Recently, the controllability of complex networks has become a hot topic in the field of network science, where the driver nodes play a key and central role. Therefore, studying their structural characteristics is of great significance to understand the underlying mechanism of network controllability. In this paper, we systematically investigate the nodal centrality of driver nodes in controlling complex networks, we find that the driver nodes tend to be low in-degree but high out-degree nodes, and most of driver nodes tend to have low betweenness centrality but relatively high closeness centrality. We also find that the tendencies of driver nodes towards eigenvector centrality and Katz centrality show very similar behaviors, both high eigenvector centrality and high Katz centrality are avoided by driver nodes. Finally, we find that the driver nodes towards PageRank centrality demonstrate a polarized distribution, i.e., the vast majority of driver nodes tend to be low PageRank nodes whereas only few driver nodes tend to be high PageRank nodes.

Many deterministic small-world network models have been proposed so far, and they have been proven useful in describing some real-life networks which have fixed interconnections. Search efficiency is an important property to characterize small-world networks. This paper tries to clarify how the search procedure behaves when random walks are performed on small-world networks, including the classic WS small-world network and three deterministic small-world network models: the deterministic small-world network created by edge iterations, the tree-structured deterministic small-world network, and the small-world network derived from the deterministic uniform recursive tree. Detailed experiments are carried out to test the search efficiency of various small-world networks with regard to three different types of random walks. From the results, we conclude that the stochastic model outperforms the deterministic ones in terms of average search steps.