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The minimum number of inputs needed to control a network is frequently usedto quantify its controllability. Control of linear dynamics through a minimumset of inputs, however, often has prohibitively large energy requirements andthere is an inherent trade-off between minimizing the number of inputs andcontrol energy. To better understand this trade-off, we study the problem ofidentifying a minimum set of input nodes such that controllabililty is ensuredwhile restricting the length of the longest control chain. The longest controlchain is the maximum distance from input nodes to any network node, and recentwork found that reducing its length significantly reduces control energy. Wemap the longest control chain-constraint minimum input problem to finding ajoint maximum matching and minimum dominating set. We show that this graphcombinatorial problem is NP-complete, and we introduce and validate a heuristicapproximation. Applying this algorithm to a collection of real and modelnetworks, we investigate how network structure affects the minimum number ofinputs, revealing, for example, that for many real networks reducing thelongest control chain requires only few or no additional inputs, only therearrangement of the input nodes.

Controllability of complex networks: input node placement restricting the longest control chain