Analytical controllability of deterministic scale-free networks and   Cayley trees
  According to the exact controllability theory, the controllability is investigated analytically for two typical types of self-similar bipartite networks, i.e., the classic deterministic scale-free networks and Cayley trees. Due to their self-similarity, the analytical results of the exact controllability are obtained, and the minimum sets of driver nodes (drivers) are also identified by elementary transformations on adjacency matrices. For these two types of undirected networks, no matter their links are unweighted or (nonzero) weighted, the controllability of networks and the configuration of drivers remain the same, showing a robustness to the link weights. These results have implications for the control of real networked systems with self-similarity. 
Ming Xu, Chuan-Yun Xu, Huan Wang, Cong-Zheng Deng, Ke-Fei Cao
physics.soc-ph cond-mat.stat-mech