It has recently been shown that the average energy required to control a subset of target nodes in a complex network scales exponentially with the cardinality of the subset. While the mean scales exponentially, the variance of the control energy over different subsets of target nodes can be large and has, as of yet, not been explained. Here, we provide an explanation of the large variance as a result of both the length of the path that connects control inputs to the target nodes and the redundancy of paths of shortest length. Our first result provides an upper bound of the control energy as a function of path length between driver node and target node along an infinite path graph for a single target node. We also show that the energy estimate is still very accurate even when finite size effects are taken into account. Our second result refines the upper bound, by an order of magnitude or more, taking into account not only the length of the path, but also the redundancy of paths. Finally, we lay out the foundations for a more accurate estimation of the control energy for the multi-target node and multi-driver node problem.

中文翻译：

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控制复杂网络的距离和能量缩放

最近已经表明，控制复杂网络中目标节点子集所需的平均能量随子集的基数呈指数增长。虽然均值呈指数级增长，但目标节点不同子集的控制能量的方差可能很大，并且迄今为止尚未得到解释。在这里，我们解释了由于将控制输入连接到目标节点的路径长度和最短路径冗余导致的大方差。我们的第一个结果提供了控制能量的上限，作为驱动器节点和目标节点之间路径长度的函数，沿着单个目标节点的无限路径图。我们还表明，即使考虑到有限尺寸效应，能量估计仍然非常准确。我们的第二个结果将上限细化了一个数量级或更多，不仅考虑了路径的长度，还考虑了路径的冗余。最后，我们为更准确地估计多目标节点和多驱动器节点问题的控制能量奠定了基础。