Graph Effective Resistance and Distributed Control: Electrical Analogy and Scalability

We introduce the concept of matrix-valued effective resistance for undirected matrix-weighted graphs. These effective resistances are defined to be the square blocks that appear in the diagonal of the inverse of the matrix-weighted Dirichlet graph Laplacian matrix. However, they can also be obtained from a “generalized” electrical network that is constructed from the graph, and for which currents, voltages and resistances take matrix values. In a recent paper [3], we showed that effective resistances have a direct physical interpretation in several distributed control problems and can be used to characterize their stability and speed of convergence. In this paper, we establish several properties of effective resistances. In particular, (i) we show that the effective resistances are monotone with respect to an appropriately defined graph embedding relation; (ii) we determine laws that characterize how effective resistances scale with the distance between nodes for regular-degree large scale graphs; and (iii) we extend these laws for irregular large scale graphs that exhibit appropriate denseness properties.