This work explores the stability properties of a class of nonlinear system, which is obtained by applying a special transformation to the vector field of a linear system. It is proved that such a nonlinear system preserves the stability of the original linear system. This property can be used for checking the stability of certain nonlinear systems, and designing novel network control protocols based on homogeneous system theory. For the network control application, finite-time convergence can be achieved in a decentralized manner by properly choosing the smooth feedback control law. The results are demonstrated by numerical examples.