  We discuss quantum network Bell nonlocality in a setting where the network
structure is not fully known. More concretely, an honest user may trust their
local network topology, but not the structure of the rest of the network,
involving distant (and potentially dishonest) parties. We demonstrate that
quantum network nonlocality can still be demonstrated in such a setting, hence
exhibiting topological robustness. Specifically, we present quantum
distributions obtained from a simple network that cannot be reproduced by
classical models, even when the latter are based on more powerful networks. In
particular, we show that in a large ring network, the knowledge of only a small
part of the network structure (involving only 2 or 3 neighbouring parties) is
enough to guarantee nonlocality over the entire network. This shows that
quantum network nonlocality can be extremely robust to changes in the network
topology. Moreover, we demonstrate that applications of quantum nonlocality,
such as the black-box certification of randomness and entanglement, are also
possible in such a setting.
