Node-weighted measures for complex networks with directed and weighted edges for studying continental moisture recycling

In many real-world networks nodes represent agents or objects of different sizes or importance. However, the size of the nodes is rarely taken into account in network analysis, possibly inducing bias in network measures and confusion in their interpretation. Recently, a new axiomatic scheme of node-weighted network measures has been suggested for networks with undirected and unweighted edges. However, many real-world systems are best represented by complex networks which have directed and/or weighted edges. Here, we extend this approach and suggest new versions of the degree and the clustering coefficient associated to network motifs for networks with directed and/or weighted edges and weighted nodes. We apply these measures to a spatially embedded network model and a real-world moisture recycling network. We show that these measures improve the representation of the underlying systems' structure and are of general use for studying any type of complex network.