  We study the sensitivity of directed complex networks to the addition and
pruning of edges and vertices and introduce the susceptibility, which
quantifies this sensitivity. We show that topologically different parts of a
directed network have different sensitivity to the addition and pruning of
edges and vertices and, therefore, they are characterized by different
susceptibilities. These susceptibilities diverge at the critical point of the
directed percolation transition, signaling the appearance (or disappearance) of
the giant strongly connected component in the infinite size limit. We
demonstrate this behavior in randomly damaged real and synthetic directed
complex networks, such as the World Wide Web, Twitter, the \emph{Caenorhabditis
elegans} neural network, directed Erd\H{o}s-R\'enyi graphs, and others. We
reveal a non-monotonous dependence of the sensitivity to random pruning of
edges or vertices in the case of \emph{Caenorhabditis elegans} and Twitter that
manifests specific structural peculiarities of these networks. We propose the
measurements of the susceptibilities during the addition or pruning of edges
and vertices as a new method for studying structural peculiarities of directed
networks.
