Scalability analysis of the synchronizability for ring or chain networks with dense clusters

It is well known that most real-world complex networks, such as the Internet and the World Wide Web, are evolving networks. An interesting fundamental question is: how do some important functions or dynamical behaviors of complex networks evolve with increasing network scale? This paper aims at investigating the scalability of the synchronizability for ring or chain networks with dense clusters as the network size increases. We discover some interesting phenomena as follows: (i) the synchronizability of ring or chain networks with clusters decreases with increasing network scale regardless of the inner structures of all communities; (ii) for the same network scale, the network synchronizability decreases more quickly with increasing number of cluster blocks than with increasing number of nodes within the cluster block; (iii) the number of rings or chains has a much more significant influence on the network synchronizability than the size of the rings or chains. Our results indicate that network synchronizability can be maintained with increasing network scale by avoiding ring and chain structures.