In this note, we study a networked system with single/multiple pinning. Given a weighted and undirected network, we derive lower and upper bounds on its algebraic connectivity with respect to the reference signal. The bounds are derived by partitioning the network in terms of distance of each node from the pinning set. Upper and lower bounds for two networks with differing topologies are computed to demonstrate the tightness of the derived results. It is shown, using the derived bounds, how requirements on the number of pinning nodes and pinning gain required for achieving stability or a specified convergence rate for the network can be easily obtained.

中文翻译：

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固定拉普拉斯矩阵的最小特征值的界限


在本文中，我们研究了具有单/多个固定的网络系统。给定一个加权无向网络，我们得出其相对于参考信号的代数连通性的下限和上限。通过根据每个节点与固定集的距离对网络进行分区来导出边界。计算具有不同拓扑的两个网络的上限和下限，以证明导出结果的紧密性。结果表明，使用导出的边界，如何可以轻松获得对网络稳定性或指定收敛速率所需的钉扎节点数量和钉扎增益的要求。