Control Distance and Energy Scaling of Complex Networks

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.

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