  To better understand the correlation between network topological features and
the robustness of network controllability in a general setting, this paper
suggests a practical approach to searching for optimal network topologies with
given numbers of nodes and edges. Since theoretical analysis seems impossible
at least in the present time, exhaustive search based on optimization
techniques is employed, firstly for a group of small-sized networks that are
realistically workable, where \textit{exhaustive} means 1) all possible network
structures with the given numbers of nodes and edges are computed and compared,
and 2) all possible node-removal sequences are considered. A main contribution
of this paper is the observation of an empirical necessary condition (ENC) from
the results of exhaustive search, which shrinks the search space to quickly
find an optimal solution. ENC shows that the maximum and minimum in- and
out-degrees of an optimal network structure should be almost identical, or
within a very narrow range, i.e., the network should be extremely homogeneous.
Edge rectification towards the satisfaction of the ENC is then designed and
evaluated. Simulation results on large-sized synthetic and real-world networks
verify the effectiveness of both the observed ENC and the edge rectification
scheme. As more operations of edge rectification are performed, the network is
getting closer to exactly satisfying the ENC, and consequently the robustness
of the network controllability is enhanced towards optimum.
