多维动态网络中的渐近一致性
Multidimensional Asymptotic Consensus in Dynamic Networks
AI总结：
本文研究了动态网络中多维渐近一致性的算法，提出两种新算法，具有恒定的收敛速度，适用于双向交互的弱连通性条件。
摘要：
我们研究了在多种人工和自然系统中出现的渐近一致性问题。特别是，我们研究了通信图可能随时间变化的系统。我们最近提出了一种新的凸组合算法家族，在一维中其权重取决于接收到的值而非仅通信拓扑。这里，我们通过引入两种新算法：极点算法和质心算法，将这种方法扩展到任意高维。与经典凸组合算法不同，这两种算法的组件收缩率在代理数量上是常数。结合凸组合算法的速度提升技术，我们得到收敛时间与代理数量线性相关的结果，这是最优的。此外，这两种算法在各自的收缩率之外，还存在差异：质心算法的更新规则不依赖于任何坐标系统，而极点算法隐含假设代理之间有共同约定的坐标系统。后者假设可能在某些人工多代理系统中是现实的，但在自然现象建模系统中则存在疑问。最后我们证明，在非常弱的连通性假设下，只要代理交互是双向的，我们的新算法也能实现渐近一致性。
英文摘要：
We study the problem of asymptotic consensus as it occurs in a wide range of applications in both man-made and natural systems. In particular, we study systems with directed communication graphs that may change over time. We recently proposed a new family of convex combination algorithms in dimension one whose weights depend on the received values and not only on the communication topology. Here, we extend this approach to arbitrarily high dimensions by introducing two new algorithms: the ExtremePoint and the Centroid algorithm. Contrary to classical convex combination algorithms, both have component-wise contraction rates that are constant in the number of agents. Paired with a speed-up technique for convex combination algorithms, we get a convergence time linear in the number of agents, which is optimal. Besides their respective contraction rates, the two algorithms differ in the fact that the Centroid algorithm's update rule is independent of any coordinate system while the ExtremePoint algorithm implicitly assumes a common agreed-upon coordinate system among agents. The latter assumption may be realistic in some man-made multi-agent systems but is highly questionable in systems designed for the modelization of natural phenomena. Finally we prove that our new algorithms also achieve asymptotic consensus under very weak connectivity assumptions, provided that agent interactions are bidirectional.
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