Laplacian Controllability of Oriented Threshold Graphs

In this work, the controllability of Laplacian networks defined over oriented threshold graphs (OTGs) is studied. Since these networks are directed, controllability conditions are derived for a system matrix that is the minus of the in-degree Laplacian associated with an OTG. In this direction, we also provide the spectrum and a modal matrix associated with an in-degree Laplacian matrix of an OTG and demonstrate that these matrices are diagonalizable. Through these results, we propose necessary and sufficient conditions ensuring the controllability of these networks. We also prove that with a binary input matrix, the minimum number of control signals, rendering the network controllable, equals the maximum geometric multiplicity of in-degree Laplacian eigenvalues.