  We investigate the concept of effective resistance in connection graphs,
expanding its traditional application from undirected graphs. We propose a
robust definition of effective resistance in connection graphs by focusing on
the duality of Dirichlet-type and Poisson-type problems on connection graphs.
Additionally, we delve into random walks, taking into account both node
transitions and vector rotations. This approach introduces novel concepts of
effective conductance and resistance matrices for connection graphs, capturing
mean rotation matrices corresponding to random walk transitions. Thereby, it
provides new theoretical insights for network analysis and optimization.
