  Despite extensive research and remarkable advancements in the control of
complex dynamical networks, most studies and practical control methods limit
their focus to time-invariant control schedules (TICS). This is both due to
their simplicity and the fact that the benefits of time-varying control
schedules (TVCS) have remained largely uncharacterized. In this paper we study
networks with linear and discrete-time dynamics and analyze the role of network
structure in TVCS. First, we show that TVCS can significantly enhance network
controllability over TICS, especially when applied to large networks. Through
the analysis of a scale-dependent notion of nodal centrality, we then show that
optimal TVCS involves the actuation of the most central nodes at appropriate
spatial scales at all times. Consequently, it is the scale-heterogeneity of the
central-nodes in a network that determine whether, and to what extent, TVCS
outperforms conventional policies based on TICS. Here, scale-heterogeneity of a
network refers to how diverse the central nodes of the network are at different
spatial (local vs. global) scales. Several analytical results and case studies
support and illustrate this relationship.
