We study the strong structural controllability (SSC) of 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 (SSCS), which is equal to the smallest possible rank of controllability matrix under admissible coupling weights among the nodes In this paper, we compare two tight lower bounds on the dimension of SSCS: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We first show that each of these two bounds can be arbitrarily better than the other in some special cases. We then show that the distance-based lower bound is usually better than the zero-forcing-based bound when the value of the latter is less than the dimensionality of the overall network state, $n$. On the other hand, we also show that any set of leaders that makes the distance-based bound equal to $n$ necessarily makes the zero-forcing-based bound equal to $n$ (the converse is not true). These results indicate that while the zero-forcing-based approach may be preferable when the focus is only on verifying complete SSC (dimension of SSCS is equal to $n$), the distance-based approach usually yields a closer bound on the dimension of SSCS when the bounds are both smaller than $n$. Furthermore, we also present a novel bound based on combining these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two original bounds. Finally, we support our analysis with numerical results on various graphs.

我们研究了网络的强结构可控性（SSC），其中外部控制输入仅注入一些节点，即领导者。对于这样的系统，可控性的一种度量是强结构可控子空间的维度(SSCS)，它等于节点间允许耦合权重下可控性矩阵的最小可能秩。其他基于图形着色的过程称为逼零。我们首先表明，在某些特殊情况下，这两个界限中的每一个都可以任意优于另一个。然后我们表明，当后者的值小于整个网络状态的维度时，基于距离的下限通常优于基于迫零的下限， $n$. 另一方面，我们还表明，任何使基于距离的界限等于的领导者集合 $n$必然使基于迫零的界限等于 $n$（反之亦然）。这些结果表明，当仅关注验证完整的 SSC（SSCS 的维度等于 $n$)，当边界都小于 $n$. 此外，我们还提出了一个基于结合这两种方法的新界限，它总是至少与两个原始界限的最大值一样好，在某些情况下严格大于最大值。最后，我们用各种图表上的数值结果来支持我们的分析。