFR 1 & 1000 & 10 & 50 & 0.1 & 20 & 100 & 100 & 2–8 \\ LFR 2 & 1000 & 10 & 50 & 0.3 & 20 & 100 & 100 & 2–8 \\ LFR 3 & 5000 & 10 & 50 & 0.1 & 20 & 100 & 500 & 2–8 \\ LFR 4 & 5000 & 10 & 50 & 0.3 & 20 & 100 & 500 & 2–8 \\ LFR 5 & 5000 & 10 & 50 & 0.3 & 10 & 50 & 500 & 2–8 \\ LFR 6 & 5000 & 10 & 50 & 0.3 & 20 & 100 & 2500 & 2–8 \\ LFR 7 & 5000 & 10 & 50 & 0.3 & 10 & 50 & 2500 & 2–8 \\ \hline \end{tabular}
\end{table} 
Table 1: The detailed parameters of \(LFR\) benchmarks are shown. \(LFR\) 1 to \(LFR\) 4 are grouped to test the performance of algorithms under different network sizes and mixing parameters, while \(LFR\) 4 to \(LFR\) 7 are governed by different community sizes and overlapping ratios. the performance of \(LELP\) as shown in Figure 1. These parameters are still retained in real-world networks. It can be seen that the performance of \(LELP\) with \(d=4\) is the worst in most cases and it is very close to that when \(d=3\). This may be due to the fact that the smaller local structure is ignored when the neighborhood is large. In addition, the ratio of shared neighbors tends to be stable as the scale of neighbors expands. The performance of \(LELP\) with \(d=1\) is slightly better than the former two cases in the face of small communities.