We study the performance of the linear consensus algorithm on strongly connected graphs using the linear quadratic (LQ) cost as a performance measure. In particular, we derive bounds on the LQ cost by leveraging effective resistance. Our results extend previous analyses -- which were limited to reversible cases -- to the nonreversible setting. To facilitate this generalization, we introduce novel concepts, termed the back-and-forth path and the pivot node, which serve as effective alternatives to traditional techniques that require reversibility. Moreover, we apply our approach to geometric graphs to estimate the LQ cost without the reversibility assumption. The proposed approach provides a framework that can be adapted to other contexts where reversibility is typically assumed.

翻译：本研究利用线性二次（LQ）成本作为性能度量，分析了强连通图上线性一致性算法的性能。特别地，我们通过引入有效电阻的概念推导出LQ成本的边界。我们的结果将先前仅限于可逆情形的分析扩展到了非可逆场景。为实现这一推广，我们提出了称为"往返路径"和"枢轴节点"的新概念，这些概念可作为传统依赖可逆性技术的有效替代方案。此外，我们将该方法应用于几何图，在无需可逆性假设的情况下估计LQ成本。所提出的方法构建了一个可适用于其他通常假设可逆性场景的理论框架。