  The graphical notion of effective resistance has found wide-ranging
applications in many areas of pure mathematics, applied mathematics and control
theory. By the nature of its construction, effective resistance can only be
computed in undirected graphs and yet in several areas of its application,
directed graphs arise as naturally (or more naturally) than undirected ones. In
part I of this work, we propose a generalization of effective resistance to
directed graphs that preserves its control-theoretic properties in relation to
consensus-type dynamics. We proceed to analyze the dependence of our algebraic
definition on the structural properties of the graph and the relationship
between our construction and a graphical distance. The results make possible
the calculation of effective resistance between any two nodes in any directed
graph and provide a solid foundation for the application of effective
resistance to problems involving directed graphs.
