content_image/399415.jpg content_image/399432.jpg content_image/399431.jpg content_image/399428.jpg content_image/399427.jpg content_image/399429.jpg content_image/399430.jpg
<img src='content_image/399415.jpg'>

homogeneous networks, heterogeneous networks are more fa- vorable for SDP. To check the influence of the average degree, we have increased the value of 〈 k 〉 from 6 to 80. The vari- ations of the critical fraction and the performance difference between the two strategies are plotted in Fig. 2(c) and (d), re- spectively. It is seen that, as 〈 k 〉 increases, both f$_{c}$ and Δ R are decreased. Besides degree exponent and average degree, we have also checked the influence of the network size, and found that, as N increases from 1000 to 5000, the values of f$_{c}$ and Δ R are decreased too (not shown). Noticing the fact that either the increase of 〈 k 〉 or the decrease of N leads to a homogeneous network structure, these additional simulations therefore confirm the finding of tuning γ , i.e., the SDP strategy for more favorable in controlling heterogeneous networks. 

The existence of f$_{c}$ can be mathematically analyzed, based on the model of star networks. When not pinned, the star net- work has three distinct eigenvalues: λ$_{1}$ = 0 , λ$_{N}$ = − N , and λ$_{i}$ = − 1 for i = 2 , . . . , N − 1 . When pinned, the eigen- value spectrum will be modified, reflected as the decreases of λ$_{1}$ , λ$_{N}$ and a few of other eigenvalues. Specifically, when n nodes are pinned, in LDP the number of modified eigenvalues is n + 1 , while in SDP this number is n + 2 . Firstly, let us consider the case of n = 1 and compare the performance of the two strategies. For LDP, only the central node is pinned, and the two modified eigenvalues, λ L 1 and λ L $_{N}$, are governed by equation 

We thus have λ L 1 ,N = [ − ( N + η ) ± √ ( N + η ) 2 − 4 η ] / 2 . For SDP, the pinned node could be any of the N 1 peripheral nodes. The three modified eigenvalues, λ S $_{1}$, λ S N − 1 and λ S $_{N}$, are governed by equation 

<img src='content_image/399432.jpg'>

<img src='content_image/399431.jpg'>

By Cardan formula, we have λ S 1 = − ( N + 1 + η + ω$_{1}$p 1 / $^{3}$+ ω$_{2}$q 1 / $^{3}$) / 3 and λ S N = − ( N + 1 + η + p 1 / 3 + q 1 / $^{3}$) / 3 , where p = ( a + √$_{a}$ 2 4 b $^{3}$) / 2 , q = ( a − √$_{a}$ 2 − 4 b $^{3}$) / 2 , a = 2( N + 1 + η ) 3 − 9 N ( N + 1 + η )(1 + η ) + 27 η , b = ( N + 1 + η ) 2 − 3 N (1 + η ) , and ω$_{1}$$_{,}$$_{2}$ = − 0 . 5 ± 0 . $_{5}$√$_{3}$$_{i}$$_{.}$ Previous studies have shown that, to control the dynamics of a complex network successfully, the pinning strength should be sufficiently large, i.e., η >> 1 [11–13]. Under this con- dition, the above eigenvalues are approximated as λ L 1 ≈ − 1 , λ L N ≈ − ( η + N ) , λ S 1 ≈ − 1 /N , λ S N ≈ − ( η + 1) . Apparently, 

i.e., LDP has the higher performance than SDP when 1 node is pinned. 

We next compare the performance of the two strategies for the situation of n = N − 1 , i.e. at the end of the comparison. For LDP, this means that except one peripheral node, all other nodes on the network are pinned. In this case, the eigenvalues λ L N − 1 and λ L N − 1 are governed by equation 

<img src='content_image/399428.jpg'>

While for SDP, only the central node is un-pinned. In this case, the eigenvalues λ S N − 1 and λ S N − 1 are governed by equa- tion 

Under the condition η >> 1 , these eigenvalues are approx- imated as λ L 1 ≈ − 1 , λ L N ≈ − ( η + 1) , λ S 1 ≈ − N , λ S N ≈ − ( η + 1) . So, when N − 1 nodes are pinned, we have 

i.e., SDP has the higher performance than LDP. Since the sit- uations n = 1 and n = N − 1 stand as the two ends of the comparison [Fig. 1], Eqs. [4] and [7] thus guarantee the exis- tence of at least one crossing in the performance curves. 

The above phenomenon of performance switching can be physically explained as follows. Still take the star-network as the model. When only one node is pinned, by pinning the largest-degree node, the pinning signal can be efficient ly propagated to the peripheral nodes, as the hub-node has the shortest average-node-distance. For this reason, LDP will have a higher performance than SDP (see Eq. [4]). How- ever, this picture is changed when N − 1 nodes are pinned. Under a strong pinning strength ( η >> 1 ), in LDP the N − 1 pinned nodes have been well controlled to the target, and they will influence the un-pinned peripheral node via a coupling o f strength ε ; on the other hand, in SDP all the peripheral nodes have been controlled, and they influence the central node to- gether via a joint coupling of strength ( N − 1) ε . Therefore in SDP the coupling between the pinned part (the peripheral nodes) and the un-pinned part is in fact amplified by N − 1 times, therefore making SDP superior to LDP. Having under- stood this mechanism, we are able to predict that, as the size of the star network increases, the advantage that SDP over LDP will be enlarged. Actually, this point can be also drawn from 

<img src='content_image/399427.jpg'>

<img src='content_image/399429.jpg'>

<img src='content_image/399430.jpg'>

