Scientific RepoRts | 6:23952 |

DOI: 10.1038/srep23952（2016）

www.nature.com/scientificreports

Effect of correlations

on controllability transition

in network control

Sen Nie , Xu-Wen Wang , Bing-Hong Wang & Luo-Luo Jiang

The network control problem has recently attracted an increasing amount of attention, owing to concerns including the avoidance of cascading failures of power-grids and the management of ecological networks. It has been proven that numerical control can be achieved if the number of control inputs exceeds a certain transition point. In the present study, we investigate the effect of degree correlation on the numerical controllability in networks whose topological structures are reconstructed from both real and modeling systems, and we find that the transition point of the number of control inputs depends strongly on the degree correlation in both undirected and directed networks with moderately sparse links. More interestingly, the effect of the degree correlation on the transition point can not be observed in dense networks for numerical controllability, which contrasts with the corresponding result for structural controllability. In particular, for directed random networks and scale-free networks,  the influence of the degree correlation is determined by the types of correlations. Our approach provides an understanding of control problems in complex sparse networks.

网络结构之度关联性(同配性）

对于网络可控性相变之影响

聂森，王旭文，汪秉宏，姜罗罗

（中国科学技术大学， 近代物理系）

复杂网络的可控性问题是近年来的一个研究热点，该问题的研究对电力网络、基因网络、脑神经网络等复杂系统的调控具有重要的指导意义。考虑到系统在实际运行中的能量消耗、运行路径等问题，对其可控性研究需要用到经典控制理论中的Gramian（格拉姆）矩阵奇异性判据。由于Gramian矩阵可控判据涉及数值计算的精度，因此称基于Gramian矩阵判据的系统可控性为数值可控性。现有的研究表明，随着驱动节点数目的增大，系统的数值可控成功率会出现相变，我们称使得系统数值可控首次成功的驱动节点数目为可控相变点。前人的工作证明，复杂网络的可控性受到网络的平均度、幂函数律的指数等结构参数的影响。特别地，网络的度度相关性对最少驱动节点数目的影响十分显著。本文中，我们研究了网络度度相关性（同配性）对模型网络和真实网络可控相变点的影响，结果显示在适度稀疏的有向和无向网络中，可控相变点数目强烈依赖于网络度相关性的变化；但是，在稠密网络中，可控性相变点大小与度相关性无关，这一现象与结构可控性中的已有结果不同。另外，在有向的随机网络和无标度网络中，度相关性对可控相变点的影响取决于入-入、入-出、出-入、出-出这些度相关性的不同类型。同时，仿真结果表明，在有向或无向的随机和无标度网络中，网络的可控相变点大小与网络的规模之间存在一定关系。可控相变点的计算，由于涉及矩阵求逆，尚未获得解析形式结果。本文结果对于复杂网络之可控性相变与网络结构之关系加深了认识。

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