Analysis of Linear Consensus Algorithm on Strongly Connected Graph Using Effective Resistance

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.