This paper discusses a popular community definition in complex network research in terms of the conditions under which a community
is minimal, that is, the community cannot be split into several smaller communities or split and reorganized with other network
elements into new communities. The result provides a base on which further optimization computation of the quantitative measure
for community identification can be realized.
Key words Community - complex network - modularity - modularity density
The research is supported by the Ministry of Science and Technology of China under Grant No. 2006CB503905. Some authors are
also supported by the National Natural Science Foundation of China under Grant Nos. 10631070 and 10701080, and the JSPS(Japan
Society for the Promotion of Science)-NSFC(National Natural Science Foundation of China) collaboration project under Grant
No. 10711140116.