Undirected Weighted Network Topologies With Best Possible Pinning Controllability

Controllability of complex networks through pinning control toward a desired trajectory is crucial for ensuring synchronization stability. Concerned with the local stability of the network's synchronization, here, the controllability problem is addressed different from the literature by incorporating weighted Laplacian and nonuniform feedback gain. This approach has led to a spectral radius minimization problem, where using its semidefinite programming (SDP) formulation leads to a unique optimal point, with uniform feedback gain, on the Pareto frontier. Using this approach, this article has systematically characterized the networks that can achieve the best possible optimal controllability measure. For such networks, it is shown that (i) network's optimal controllability measure is expressed in terms of the number of pinned and free nodes, (ii) the optimal feedback gains of pinned nodes are uniform, (iii) sum of optimal weight linked to a node is determined in terms of its type and the optimal spectral radius of network, (iv) nodes of same type have the same value for the optimal dual SDP variables.