The structural controllability of edge dynamics in complex networks

In the physical, social, biological and technological systems, the interaction of different individuals forms a complex networked structures. In the past decade, various networks have been continuously emerged and remarkable progress has been made in the study of their structure and dynamic properties[1, 2]. However, most of the studies we have done are nodal dynamic processes in the past, so here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics, In addition, the controllable subspace and its dimension of a structured linear system vary as a function of the free parameter. However, the dimension is stable in the sense that it takes, for almost any system parameters, some maximal constant which is the generic rank of the controllability matrix, and here this maximal constant is called the generic dimension of the controllable subspace. In this paper, we propose a theoretical framework to determine the controllable subspace and calculate its generic dimension for the edge dynamic system, and give the methods to analyze the structural controllability of the system,