Enhancing speed of pinning synchronizability: low-degree nodes with high feedback gains

Controlling complex networks is of paramount importance in science and engineering. Despite recent efforts to improve controllability and synchronous strength, little attention has been paid to the speed of pinning synchronizability (rate of convergence in pinning control) and the corresponding pinning node selection. To address this issue, we propose a hypothesis to restrict the control cost, then build a linear matrix inequality related to the speed of pinning controllability. By solving the inequality, we obtain both the speed of pinning controllability and optimal control strength (feedback gains in pinning control) for all nodes. Interestingly, some low-degree nodes are able to achieve large feedback gains, which suggests that they have high influence on controlling system. In addition, when choosing nodes with high feedback gains as pinning nodes, the controlling speed of real systems is remarkably enhanced compared to that of traditional large-degree and large-betweenness selections. Thus, the proposed approach provides a novel way to investigate the speed of pinning controllability and can evoke other effective heuristic pinning node selections for large-scale systems.


Equation 1 can be stabilized to s(t) on condition that the matrix [∂f (x) + aΓ] is Hurwitz
matrix [4], which requires that the real part of [∂f (x) + aΓ] is negative: where a manifold region S exists that equation 6 is negative if a ∈ S; otherwise, the system is chaos.

II. SI NOTES: SPEED OF CONTROLLABILITY
The speed of controllability characterizes the rate of convergence in pinning control. To investigate the speed of controllability, a precondition exists that the system is stable. Under this condition, the convergence speed of equation 3 is determined by the largest eigenvalue Equation 12 is a general metric to evaluate the speed of controllability. The aim is selecting appropriate pinning nodes to minimize the largest eigenvalue of [∂f (x) + cλ k Γ], i.e., minimize v.
If the stable region is S = S 2 = (α 2 , α 1 ), conventional method usually use Eq. 11 to characterize the strength of stability. Previous research mostly focuses on minimizing R to extend the coupling interval (σ 1 , σ 2 ) of c [1,2]. Low R requires large λ N , but small λ 1 . Smaller R usually indicates larger stable range and better stability. However the minimal R don't always represent minimal v of Eq. 12. For example, we suppose that λ N = Const. As λ 1 decreases, R decreases as a result. But the speed v depends on not only . So the speed of controllability v may not change. Thus R is not enough to evaluate the control speed of the system when S = S 2 , and previous research about enhancing stability is not suitable for the speed of controllability. Only Equation 12 is appropriate for evaluating the speed of controllability.
If cλ k < ρ (∀i, i = 1, ..., N ), we can ensure that ∂f (x) + cλ k Γ are Hurwitz matrix and the controlled network is exponentially stable [4]. so the stable condition is where ρ is a constant. The stable range of c is c ∈ (c min , +∞).
Under the stable condition, the speed are mainly determined by the largest eigenvalue of lower λ 1 (B) represents lower negative eigenvalues of [∂f (x) + cλ 1 Γ] and higher speed of controllability. Thus, λ 1 (B) is also utilized to characterize the speed of controllability for a network in the paper. performs better than degree pinning control in other three real networks and the differences are due to the special topologies of real networks. Note that, the only difference is the lines C = 1 in Fig. S1(b) and S1(d), and these lines have similar performance with those of large-degree pinning control. When C is small, restricted by E sum = C = 1, only some high degree nodes obtain high feedback gains and are selected as pinning nodes. Therefore, both approaches have similar speed of pinning controllability. In practice, we can increase properly C to avoid the trap of performance.

IV. SI NOTES: SPEED OF CONTROLLABILITY BASED ON BETWEENNESS
Since network topology has a great influence on the controllability [7] and the importance of a node relates much to the structure of the network [8], apart from degree of nodes, the importance of a node could also be evaluated by its betweenness [9,10]. A node's betweenness is defined as the number of shortest paths from all vertices to all others that pass through that node. A node with high betweenness has a large influence on the transfer of information through the network, under the assumption that information transfer follows the shortest paths. Here, we suppose that w i = g i , where g i is the betweenness of node i. The results are shown in Fig. S4-S11.
In Figure S4 and Figure S5, the results are similar to that of w i = k i . The proposed method also enhances the speed of controllability a lot compared to betweenness control in Fig. S6 and Fig. S7. The reason is that proposed method selects sparser pinning nodes than that of betweenness selection(See Fig. S8 and Fig. S9). However, the performances ofL min are almost the same for both methods(See Fig. S10 and Fig. S11), which are due to the boundary restriction ofL min (L min >= 1). Both methods arrive at theL min = 1. Though L min of both methods have similar performance, the sparsity could be distinguished by Fig.   S8 and Fig. S9.