# Detect Overlapping Community Based on the Combination of Local Expansion and Label Propagation
## 4 Experimental Result
### Synthetic Networks
In the cases where the proportion of the external edge is small (\(\mu=0.1\)), there is little difference among \(LELP\) with different \(d\) (the four curves are very close in Figure 1a,c). In the networks where there are more external edges of nodes (\(\mu=0.3\)), the \(LELP\) with \(d=2\) performs far better than others, especially when the community size is large and the membership is complex (the red curve is much higher than the others in Figure 1b,d, respectively). In general, the \(LELP\) with \(d=2\) performs better in most cases. Therefore, in the following of this paper, we choose the result of our algorithm with \(d=2\). We first detect the performance of algorithms in the cases where there are different mixing parameter \(\mu\) and network size \(N\). The former controls the proportion of the internal edge to the external edge of nodes. Generally, the larger the \(\mu\), the fuzzier the community structure. When \(\mu\) is close to \(0.5\), the community structure is not prominent at all. Obviously, the experimental results are in line with this fact. When \(\mu=0.1\) but other conditions are the same, the \(NMI\) of all algorithms are almost better than those when \(\mu=0.3\) (the yellow and blue curves are at the top of most \(LFR\)s). For \(LELP\), high \(\mu\) does not weaken its performance greatly (the curves are very close in Figure 2a). Parameter \(N\) determines the network size, that is, the total number of nodes in the network. The performance of algorithms in larger networks is slightly better than that of small networks, except \(EAGLE\), which deteriorates greatly with large \(N\). \(O_{m}\) represents the membership of nodes. When \(O_{m}\) is large, the task of the algorithm is difficult. As shown in Figure 2, the majority of the \(NMI\) curves of each algorithm gradually decreases with the increase in \(O_{m}\). Then, the \(NMI\) of the algorithms is measured in the cases where the number of overlapping nodes \(O_{n}\) and community size are different. As expected, there is a serious decline in the performance of algorithms when there are a large number of overlapping nodes in networks (yellow and green curves are much lower than blue and red curves for most \(LFR\)s in Figure 3). This is because \(O_{n}\) can indirectly determine the overlapping size among communities. A high overlapping size will hinder the accuracy of algorithms in detecting a community. Then, most algorithms perform better in the \(LFR\) with smaller communities. The reason may be that the structural characteristics of a community become blurred as the community size increases, which is also mentioned in [31; 32]. This feature has a great influence on simple local expansion or label propagation. For the former, the sensitivity of the fitness function lowers as the community size increases. For the latter, there are so many dense structures within a large community that  
Figure 1: In the sub figure (**a**–**g**), the curves with different colors represent the NMI of LELP when \(d\) is different. stability at many places, which leads to the detection failure. Unlike others, \(LELP\) is less affected by the community size than ther other algorithms as shown in Figure 3a. This is because local expansion is only used in the first stage of \(LELP\) to locate dense regions, not the final community. These dense regions are small but have high density of edges, which prevents accuracy degradation. These dense regions determine the initial seed of label propagation in the following process, which can ensure that the label propagates from the dense to sparse regions. Usually, these two regions are likely to be the center and border of the expected community, respectively. Using the dense region instead of single node as the seed makes the labels reach stability more quickly in the early propagation. Figure 3: Sub figure (**a-j**) represent the NMI of different algorithms in the \(LFR\) 4 to \(LFR\) 7, where there are different overlapping nodes \(O_{n}\) and community size.