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We study the strong structural controllability (SSC) of diffusively couplednetworks, where the external control inputs are injected to only some nodes,namely the leaders. For such systems, one measure of controllability is thedimension of strong structurally controllable subspace, which is equal to thesmallest possible rank of controllability matrix under admissible (positive)coupling weights. In this paper, we compare two tight lower bounds on thedimension of strong structurally controllable subspace: one based on thedistances of followers to leaders, and the other based on the graph coloringprocess known as zero forcing. We show that the distance-based lower bound isusually better than the zero-forcing-based bound when the leaders do notconstitute a zero-forcing set. On the other hand, we also show that any set ofleaders that can be shown to achieve complete SSC via the distance-based boundis necessarily a zero-forcing set. These results indicate that while thezero-forcing based approach may be preferable when the focus is only onverifying complete SSC, the distance-based approach is usually more informativewhen partial SSC is also of interest. Furthermore, we also present a novelbound based on the combination of these two approaches, which is always atleast as good as, and in some cases strictly greater than, the maximum of thetwo bounds. We support our analysis with numerical results for various graphsand leader sets.

Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing