Controllable subspace of edge dynamics in complex networks

For the edge dynamics in some real networks, it may be neither feasible nor necessary to be fully controlled. An accompanying issue is that, when the external signal is applied to a few nodes or even a single node, how many edges can be controlled? In this paper, for the edge dynamics system, we propose a theoretical framework to determine the controllable subspace and calculate its generic dimension based on the integer linear programming. This framework allows us not only to analyze the control centrality, i.e., the ability of a node to control, but also to uncover the controllable centrality, i.e., the propensity of an edge to be controllable. The simulation results and analytic calculation show that dense and homogeneous networks tend to have larger control centrality of nodes and controllable centrality of edges, but the negatively correlated in- and out-degrees of nodes or edges can reduce the two centrality. The positive correlation between the control centrality of node and its out-degree leads to that the distribution of control centrality, instead of that of controllable centrality, is encoded by the out-degree distribution of networks. Meanwhile, the positive correlation indicates that the nodes with high out-degree tend to play more important roles in control.