Physical controllability of complex networks

Abstract

A challenging problem in network science is to control complex networks. In existing frameworks of structural or exact controllability, the ability to steer a complex network toward any desired state is measured by the minimum number of required driver nodes. However, if we implement actual control by imposing input signals on the minimum set of driver nodes, an unexpected phenomenon arises: due to computational or experimental error there is a great probability that convergence to the final state cannot be achieved. In fact, the associated control cost can become unbearably large, effectively preventing actual control from being realized physically. The difficulty is particularly severe when the network is deemed controllable with a small number of drivers. Here we develop a physical controllability framework based on the probability of achieving actual control. Using a recently identified fundamental chain structure underlying the control energy, we offer strategies to turn physically uncontrollable networks into physically controllable ones by imposing slightly augmented set of input signals on properly chosen nodes. Our findings indicate that, although full control can be theoretically guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and control cost to achieve physical control.

Enhancing Controllability Robustness of q-Snapback Networks through Redirecting Edges

“…Recently, the network controllability has become a focal topic in network science investigation [5][6][7][8][9][10][11][12][13], where the controllability means the ability of the network in changing its state from any initial position to any targeted position under a certain control input within a finite duration of time. Meanwhile, malicious attacks on complex networks are another concerned issue in network science studies [14][15][16][17][18].…”

Irrelevance of linear controllability to nonlinear dynamical networks

“…A key development is the systematic adoption of the linear structural controllability theory to complex networks with directed interactions [4]. Since then, there has been a great deal of effort in investigating the linear controllability of complex networks [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…Using the exact controllability theory [9], we calculate the minimal control set. A key feature of linear network control, which was usually not emphasized in most existing literature on linear controllability [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] but was mentioned in a recent paper [60], is that the minimal control set of nodes is not unique. For a reasonably large network (e.g., of size of a few hundred), there can be vastly many such sets that are equivalent to each other in terms of control realization.…”

Topological Controllability of Undirected Networks of Diffusively-Coupled Agents

“…However, in certain networks, it may be hard to know the coupling strengths between nodes or there may be uncertainties in the measure or identification of the coupling strengths. Consequently, if the controllability of a network can be evaluated only using the topology or structure, it will be beneficial in some applications, including control of digital and electric circuits [3], [4], power electronics [5], large scale networks [6], complex networks [7], [8], and brain networks [9].…”