  We study the strong structural controllability (SSC) of diffusively coupled
networks, where the external control inputs are injected to only some nodes,
namely the leaders. For such systems, one measure of controllability is the
dimension of strong structurally controllable subspace, which is equal to the
smallest possible rank of controllability matrix under admissible (positive)
coupling weights. In this paper, we compare two tight lower bounds on the
dimension of strong structurally controllable subspace: one based on the
distances of followers to leaders, and the other based on the graph coloring
process known as zero forcing. We show that the distance-based lower bound is
usually better than the zero-forcing-based bound when the leaders do not
constitute a zero-forcing set. On the other hand, we also show that any set of
leaders that can be shown to achieve complete SSC via the distance-based bound
is necessarily a zero-forcing set. These results indicate that while the
zero-forcing based approach may be preferable when the focus is only on
verifying complete SSC, the distance-based approach is usually more informative
when partial SSC is also of interest. Furthermore, we also present a novel
bound based on the combination of these two approaches, which is always at
least as good as, and in some cases strictly greater than, the maximum of the
two bounds. We support our analysis with numerical results for various graphs
and leader sets.
