复杂网络严格控制相关问题研究

Controlling collective dynamics of complex networks is our ultimate goal of studying them and has a wide range of applications in many complex systems and dynamical processes in nature and society, such as cascading failures, epidemic and rumor spreading, synchronization and etc. However, the field of controlling complex networked systems is still in its infancy, due to the lack of general theoretical tools to address the controllability of arbitrary networks, despite the recent development of structural controllability theory for complex networks. We plan to improve the exact controllability theory developed by us in the aspect of identifying driver nodes and independent input signals exactly. The modified theory allows us to explore the effects of a variety of structural properties on the controllability and identify the properties that play the key role in the controllability of complex networks. Subsequently, we attempt to offer a general theory to measure the area of control by imposing signals to a single node, which is not only important for quantifying the importance of nodes pertaining to control, but also has practical implications. Finally, we aim to achieve the classification of nodes in three categories: driver nodes, driven nodes and ordinary nodes. Moreover, we plan to explore the relationship between the node categories and the characteristics of nodes and, the distribution of different types of nodes. Our work will offer new insight into the fundamental problem of controlling complex networked systems, especially for improving the exact controllability framework and providing theoretical support for devising optimal control.

复杂网络的控制研究是研究复杂网络的终极目标。对复杂网络控制的研究有利于完善对复杂系统及其动力学行为的研究，比如级联效应，病毒传播，谣言传播，同步等。但是，复杂网络控制的研究还处在起步阶段，已有的结构控制研究仅解决了一部分的问题，还缺乏对网络严格控制的一般性理论研究。本项目将基于我们的复杂网络严格可控性研究结果，首先研究网络严格可控能力与网络结构的关系，探讨影响网络严格可控能力的决定性因素；然后为了探讨严格控制中的重要性节点，研究一般网络节点的控制范围问题，为选择具有最大控制范围的驱动节点打下坚实的基础；最后为了分析输入矩阵特点，我们将提出甄别网络的独立控制节点和非独立控制节点以及剩下的不控节点的方法，分析三类节点的属性特征和在不同网络中的分布。本项目的研究结果将扩充复杂网络控制研究思路和方法，对复杂网络的严格控制和优化控制提供有力的理论支撑。

如何控制复杂网络的动力学行为是复杂网络研究的一个重要课题。本项目首先研究了复杂网络可控能力与网络结构之间的关系，主要研究了可控能力与网络平均度、集聚系数、Pearson系数之间的关系；然后我们对复杂网络节点的控制范围展开研究，研究了一般的理论情况和主链影响等情况；最后我们对复杂网络的控制节点进行分析，通过叶子节点删除法获取对网络节点的分类，然后分析节点的特性。本项目的研究成果将为我们设置满足可控性的控制输入矩阵提供理论基础和方法指导。