node i thus reads k$_{i}$ = ∑ N j $_{=1}$a$_{ij}$ . The target orbit to which the whole network is assumed to be controlled is denoted by x$_{T}$ . The total number of pinned nodes is n , and the set of pinned nodes is represented by V = { m } . Specifically, if node i is pinned in the network, we have δ$_{im}$ = 1 in Eq. [1], otherwise δ$_{im}$ = 0 . Without loss of generality, we set the target (con- troller) to be having the same dynamics as the network node, i.e. ˙ x$_{T}$ = F ( x$_{T}$ ) . Regarding the controller as an additional node of the net- work, we then are able to treat the pinning problem under the framework of network synchronization [19]. In the enlarged network, the controller has degree n , and is unidirectionally coupled to the pinned nodes with strength η . So the enlarged system can be regarded as a special example of weighted and directed network. The enlarged network can be written as in which the controller is represented by the node of index N +1 . The pinnings are represented in the last column of W , where w$_{i}$$_{(}$$_{N}$$_{+1)}$ = η if i ∈ V , and w$_{i}$$_{(}$$_{N}$$_{+1)}$ = 0 otherwise. Based on the method of master-stability function [20, 21], we know that whether the network can be “synchronized” to the controller is determined jointly by the node dynamics, F , and the eigenvalue spectrum of the coupling matrix G = W − DI . Here, D = ( d$_{1}$, d$_{2}$, . . . , d$_{N}$$_{+1}$ ) T ( d$_{i}$ = ∑ N +1 j =1 w$_{ij}$ is the cou- pling intensity of node i ) and I is the identity matrix of dimen- sion N + 1 . Previous studies have shown that, for the general types of node dynamics, the network is more synchronizable when the spread of the eigenvalue spectrum is narrower [20– 22]. In particular, let 0 = λ$_{1}$ > λ$_{2}$ . . . λ$_{N}$$_{+1}$ be the eigen- value spectrum of the coupling matrix G . Then the smaller the eigenratio R = λ$_{N}$$_{+1}$/λ$_{2}$ , the more likely synchronous dynamics is to occur on the network. Since here we treat the pinning problem as a special case of synchronization, smaller eigenratio thus also means higher controllability. Having specified the pinning model, we now compare the performance of the two pinning strategies, namely LDP and SDP, based on numerical simulations. Without loss of gen- erality, we generate a complex network by the standard BA algorithm [17], which consists of N = 1000 nodes and has average degree 〈 k 〉 = 8 . The node degree of the generated net- work follows roughly a power-law distribution p ( k ) ∼ k − $^{γ}$, with γ = 3 . The LDP and SDP strategies are implemented as follows. We first rearrange the nodes by a descending order of their degree, i.e. k$_{1}$ > k$_{2}$ . . . > k$_{N}$ . Then, in LDP the n largest-degree nodes are pinned all together, i.e., nodes of index 1 ≤ m ≤ n are pinned; while in SDP, the n smallest-degree nodes are pinned i.e., nodes of index N − n + 2 ≤ m ≤ N + 1 are pinned. The eigenratio of the two pinned networks are denoted by R L (LDP) and R S (SDP). The variations of R L and R S as a function of the frac- tion of pinned nodes, f = n/N , are plotted in Fig. 1. In- terestingly, we find that, when the number of pinned nodes is large enough, R S is smaller than R $^{L}$, i.e., the SDP strat- egy has a higher performance than the LDP strategy. More specifically, there exists a critical fraction, f$_{c}$ ≈ 0 . 4 , in the number of pinned nodes. When f is small, as tacitly believed, the LDP strategy has the higher performance than the SDP strategy. However, as f increases, the advantage of LDP over SDP is gradually narrowed and, at the critical value f$_{c}$ , the two strategies give the same performance. After that, the SDP strategy will outperform the LDP strategy. Moreover, as f in- creases from f$_{c}$ , the advantage of SDP over LDP is gradually enlarged. The difference between the two strategies reaches its maximum at f = 1 − 1 /N , where only the small-degree (largest-degree) node is un-pinned in LDP (SDP). Of course, at the point f = 1 the network is globally controlled, and the two strategies give the same performance again. (The switch of control performance is also observed in the variation of λ$_{2}$ , which characterizes the system controllability for the situation of non-bounded MSF function [22].) Is the above phenomenon general for complex networks? To answer this question, we have compared the performance of the two strategies on a variety of networks, including changing the degree exponent, the average degree, the net- work size and the pinning strength of the network model used in Fig. 1. In exploring the influence of the degree exponent γ , we have adopted the network model of Ref. [23], where γ can be increased from 3 to a large value by tuning a parameter. In doing this, the network topology is changing from heteroge- nous to homogeneous gradually. The variation of the critical fraction f$_{c}$ as a function of γ is plotted in Fig. 2(a), where it is found that the value of f$_{c}$ is monotonically decreased. Mean- while, with the increase of γ , the performance difference be- tween the two strategies, Δ R = R L − R $^{S}$, is narrowed, e.g, the variation of Δ R at n = N − 1 [Fig. 2(b)]. The narrowed difference in Fig. 2(b) seems to implies that, comparing to GLYPH<s48> GLYPH<s46> GLYPH<s48> GLYPH<s48> GLYPH<s46> GLYPH<s50> GLYPH<s48> GLYPH<s46> GLYPH<s52> GLYPH<s48> GLYPH<s46> GLYPH<s54> GLYPH<s48> GLYPH<s46> GLYPH<s56> GLYPH<s49> GLYPH<s46> GLYPH<s48> GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s49> GLYPH<s48> GLYPH<s50> GLYPH<s49> GLYPH<s48> GLYPH<s51> GLYPH<s49> GLYPH<s48> GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s56> GLYPH<s49>GLYPH<s48> GLYPH<s45> GLYPH<s51> GLYPH<s49>GLYPH<s48> GLYPH<s45> GLYPH<s49> GLYPH<s49>GLYPH<s48> GLYPH<s49> GLYPH<s102> GLYPH<s99> GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s102> GLYPH<s99> GLYPH<s61> GLYPH<s48> GLYPH<s46> GLYPH<s52> GLYPH<s32> GLYPH<s32> GLYPH<s32>GLYPH<s82> GLYPH<s76> GLYPH<s44>GLYPH<s32>GLYPH<s76> GLYPH<s68> GLYPH<s80> GLYPH<s32>GLYPH<s82> GLYPH<s115> GLYPH<s44>GLYPH<s32>GLYPH<s83> GLYPH<s68> GLYPH<s80> GLYPH<s124> GLYPH<s124> GLYPH<s82> GLYPH<s32> GLYPH<s102> 2 FIG. 1: (Color online) With pinning strength η $^{= 25}$, the comparison of the performance between the LDP and SDP strategies in control- ling a scale-free network. Inset: the variations of the largest non-zero eigenvalue as a function of the fraction of pinned nodes. Each data is averaged over $^{20}$network realizations. W =     a$_{11}$ a$_{12}$ . . . a$_{1}$$_{N}$ w$_{1(}$$_{N}$$_{+1)}$ . . . . . . . a$_{i}$$_{1}$ a$_{i}$$_{2}$ . . . a$_{iN}$ w$_{i}$$_{(}$$_{N}$$_{+1)}$ . . . . . . . a$_{N}$$_{1}$ a$_{N}$$_{2}$ . . . a$_{NN}$ w$_{N}$$_{(}$$_{N}$$_{+1)}$ 0 0 . . . 0 0    $^{,}$