In this paper we introduce two types of simulations (forward and backward) and five types of bisimulations (forward, backward, forward-backward, backward-forward, and regular) for weighted networks with weights taken in a quntale, we describe their basic properties and provide procedures for testing the existence of a simulation or bisimulation of a given type and computing the greatest one, in cases when such simulations or bisimulations exist. We also describe basic properties of homogeneous simulations and bisimulations, which relate actors within the same weighted network, and establish relationships between heterogeneous bisimulations between two different networks and homogeneous bisimulations on these networks. In addition, we characterize the greatest forward, backward and regular bisimulations between weighted networks by means of multi-valued multimodal logics.