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.

翻译：我们研究了连接图中有效电阻的概念，将其从无向图的传统应用拓展至更广泛的场景。通过聚焦连接图上狄利克雷型与泊松型问题的对偶性，我们提出了连接图中有效电阻的稳健定义。此外，我们深入探讨了随机游走过程，同时考虑节点转移与向量旋转。该方法引入了连接图有效电导矩阵与有效电阻矩阵的新概念，捕捉了随机游走转移对应的平均旋转矩阵。由此，为网络分析与优化提供了新的理论洞见。