Minimum energy control for complex networks

Abstract

The aim of this paper is to shed light on the problem of controlling a complex network with minimal control energy. We show first that the control energy depends on the time constant of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for complete controllability. In the limit case of networks having all purely imaginary eigenvalues (e.g. networks of coupled harmonic oscillators), several constructive algorithms for minimum control energy driver node selection are developed. A general heuristic principle valid for any directed network is also proposed: the overall cost of controlling a network is reduced when the controls are concentrated on the nodes with highest ratio of weighted outdegree vs indegree.

Combining centrality measures for control energy reduction in network controllability problems

“…Also for these networks, r quot gives the best λ min (W ) while r diff is better for Tr(W ). These results are coherent with what obtained in Lindmark and Altafini (2018) based only on numerical evidence. In fact, choosing power laws for the degree distributions means that the amount of non-normality of the corresponding adjacency matrix is increased as a significant fraction of overall outgoing edge weights is concentrated at a few nodes, and similarly for the overall incoming edge weights, thereby resulting into skewed distribution of p i and q i , compare Figure 4 with Figure 7.…”

“…In Lindmark and Altafini (2018), we showed numerically that there are at least two factors influencing the energy required to control a network. One is the location of the eigenvalues (fast modes require more control energy than slow modes, see also Yan et al (2015)).…”

“…One is the location of the eigenvalues (fast modes require more control energy than slow modes, see also Yan et al (2015)). The second is instead a connectivity property, which in Lindmark and Altafini (2018) was expressed as a ratio between the weighted outdegree of the nodes and their weighted indegree. We showed that the higher this ratio is, the lower is the control energy needed to steer the system.…”

Functional Target Controllability of Networks: Structural Properties and Efficient Algorithms

“…Most of the papers in literature study controllability according to the most common definition in systems theory, that is, the ability to steer the state to a target point: we shall refer to this definition as point-wise controllability. This definition (and the criticism of its limits [11], [6]) have also been the starting point of a series of works that approached more advanced questions like quantifying the energy required for control [12], [13], [10] and the robustness of the controllability properties [14], [8], [15], [16] in the context of networks.…”

Topology Effects on Sparse Control of Complex Networks with Laplacian Dynamics

“…a norm of the control signals that move the state of the network to a desired point in state space 15–28 . Different metrics of controllability have been used to characterize control energy based on the controllability Gramian, including its minimum eigenvalue 15–19 , its trace 20 , the trace of its inverse 21–24 , its condition number 25,26 , and even mixed properties 27 . Some works aimed at the optimal placement of the control nodes to maximize practical controllability of complex networks 20–22,24,28 , while others have focused on relating network structure to the minimum number of control nodes necessary to achieve energetically implementable control profiles 15–19,23,25–27 .…”