Today, there exist many centrality measures for assessing the importance of nodes in a network as a function of their position and the underlying topology. One class of such measures builds on eigenvector centrality, where the importance of a node is derived from the importance of its neighboring nodes. For directed and weighted complex networks, where the nodes can carry some intrinsic property value, there have been centrality measures proposed that are variants of eigenvector centrality. However, these expressions all suffer from shortcomings. Here, an extension of such centrality measures is presented that remedies all previously encountered issues. While similar improved centrality measures have been proposed as algorithmic recipes, the novel quantity that is presented here is a purely analytical expression, only utilizing the adjacency matrix and the vector of node values. The derivation of the new centrality measure is motivated in detail. Specifically, the centrality itself is ideal for the analysis of directed and weighted networks (with node properties) displaying a bow-tie topology. The novel bow-tie centrality is then computed for a unique and extensive real-world data set, coming from economics. It is shown how the bow-tie centrality assesses the relevance of nodes similarly to other eigenvector centrality measures, while not being plagued by their drawbacks in the presence of cycles in the network.