  Genuine multipartite nonlocality and nonlocality arising in networks composed
of several independent sources have been separately investigated. While some
genuinely entangled states cannot be verified by violating a single Bell-type
inequality, a quantum network consisting of different sources allows for the
certification of the non-classicality of all sources. In this paper, we propose
the first method to verify both types of nonlocality simultaneously in a single
experiment. We consider a quantum network comprising a bipartite source and a
tripartite source. We demonstrate that there are quantum correlations cannot be
simulated if the tripartite source distributes biseparable systems while the
bipartite source distributes even stronger-than-quantum systems. These
correlations can be used to verify both the genuine multipartite nonlocality of
generalized Greenberger-Horne-Zeilinger states and the full network nonlocality
that is stronger than all the existing results. Experimentally, we observe both
types of nonlocality in a high fidelity photonic quantum network by violating a
single network Bell inequality.
