We study the rise dynamics of a large particle in a granular bed under vertical vibration using molecular dynamics simulations. Systematic variation of the particle properties and external wall friction in the simulations shows that the large particle rising is very sensitive to external wall friction. The dynamical response of the granular bed with wall friction is shown to include an expansion stage and a compression stage within one cycle. With wall friction, large-scale force networks bearing larger-than-average forces are found in the compression stage. However, without wall friction, large strong force networks do not exist. The distribution of normal contact forces in the force networks is found to have an exponential tail similar to those in packing experiments. Numerical estimation of the two-point spatial correlation of normal contact force reveals predominantly short-range force correlation persisting over only 2–3 particle diameters. The structural properties of the force network are analyzed using a graph-theoretic approach, which is a modification of a minimum spanning tree (MST) constructed on the particles in physical space. The modified MST algorithm, which identifies local structures such as nearest neighbors, asymptotically recovers all particle contacts, the force distribution, and spatial force correlation of the force network. This indicates that although wall friction strongly affects the rising dynamics of the Brazil nut over the device scale (tens of particle diameters) through the force network, it does so through local short-range interactions. Thus this study affirms the basis for local constitutive models in continuum descriptions of segregation.