Simplicial networks and effective resistance

The effective resistance Rσ of a current generator σ shall be defined as a ratio of the σ-components of Vσ and Iσ. By introducing potential for voltage vectors, we present a formula for Rσ via the inverse of the weighted combinatorial Laplacian of X in codimension one. We also derive a formula for Rσ via weighted high-dimensional tree-numbers for X, providing a combinatorial interpretation for Rσ. As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples.