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On Convergence Rate of Weighted-Averaging Dynamics for Consensus Problems

机译：关于共识问题的加权平均动力学的收敛速度

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摘要

This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of 1-O(1/(mlog_{2}m)) for special tree-like regular graphs. Also, the exponential convergence rate is established for constrained consensus over time-varying graphs, which nontrivially extends the existing result limited to the static graph case and the use of uniform weight matrices. Our main results are developed for directed graphs that are weakly connected, and we also provide statements regarding the rates in case of joint connectivity.
机译：本文研究了无约束和约束共识问题的加权平均动态。通过使用适当定义的伴随动态，构造了二次Lyapunov比较函数来分析加权平均动态的行为。结果，获得了新的收敛速度结果，其以新颖的方式捕获了图结构。特别是，对于特殊树状正则图，针对无约束共识建立了指数收敛速率，其指数范围为1-O（1 /（mlog_ {2} m））。同样，建立了随时间变化图的约束共识的指数收敛速度，这极大地扩展了限于静态图情况和使用统一权重矩阵的现有结果。我们的主要结果是针对弱连接的有向图开发的，并且在联合连接的情况下，我们还提供了有关速率的声明。

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