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On Optimality of Sparse Long-Range Links in Circulant Consensus Networks

机译：循环共识网络中稀疏远程链接的最优性

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We consider spatially invariant consensus networks in which the directed graph describing the interconnection topology, the link weights, and the temporal dynamics, are all characterized by circulant matrices. We seek the best new links, subject to budget constraints, whose addition to the network maximally improves its rate of convergence to consensus. Motivated by small-world networks, we apply the optimal link creation problem to circulant networks with local communication links. We observe that the optimal new links are sparse and long-range, and have an increasingly pronounced effect on the convergence rate of the network as its size grows. To further investigate the properties of optimal links analytically, we restrict attention to the creation of links with small weights, referred to as weak links. We employ perturbation methods to reformulate the problem of optimal weak link creation, and uncover conditions on the network architecture which guarantee sparse and long-range solutions to this optimization problem.
机译：我们考虑空间不变的共识网络，其中描述互连拓扑，链接权重和时间动态的有向图均以循环矩阵为特征。我们寻求受预算限制的最佳新链接，将其添加到网络中可最大程度地提高其达成共识的融合率。受小世界网络的激励，我们将最佳链路创建问题应用于具有本地通信链路的循环网络。我们观察到，最佳的新链路稀疏且远距离，并且随着网络规模的增长，对网络的收敛速度的影响越来越明显。为了通过分析进一步研究最佳链接的属性，我们将注意力集中在创建权重较小的链接（称为弱链接）上。我们采用摄动方法来重新构造最佳的弱链接创建问题，并揭示网络体系结构的条件，这些条件保证了该优化问题的稀疏和远程解决方案。

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