Energy cost for controlling complex networks with linear dynamics

Abstract

The controllability of complex networks has received much attention recently, which tells whether we can steer a system from an initial state to any final state within finite time with admissible external inputs. In order to accomplish the control in practice at the minimum cost, we must study how much control energy is needed to reach the desired final state. At a given control distance between the initial and final states, existing results present the scaling behavior of lower bounds of the minimum energy in terms of the control time analytically. However, to reach an arbitrary final state at a given control distance, the minimum energy is actually dominated by the upper bound, whose analytic expression still remains elusive. Here we theoretically show the scaling behavior of the upper bound of the minimum energy in terms of the time required to achieve control. Apart from validating the analytical results with numerical simulations, our findings are feasible to the scenario with any number of nodes that receive inputs directly and any types of networks. Moreover, more precise analytical results for the lower bound of the minimum energy are derived in the proposed framework. Our results pave the way to implement realistic control over various complex networks with the minimum control cost.

Investigating the energy cost for controlling complex social networks

“…Refs. [3,5] study how the final control time t f affects the energy cost (assuming t 0 = 0 for convenience). The energy cost was characterized by their lower and upper bound scaling laws with respect to t f .…”

“…, lower bound and small t f t −θ f , upper bound and small t f , (1.11) where θ 1, which is to be determined numerically. It turns out that [5] θ ≈ N 0 − N min when using less than N number of drivers, where N 0 and N min are numerical values related to the invertibility of the controllability Gramian, and when using N number of drivers such that each node directly receives a control signal, the upper bound also scales as t −1 f in the small t f regime. What this means is that demanding the control signals the drive the network node states in a small amount of time requires the most control energy, and by relaxing the demand, energy cost is reduced.…”

“…[3] and Ref. [5] differ because one studies the scaling laws with assumption that x f = 0, and the other x 0 = 0 (also known as controllability-to-zero and reachability-from-zero [23]). Essentially, x 0 = 0 or x f = 0 are just the reverse direction of each other, as far as system stability/instability are concerned.…”

“…Networks which are negative semi-definite (NSD) or positive semi-definite (PSD) have respectively the lower and upper bound scale as t −1 f . Networks which are not ND or are PD have respectively the lower and upper bound scale as ∼ e −λnt f , or ∼ e −λ 1 t f , where λ n and λ 1 are the largest and smallest eigenvalue of A [5]. All of these scaling laws represent the different implications about how the energy cost varies with respect to t f and the inherent system stability modes.…”

“…• Secondly, Chapter 3 revisits the energy cost scaling laws with respect to final control time t f when conformity behavior [27] is modelled into the network dynamics, where each node mimics the opinions of their nearest neighbors over time. Previous works [3,5] have derived the scaling laws with respect to t f for networks with continuous-time linear dynamics. However, continuous-time linear dynamics is unable to capture conformity behavior, which the discrete-time modelling [27] can.…”