Graphene and fullerenes are impressive nanostructures having great potential be used in many areas due to their unique physical and chemical properties. By hybridizing these nanostructural units, it is possible to obtain superior composite nanostructures with novel properties. In this study, a new hybrid nanostructure that will be named as graphene nanoribbon-fullerene (GNR-F) network is introduced by providing its tensile and compressive mechanical characteristics through molecular dynamics simulations. The proposed nanostructure is mainly an interconnected network of GNR units on which fullerenes are covalently grafted. Three different atomistic models are generated using different types of fullerenes (i.e. C60, C180 and C320) to investigate the effects of the fullerene types on the mechanical properties. After examining the thermodynamic stability of the specimens -by monitoring the free energy profiles and morphologic evolution within a sufficiently long period of time, deformation mechanisms of the specimens subjected to tensile and compression tests are investigated. Simulation results indicate that the proposed GNR-F network nanostructure has a ductile nature in tensile loading while its compressive response is foam-like with remarkable energy absorbing capacity. In addition to the expected effect of the fullerenes on the compressive behavior, a significant effect is also observed for the tensile response.