The Controllability and Strong Structural Controllability of Laplacian Dynamics

In this paper, we analyze the controllability of two different protocols. It is found that the absolute value of diagonal entries of Laplacian matrix induces linear outputs of agents. The multiplicity of eigenvalue 0 of general Laplacian matrix is related to the number of zero circles, identical nodes and opposite pairs of nodes, while the eigenvalue 0 is always simple for absolute L. For unweighted graphs, if the topology of structural balance is fixed, the controllable subspace will never change. For weighted graphs, we reveal the effects of topologies to strong structural controllability. The substructures of paths in a topology determine the strong structural controllability of systems. The connection between father nodes and children nodes can affect the strong structural controllability, which determines the linearity relationship of the control information from father nodes to children nodes. And we first give the sufficient and necessary condition for strong structural controllability of multi-agent systems.