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

TL;DR

This paper explores how to select input nodes in complex networks to ensure controllability while limiting the longest control chain, balancing control energy and input minimization.

Contribution

It formulates the problem as a graph combinatorial optimization, proves its NP-completeness, and proposes a heuristic solution, demonstrating its effectiveness on real and model networks.

Findings

Reducing the longest control chain often requires few additional inputs.

Network structure influences the number of inputs needed for control.

Rearranging input nodes can achieve control with minimal additional inputs.

Abstract

The minimum number of inputs needed to control a network is frequently used to quantify its controllability. Control of linear dynamics through a minimum set of inputs, however, often has prohibitively large energy requirements and there is an inherent trade-off between minimizing the number of inputs and control energy. To better understand this trade-off, we study the problem of identifying a minimum set of input nodes such that controllabililty is ensured while restricting the length of the longest control chain. The longest control chain is the maximum distance from input nodes to any network node, and recent work found that reducing its length significantly reduces control energy. We map the longest control chain-constraint minimum input problem to finding a joint maximum matching and minimum dominating set. We show that this graph combinatorial problem is NP-complete, and we introduce and validate a heuristic approximation. Applying this algorithm to a collection of real and model networks, we investigate how network structure affects the minimum number of inputs, revealing, for example, that for many real networks reducing the longest control chain requires only few or no additional inputs, only the rearrangement of the input nodes.