Optimal sparse network topology under sparse control in Laplacian networks

Understanding the effect of network topology on its controllability or control performance helps to explain the emergence and evolution of different topological features in biological or social networks, and to provide principles of designing networked engineering systems. In natural and artificial systems, networks have sparse topology and are controlled by sparse feedback controllers. Therefore, the optimal network topology and controllers are specified by a trade-off between the control performance and the sparsity of the network and the controllers. In this work, by considering Laplacian networks and adopting an H₂-optimal control framework that incorporates the network Laplacian matrix as an optimization variable and promotes the network sparsity and controller sparsity, we simultaneously optimize the network topology and the feedback gain matrix through the alternating direction method of multipliers (ADMM). We observe that with increasing sparsity, modular and hierarchical topological features emerge along with distributed and hierarchical control architectures, which is consistent with observations in biological networks and allows an evolutionary interpretation of their emergence.