Graph states are a large class of multipartite entangled quantum states that form the basis of schemes for quantum computation, communication, error correction, metrology, and more. In this work, we consider verification of graph states generated by an untrusted source and shared between a network of possibly dishonest parties. This has implications in certifying the application of graph states for various distributed tasks. We first provide a general protocol and analysis for the verification of any graph state in such a network, and then adapt it to reduce the resources required for specific examples such as cluster states, complete and cycle graph states. In each case, we demonstrate how parties in the network can efficiently test and assess the closeness of their shared state to the desired graph state, even in the presence of any number of dishonest parties.