  We investigate to what extent the degree sequence of a directed network
constrains the number of driver nodes. We develop a pair of algorithms that
take a directed degree sequence as input and aim to output a network with the
maximum or minimum number of driver nodes. We find an upper bound for the
maximum and a lower bound for the minimum, and show that the algorithms achieve
these bounds for all real and model networks, with few exceptions characterized
by tiny system size and heterogeneous degree distributions. Applying these
algorithms to a broad range of real networks, we show the gap between the upper
and lower bounds can vary dramatically across different degree sequences. Thus,
we introduce the notion of \bl{structural control} complexity to capture how
much more difficult it is to control a specific network beyond what is required
given its degree sequence, suggesting additional structure is present. Using
model networks, we numerically and analytically investigate how typical
features of the degree distribution affect the range of required driver nodes.
We find that the minimum is determined by the number of sources or sinks, while
the maximum is strongly affected by the presence of hubs.
