  Quantum nonlocality can be observed in networks even in the case where every
party can only perform a single measurement, i.e. does not receive any input.
So far, this effect has been demonstrated under the assumption that all sources
in the network are fully independent from each other. Here we investigate to
what extent this independence assumption can be relaxed. After formalizing the
question, we show that, in the triangle network without inputs, quantum
nonlocality can be observed, even when assuming only an arbitrarily small level
of independence between the sources. This means that quantum predictions cannot
be reproduced by a local model unless the three sources can be perfectly
correlated.
