Research On The Controllability Of Complex Networks Under Adjacency Matrix

Complex network is an emerging interdisciplinary, which has caused great attention of scholars both at home and abroad. The controllability problem is a focus in complex network research, and it is also the foundation and core of complex network system. In this paper, the main research contents and contributions are as follows:First, the thesis studies the exact controllability of complex network under the adjacency matrix, exact controllability theory of complex network is proved. And we determine the minimum number of driver nodes required to achieve full control of networks.Second, the exact controllability framework is applied to the path graph, circle graph, star graph and connected graph. the research finds out the location of the driver nodes in different topology. At last the exact controllability theory is verified by examples.Third, the adjacency matrix is divided into a leader-follower framework. Under the leader-follower framework, the thesis summarizes algebraic conditions for controllability of complex networks. As the system matrix is regard as adjacency matrix and adjacency matrix is divided into a leader-follower framework, we can get the controllability algebraic conditions. Secondly, the system matrix is divided into a leader-follower structure and regard the follower matrix as the adjacency matrix, then get the algebraic conditions. The thesis also prove the algebraic conditions. The path is analyzed in theory. At last the conclusions of this thesis are proved through concrete examples.Fourth, the system of adjacency matrix and Laplacian are compared.