Controllability Transition and Nonlocality in Network Control

Abstract

A common goal in the control of a large network is to minimize the number of driver nodes or control inputs. Yet, the physical determination of control signals and the properties of the resulting control trajectories remain widely under-explored. Here we show that: (i) numerical control fails in practice even for linear systems if the controllability Gramian is ill-conditioned, which occurs frequently even when existing controllability criteria are satisfied unambiguously; (ii) the control trajectories are generally nonlocal in the phase space, and their lengths are strongly anti-correlated with the numerical success rate and number of control inputs; (iii) numerical success rate increases abruptly from zero to nearly one as the number of control inputs is increased, a transformation we term numerical controllability transition. This reveals a trade-off between nonlocality of the control trajectory in the phase space and nonlocality of the control inputs in the network itself. The failure of numerical control cannot be overcome in general by merely increasing numerical precision-successful control requires instead increasing the number of control inputs beyond the numerical controllability transition.

Causal Network Inference by Optimal Causation Entropy

“…Research of dynamic processes on large-scale complex networks has attracted considerable interest in recent years with exciting developments in a wide range of disciplines in social, scientific, engineering, and medical fields [48,49,74]. One important line of research focuses on exploring the role of network structure in determining the dynamic properties of a system [6,17,18,19,27,55,67,79] and utilizing such knowledge in controlling network dynamics [15,70] and optimizing network performance [13,38,50,56,72]. In applications such as the study of neuronal connectivity or gene interactions, it is nearly impossible to directly identify the network structure without severely interfering with the underlying system whereas time series measurements of the individual node states are often more accessible [68].…”

Synchronization in complex networks and its application – A survey of recent advances and challenges

“…The results in Tang et al (2012cTang et al ( , 2014c do not show their generality in other networks, which is a little bit different from the statistical results in Liu et al (2011). Based on this seminal work Liu et al (2011), there are intensive and extensive works focusing on controllability of complex networks Menichetti, Asta, & Bianconi, 2014;Nepusz & Vicsek, 2012;Ruths & Ruths, 2014;Sun & Motter, 2013;Suweis, Simini, Banavar, & Maritan, 2013;Yan, Ren, Lai, Lai, & Li, 2012;Yuan, Zhao, Di, Wang, & Lai, 2013) and their applications (Csermely, Korcsmáros, Kiss, London, & Nussinov, 2013;Notarstefano & Parlangeli, 2013). For instance, a dynamical process of controllability defined on the edges of a network is discussed, and it is shown that the controllability properties of this process is significantly different from simple nodal dynamics (Nepusz & Vicsek, 2012), as shown in Liu et al (2011).…”

“…1(b) a network's mean basin stability hS B i is determined primarily by the location of its stability interval I s (Menck et al, 2013). Actually, the concept of region of attraction has been used in controllability of complex networks (Cornelius, Kath, & Motter, 2013;Sun & Motter, 2013), in which the perturbations are considered as a tool to drive the states of a system to the region of attraction of a desired state in contrast to Menck et al (2013). This way, the control objective of complex networks can be achieved.…”

On controlling networks of limit-cycle oscillators

“…For instance, the concept of controllability of complex networks has been established using control theory for linear dynamical systems [14][15][16][17][18] . Significant advances have also been made in the control of several a) Electronic mail: persebastian.skardal@trincoll.edu nonlinear networks-connected dynamical systems [19][20][21][22][23] .…”