  In practical terms, controlling a network requires manipulating a large
number of nodes with a comparatively small number of external inputs, a process
that is facilitated by paths that broadcast the influence of the
(directly-controlled) driver nodes to the rest of the network. Recent work has
shown that surprisingly, temporal networks can enjoy tremendous control
advantages over their static counterparts despite the fact that in temporal
networks such paths are seldom instantaneously available. To understand the
underlying reasons, here we systematically analyze the scaling behavior of a
key control cost for temporal networks--the control energy. We show that the
energy costs of controlling temporal networks are determined solely by the
spectral properties of an "effective" Gramian matrix, analogous to the static
network case. Surprisingly, we find that this scaling is largely dictated by
the first and the last network snapshot in the temporal sequence, independent
of the number of intervening snapshots, the initial and final states, and the
number of driver nodes. Our results uncover the intrinsic laws governing why
and when temporal networks save considerable control energy over their static
counterparts.
