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 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.

中文翻译：

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走向网络可控性的最优鲁棒性：一个经验\endgraf必要条件

为了更好地理解网络拓扑特征与一般环境中网络可控性的鲁棒性之间的相关性，本文提出了一种实用的方法来搜索具有给定节点和边数的最佳网络拓扑。由于理论分析至少在目前看来是不可能的，因此采用基于优化技术的穷举搜索，首先针对一组实际可行的小型网络，其中穷举意味着 1) 具有给定节点数的所有可能的网络结构和边被计算和比较，并且 2) 考虑所有可能的节点移除序列。本文的一个主要贡献是从穷举搜索的结果中观察到一个经验必要条件（ENC），这会缩小搜索空间以快速找到最佳解决方案。ENC 表明最优网络结构的最大和最小入度和出度应该几乎相同，或者在一个非常窄的范围内，即网络应该是非常均匀的。然后设计和评估满足 ENC 要求的边缘校正。大型合成网络和真实世界网络的仿真结果验证了观察到的 ENC 和边缘校正方案的有效性。随着执行更多的边缘校正操作，网络越来越接近精确满足 ENC，因此网络可控性的鲁棒性向最优方向增强。或者在一个很窄的范围内，即网络应该是非常同质的。然后设计和评估满足 ENC 要求的边缘校正。大型合成网络和真实世界网络的仿真结果验证了观察到的 ENC 和边缘校正方案的有效性。随着执行更多的边缘校正操作，网络越来越接近精确满足 ENC，因此网络可控性的鲁棒性向最优方向增强。或者在一个很窄的范围内，即网络应该是非常同质的。然后设计和评估满足 ENC 要求的边缘校正。大型合成网络和真实世界网络的仿真结果验证了观察到的 ENC 和边缘校正方案的有效性。随着执行更多的边缘校正操作，网络越来越接近精确满足 ENC，因此网络可控性的鲁棒性向最优方向增强。