When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network's communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, L. Reichel, Communication in complex networks, Appl. Numer. Math., 172, pp. 186-205, 2022], which allows to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straight-forward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture the qualitative behavior well and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fr\'echet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for other applications than network analysis.

翻译：在分析复杂网络时，识别那些对网络整体通信能力起主导作用的节点是一项重要任务。在修改网络（或使其对针对性攻击或故障具有鲁棒性）的背景下，了解网络通信能力对某些节点或边变化的敏感性也具有相关性。近期，O. De la Cruz Cabrera、J. Jin、S. Noschese和L. Reichel在《复杂网络中的通信》(Appl. Numer. Math., 172, pp. 186-205, 2022)一文中引入了总网络敏感性的概念，该概念允许衡量网络总通信能力对添加或移除特定边的敏感程度。该概念的一个缺点在于，当采用直接计算方法时，敏感性计算成本极高（比相应通信能力度量的计算成本高出数个数量级）。本研究提出了一种计算程序，可在许多现实世界复杂网络中，以与节点数基本呈线性的成本估算网络敏感性。此外，我们将敏感性概念进行扩展，使其涵盖子图中心性和Estrada指数的敏感性，并讨论了节点移除的情况。我们提出了这些敏感性的先验界限，这些界限能够很好地捕捉定性行为，并深入理解基于矩阵函数的网络指标在扰动下的总体行为。这些界限基于具有结构化低秩方向项的矩阵函数Fréchet导数的衰减结果，该结果可能对除网络分析之外的其他应用也具有独立参考价值。