Theoretical progress and practical challenges in controlling complex networks

Abstract

The issue of control has gained significant prominence in engineering, useful for aircraft control, manufacturing processes, communication systems and so on. Yet, the control of complex networks was barely studied, as we lack powerful theories to address it in a quantitative fashion. Fair recently, considerable efforts were made to address the controllability issue of complex networks [1]. As a key notion in control theory, controllability concerns our ability to drive a dynamic system from any initial state to any final state in finite time [2], which agrees well with our intuitive notion of control. To model complex networks as dynamic systems, we normally adopt the so-called nodal dynamics, i.e. we associate each node in a network with a state variable, whose time evolution crucially depends on the state variables of the node itself and/or its neighbors. Nodal dynamics is natural for modeling dynamic processes on many real systems, where the state variable of a node often has a clear physical meaning, e.g. the expression level of genes in transcriptional regulatory networks or the population of species in food webs. Thanks to a combination of tools from graph theory, control theory and statistical physics, a series of results about network controllability with nodal dynamics were obtained (see Fig. 1). For example, the minimum number of driver nodes, whose time-dependent control can guide the system’s dynamics, is mainly determined by the degree distribution of the network. And surprisingly, driver nodes tend to avoid hubs, i.e. the nodes with high connectivity. Sparse and heterogeneous networks are harder to be controllable than dense and homogeneous networks [1]. These