  This paper provides an $H_2$ optimal scheme for reducing diffusively coupled
second-order systems evolving over undirected networks. The aim is to find a
reduced-order model that not only approximates the input-output mapping of the
original system but also preserves crucial structures, such as the second-order
form, asymptotically stability, and diffusive couplings. To this end, an $H_2$
optimal approach based on a convex relaxation is implemented to reduce the
dimension, yielding a lower order asymptotically stable approximation of the
original second-order network system. Then, a novel graph reconstruction
approach is employed to convert the obtained model to a reduced system that is
interpretable as an undirected diffusively coupled network. Finally, the
effectiveness of the proposed method is illustrated via a large-scale networked
mass-spring-damper system.
