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One of the properties of interest in the analysis of networks is \emph{global communicability}, i.e., how easy or difficult it is, generally, to reach nodes from other nodes by following edges. Different global communicability measures provide quantitative assessments of this property, emphasizing different aspects of the problem. This paper investigates the sensitivity of global measures of communicability to local changes. In particular, for directed, weighted networks, we study how different global measures of communicability change when the weight of a single edge is changed; or, in the unweighted case, when an edge is added or removed. The measures we study include the \emph{total network communicability}, based on the matrix exponential of the adjacency matrix, and the \emph{Perron network communicability}, defined in terms of the Perron root of the adjacency matrix and the associated left and right eigenvectors. Finding what local changes lead to the largest changes in global communicability has many potential applications, including assessing the resilience of a system to failure or attack, guidance for incremental system improvements, and studying the sensitivity of global communicability measures to errors in the network connection data., Comment: 28 pages, 7 figures