Optimal symmetric networks in terms of minimizing average shortest path length and their sub-optimal growth model

Homogeneous entangled networks characterized by small world, large girths, and no community structure have attracted much attention due to some of their favorable performances. However, the optimization algorithm proposed by Donetti et al. is very time-consuming and will lose its efficiency when the size of the target network becomes large. In this paper, an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length. It is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds, which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks. In order to overcome the time-consuming obstacle of the optimization algorithms proposed by us and Donetti et al., a growth model is proposed to get large scale sub-optimal symmetric networks. Numerical simulations show that the symmetric networks derived by our growth model will have small-world property, and besides, these networks will have many other similar favorable performances as entangled networks, e.g., robustness against errors and attacks, very good load balancing ability, and strong synchronizability.