Edges
We explore the relationship between the number of nodes in the maximal commu- nities and the number of their corresponding edges using a relationship chart to investigate the character of community structure (Fig.6.3).
According to the study of the relationship between the number of nodes in maximal communities and the number of communities in the corresponding net- works, we find that there exists a linear correlation between them.
We now explore the relationship between the number of nodes of the maximal communities and the number of edges in the corresponding maximal communities. Figure6.3displays a chart showing the relationship between the five data sets. The figure shows that the number of edges rapidly increases while the number of nodes initially increases. When the number of nodes reaches a certain level, the increase in the number of edges tends to smooth out while the number of nodes increases. To examine this property, we quantified the five data sets and compared them. Fig- ure 6.4 demonstrates that there are differences in the five data sets; the trend changes in some data sets, such as electronic collaboration networks, are steep, Fig. 6.3 Visualization of relationship of five data sets. (a) MATLAB help document. (b) Chinese characters. (c) Yeast. (d) P2P file-sharing networks. (e) Electronic collaboration networks.N_N number of nodes of maximal communities,E_Nnumber of edges in corresponding maximal communities
whereas other data sets, such as peer-to-peer file-sharing networks, change slowly. When the increase in the number of nodes reaches a certain level, the number of edges does not increase but achieves a flat state. Based on the preceding analysis, we may conclude that complex networks have a sparsity feature.
Discussion and Conclusion
Community structure detection plays a significant role in studying the struc- tural properties, functions, and evolutionary mechanisms of complex net- works by breaking down the nodes, which is helpful for properly disclosing network construction principles and topology functions.
This paper focused on several data sets from various areas to explore the features of community structure. To begin with, we calculated the modularity of five complex networks, proposing the conclusion that the modularity and density of the communities in the whole networks are positively related. Then, after fitting the number of nodes in the maximum communities to the number of communities in the corresponding networks, we observed a linear correlation between them. We could extend this linear correlation to other kinds of networks and predict their scale. Finally, examination of the relation- ships between the number of nodes of maximal communities and the number of edges in the corresponding maximal communities revealed the sparse character of the community structure in networks and which revealed the generality feature of complex networks.
(continued) Fig. 6.4 Quantification of
five data sets and their relationships.N_Nnumber of nodes of maximal communities,E_Nnumber of edges in corresponding maximal communities
(continued)
However, a few problems remain. First, there is no common definition of community, which is justified by the nature of the problem itself. What’s more, real-world networks are dynamic. An increasing number of studies on complex networks are focusing on excavating the hidden relations and fea- tures in real networks such as social networks. Therefore, improving the methods of researching ever-changing dynamic networks represents an inno- vative and challenging avenue for future work.
Acknowledgements The author would like to thank Prof. Li for helpful suggestions and com- ments. This research work was supported by Key Research Institute in the University of China, National Center for Radio and Television Studies 2013GDYB07, and National Key Technology R&D Program 2013BAH66F02-02.
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