# Detect Overlapping Community Based on the Combination of Local Expansion and Label Propagation
## 4 Experimental Result
### Real-World Networks  
We tested on a series of real-world networks, whose main information is listed in Table 2. The dolphins and football network are commonly used social network. Email is the core of e-mail communication network from a large European research institution. CA-GrQc is a co-authorship network based on papers in General Relativity publishing in Arxiv. cit-HepTh is the citation graph of high energy physics theory in Arxiv. P2P is the Gnutella peer-to-peer file sharing network. Due to the unknown ground truth of these real-world networks, the algorithms' performance is measured by \(Q_{ov}^{e}\), as shown in Figure 6a. In each sub-figure of Figure 6, networks are arranged from left to right along the \(x\)-axis, according to the size and symbols of points representing different algorithms. The number of overlapping nodes and the average membership of the nodes are illustrated in Figure 6b,c, respectively. It can be seen that the performance of algorithms in different real-world networks is not constant, but the ranking of algorithms is relatively stable in each network. In most cases, \(Q_{ov}^{e}\) of \(LELP\) and \(SLPA\) are generally the best two, especially in large networks, while \(EAGLE\) and \(CIS\) are always the worst two. In addition, the performance of \(COPRA\) is also in the forefront. It illustrates that the algorithms utilizing label propagation are effective because the label propagation is similar to a simulation of information spreading in social networks. Given a good source of propagation, labels are more easily and accurately stable in a community through iterative propagation. In \(LELP\), local expansion can generate \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & \(N\) & **E** & \(\overline{\mathbf{k}}\) & **CC** & **Ref.** \\ \hline dolphins & 62 & 3162 & 51 & 0.258 & [33] \\ football & 115 & 1219 & 10.6 & 0.403 & [34] \\ Email & 1005 & 25, 571 & 25.4 & 0.399 & [35] \\ CA-GrQc & 5242 & 14,496 & 2.8 & 0.530 & [36] \\ cit-HepTh & 27,770 & 352,807 & 12.7 & 0.312 & [37] \\ P2P & 62,586 & 147,892 & 2.4 & 0.006 & [36] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The basic features of real-world networks in the experiments are shown in the Table 2, including the number of nodes and edges, average degree, average clustering coefficient and reference. Figure 5: Sub figure (**a**,**d**,**g**,**i**) represent the F-score of algorithms in \(LFR\) 4 to \(LFR\) 7. Sub figure (**b**,**e**,**h**,**k**) represent the _precision_ of algorithms in \(LFR\) 4 to \(LFR\) 7. Sub figure (**c**,**f**,**i**,**i**) represent the _recall_ of algorithms in \(LFR\) 4 to \(LFR\) 7. In \(LFR\) 4 to \(LFR\) 7, there are different overlapping nodes \(O_{n}\) and community sizes but other conditions are the same. some dense regions closer to the center of expected communities as the source of label propagation. This is why our algorithm performs well. As can be seen in Figure 6b, some algorithms may over-detect the overlap, such as \(CFinder\), \(CIS\) and \(EAGLE\). The proportion of overlapping nodes that they detect in all nodes is far greater than that in other algorithms, and even more than 40% in some cases. This is why these algorithms do not perform well in \(Q^{e}_{ov}\). Too many nodes are detected as overlapping nodes, which indicates that these algorithms fail. In contrast, this proportion of other algorithms is usually below 20%, and even \(COPRA\) is often lower than 5%. For a detection algorithm, too high of an overlapping proportion detected means that the precision is too rough; too low of a proportion usually means that it does not work. The overlap detected by \(LELP\) can be considered meaningful because the proportion of overlapping nodes detected in all nodes is appropriate at about 8%. Meanwhile, \(LELP\), \(LFM\), \(NISE\) and \(SLPA\) fluctuate a little as a result of insensitivity to specific structures, and all remain around 10%, except for the small networks. These algorithms also perform well in \(LFR\).