On maximizing algebraic connectivity of networks for various engineering applications

We discuss a simplified version of an open problem in system realization theory which has several important practical applications in complex networks. Particularly, we focus on algebraic connectivity (λ₂) of the Laplacian of the network as an objective for maximization. We show that the maximum value of forced response of a linear mechanical (spring-mass) system can be minimized when the λ₂ of the corresponding stiffness matrix is maximized. In the case of motion control problems related to vehicle localization with noisy measurements, we show that the λ₂ plays a vital role to control their positions towards a desired formation. In UAV adhoc infrastructure networks, we show that the λ₂ of the information flow graph governs the stability of the rigid formation with respect to disturbance attenuation. In the context of UAV adhoc communication networks, we also provide a physical interpretation for the maximization of λ₂ via the closely related Cheeger constant or the isoperimetric number. We further discuss a Fiedler vector based mixed-integer linear programing formulation for the problem of maximizing λ₂ and obtain optimal solutions and upper bounds based on cutting plane methods. Computational results corroborate the performance of proposed algorithms.