  Quantum networks allow for novel forms of quantum nonlocality. By exploiting
the combination of entangled states and entangled measurements, strong nonlocal
correlations can be generated across the entire network. So far, all proofs of
this effect are essentially restricted to the idealized case of pure entangled
states and projective local measurements. Here we present noise-robust proofs
of network quantum nonlocality, for a class of quantum distributions on the
triangle network that are based on entangled states and entangled measurements.
The key ingredient is a result of approximate rigidity for local distributions
that satisfy the so-called ``parity token counting'' property with high
probability. Considering quantum distributions obtained with imperfect sources,
we obtain noise robustness up to $\sim 80\%$ for dephasing noise and up to
$\sim 0.67\%$ for white noise. Additionally, we can prove that all
distributions in the vicinity of some ideal quantum distributions are nonlocal,
with a bound on the total-variation distance. Our work opens interesting
perspectives towards the practical implementation of quantum network
nonlocality.
