The ability to modify the structure of network systems offers great opportunities to enhance their operation, improve their efficiency, and increase their resilience against failures and attacks. This paper focuses on the edge modification problem, i.e., improving network controllability by adding and/or re-weighting interconnections while keeping the actuation structure fixed. We consider a network system following linear dynamics and propose a novel edge centrality measure that captures the extent to which an edge facilitates energy exchange across the network through its defining nodes. We analyze the effectiveness of the proposed measure by characterizing its relationship with the gradient (with respect to edge weights) of trace, logdet and inverse of the trace inverse of the Gramian. We show that the optimal solution of the edge modification problem lies on the boundary of the feasible search space when the objective is the trace of the Gramian or the network has a diagonal controllability Gramian and the objective is either logdet or the inverse of the trace inverse of the Gramian. Finally, using the proposed edge centrality measure we design two network modification algorithms that restrict the search space to a smaller subset of all possible edges and numerically demonstrate their efficacy.

中文翻译：

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能量传输边缘中心性及其在增强网络可控性中的作用

修改网络系统结构的能力为增强其运行、提高其效率和增加其对故障和攻击的恢复能力提供了巨大的机会。本文重点讨论边缘修改问题，即在保持驱动结构固定的同时，通过添加和/或重新加权互连来提高网络可控性。我们考虑了一个遵循线性动力学的网络系统，并提出了一种新颖的边缘中心性度量，该度量可以捕捉边缘通过其定义节点促进跨网络能量交换的程度。我们通过表征其与迹的梯度（相对于边缘权重）、logdet 和 Gramian 的迹逆的逆的关系来分析所提出的度量的有效性。我们表明，当目标是 Gramian 的迹或网络具有对角可控 Gramian 并且目标是 logdet 或迹逆的逆时，边缘修改问题的最优解位于可行搜索空间的边界上的格拉米。最后，使用所提出的边缘中心性度量，我们设计了两种网络修改算法，将搜索空间限制为所有可能边缘的较小子集，并以数值方式证明其有效性。