arXiv:1001.1393v1[physics.soc-ph]9 Jan 2010 On the pinning strategy of complex networks Lei Yang, 1 Xingang Wang, 1,2, ∗ Yao Li, 1 and Zhengmao Sheng 1 $^{1}$Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, China 310027 $^{2}$Department of Physics, Zhejiang University, Hangzhou, China 310027 (Dated: October 23, 2018) In pinning control of complex networks, a tacit believing is that the system dynamics will be better controlled by pinning the large-degree nodes than the small-degree ones. Here, by changing the number of pinned nodes, we find that, when a significant fraction of the network nodes a re pinned, pinning the small-degree nodes could generally have a higher performance than pinning the large-degree nodes. We demonstrate this interesting phenomenon on a variety of complex networks, and analyze the underlying mechanisms by the model of star networks. By changing the network properties, we also find th at, comparing to densely connected homogeneous networks, the advantage of the small-degree pinning strategy is more distinct in sparsely connected heterogenous networks. PACS numbers: 05.45.Xt,89.75.Hc degree pinning (LDP) and random pinning. In LDP, the nodes are pinned by the decreasing order of their degree, while in random pinning the nodes are randomly selected. The authors found that, with the same number of pinned nodes, the LDP strategy has a clear advantage over the random pinning strat- egy. After the work of Ref. [11], pinning control of complex networks has received continuous interest, and the advantage of LDP has been well addressed [13–16]. Comparing to the regular networks, optimization of pinning scheme according to topology properties is more significantin complex networks [11–16]. For practical networks, a genera l feature is that the number of links (degree) associated to the network node has a large variation, e.g. the type of scale- free networks (SFN) [17]. In networks like SFN, the ma- jority of the network nodes have small degree, but a few of the network nodes could have distinctively large degree. No- ticed the important roles of the large-degree nodes played in other types of network dynamics, e.g. network synchroniza- tion [18], it is natural to expect that the spatiotemporal chaos of complex networks will be better controlled by pinning the large-degree nodes than pinning the small-degree ones or pin- ning randomly. To check this, in Ref. [11] the authors com- pared the performance of two pinning schemes on SFN: large- where i, j = 1 , 2 , . . . , N are the node indices, F is the func- tion of the node dynamics, H is the coupling function. ε and η represent the uniform coupling strength and pinning strength, respectively. The network structure is captured by the adja- cency matrix A = { a$_{ij}$ } , in which a$_{ij}$ = 1 if nodes i and j are directly connected, and a$_{ij}$ = 0 otherwise. The degree of Our model of network pinning is the following In a practical situation, due to the limited pinning strength, to control a large-size complex network, a number of the net- work nodes should be pinned simultaneously [11, 12]. For instance, with a reasonable pinning strength (even with the order of the dynamics variables), it is shown that the number of pinned nodes necessary for controlling a SFN to a homo- geneous state could take up to 30 or 50 percent of the total network nodes [11]. With such a significant fraction of nodes be pinned, it is questionable whether the LDP strategy will be still the best option, as now many of the pinned nodes do not have large degree. In this paper, we will re-evaluate the performance of LDP by comparing it with its opposite case, namely the small-degree pinning (SDP) strategy. In contrast to LDP, in SDP the network nodes are pinned by the increas- ing order of their degree. Very interestingly, we found that when the fraction of pinned nodes is large enough (depending on the network details, this fraction could be ranging from 30 to 50 percent), the SDP strategy, which has been tacitly be- lieved as having the worst performance, will outperform the LDP strategy. We demonstrate this phenomenon in various complex networks, and investigate its underlying mechanism by a simplified network model. Pinning control refers to the controlling of system dynam- ics to a target state by perturbing only partial of the system variables [1, 2]. Due to its convenient operation and high per- formance, this technique has been widely used in controlling spatiotemporal dynamics in complex systems, including op- tical systems [3], plasma [4], neural systems [5], and chem- ical and flow turbulence in various systems [6–9]. In pin- ning control, one of the focusing issues is about how to op- timize the pinning scheme according to the system topology [6, 7]. In controlling the dynamics of coupled map lattices, it is found that, with the same pinning effort, the controllabil- ity can be significantly improved if the pinnings are arrange d in an asymmetric fashion [6]. The underlying mechanism for this improvement is that, by asymmetric pinning scheme, the topology of the lattice network is actually modified from sym - metric to asymmetric, which causes a more uniform suppres- sion of the unstable modes [6]. Still in lattice network, based on the special property of coupling gradient, it is even found that the spatiotemporal chaos can be successfully controlled by pinning a single node [10]. In this application, to iden- tify the location of the pinned node, we need to analyze the topology of the weighted and directed lattice [10]. ˙ x$_{i}$ = F ( x$_{i}$ )+ ε N ∑ j =1 a$_{ij}$ [ H ( x$_{j}$ ) − H ( x$_{i}$ )]+ η ∑ m ∈ V δ$_{im}$ [ H ( x$_{T}$ ) − H ( x$_{i}$ )] , (1) $^{∗}$Corresponding author. Email address: wangxg@zju.edu.cn