Networks are used to model many real-world systems, including molecules, transportation systems, social networks, the World Wide Web and communication networks. Some applications require counting network substructures of many different types. The Tutte polynomial is a tool that is widely used for counting substructures in networks. We study several counting functions related to the Tutte polynomial. We focus on networks where every link has a fixed direction. These networks are more complex than undirected networks. We establish fundamental properties of these functions for some networks drawn on surfaces, and networks with one principal node.