A New Notion of Effective Resistance for Directed Graphs---Part I: Definition and Properties

George Forrest Young, Luca Scardovi, Naomi Ehrich Leonard

(2016), IEEE Transactions on Automatic Control, Vol. 61, No. 7, pp. 1727-1736.

Part I Paper PDF
arXiv:1310.5163v2 [math.OC]
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.

A New Notion of Effective Resistance for Directed Graphs---Part II: Computing Resistances

George Forrest Young, Luca Scardovi, Naomi Ehrich Leonard

(2016), IEEE Transactions on Automatic Control, Vol. 61, No. 7, pp. 1737-1751.

Part II Paper PDF
arXiv:1310.5168v2 [math.OC]
In Part I of this work we defined a generalization of the concept of effective resistance to directed graphs, and we explored some of the properties of this new definition. Here, we use the theory developed in Part I to compute effective resistances in some prototypical directed graphs. This exploration highlights cases where our notion of effective resistance for directed graphs behaves analogously to our experience from undirected graphs, as well as cases where it behaves in unexpected ways.

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