We study the performance of the linear consensus algorithm on strongly connected directed graphs using the linear quadratic (LQ) cost as a performance measure. In particular, we derive bounds on the LQ cost by leveraging effective resistance and reversiblization. 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 Cayley graphs and random 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：本文采用线性二次（LQ）成本作为性能指标，研究了强连通有向图上线性一致性算法的性能。特别地，我们通过利用有效电阻与可逆化技术，推导了LQ成本的上下界。我们的结果将先前仅限于可逆情形的分析推广到了非可逆场景。为实现这一推广，我们引入了称为“往返路径”与“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法。此外，我们将所提方法应用于Cayley图与随机几何图，以在无需可逆性假设的情况下估计LQ成本。所提出的框架可适用于其他通常假设可逆性的研究场景。