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：分析复杂网络时，一项重要任务是识别那些对网络整体通信性起主导作用的节点。在修改网络（或使其能抵御针对性攻击或故障）的背景下，了解网络通信性对特定节点或边变化的敏感程度也具有重要意义。最近，[O. De la Cruz Cabrera, J. Jin, S. Noschese, L. Reichel, Communication in complex networks, Appl. Numer. Math., 172, pp. 186-205, 2022]引入了总网络敏感性的概念，该概念可用于衡量网络总通信性对添加或移除某些边的敏感程度。该概念的一个缺陷在于，当采用直接计算方法时，敏感性的计算成本极高（比相应的通信性量测度高出数个数量级）。本文提出了估算网络敏感性的计算流程，对于许多真实世界复杂网络，其计算成本基本与节点数呈线性关系。此外，我们扩展了敏感性概念，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，这些界能很好地捕捉定性行为，并揭示基于矩阵函数的网络指标在扰动下的总体行为特征。这些界基于具有结构化、低秩方向项的矩阵函数Fréchet导数的衰减结果，该结果可能对网络分析之外的其他应用也具有独立价值。