Random Walks, Conductance, and Resistance for the Connection Graph Laplacian

TL;DR

This paper extends the concept of effective resistance to connection graphs, integrating random walks with node transitions and vector rotations to develop new theoretical tools for network analysis.

Contribution

It introduces a robust definition of effective resistance in connection graphs and develops novel conductance and resistance matrices incorporating vector rotations.

Findings

New theoretical framework for connection graphs

Effective resistance and conductance matrices for connection graphs

Insights into network analysis and optimization

Abstract

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.