  In this paper the effects of external links on the synchronization
performance of community networks, especially on the competition between
individual community and the whole network, are studied in detail. The study is
organized from two aspects: the number or portion of external links and the
connecting strategy of external links between different communities. It is
found that increasing the number of external links will enhance the global
synchronizability but degrade the ynchronization performance of individual
community before some critical point. After that the individual community will
synchronize better and better as part of the whole network because the
community structure is not so prominent. Among various connection strategies,
connecting nodes belonging to different communities randomly rather than
connecting nodes with larger degrees is the most efficient way to enhance
global synchronization of the network. However, a preferential connection
scheme linking most of the hubs from the communities will allow rather
efficient global synchronization while maintaining strong dynamical clustering
of the communities. Interestingly, the observations are found to be relevant in
a realistic network of cat cortex. The synchronization state is just at the
critical point, which shows a reasonable combination of segregated function in
individual communities and coordination among them. Our work sheds light on
principles underlying the emergence of modular architectures in real network
systems and provides guidance for the manipulation of synchronization in
community networks.
