  Adaptive transport networks are known to contain loops when subject to
hydrodynamic fluctuations. However, fluctuations are no guarantee that a loop
will form, as shown by loop-free networks driven by oscillating flows. We
provide a complete stability analysis of the dynamical behaviour of any loop
formed by fluctuating flows. We find a threshold for loop stability that
involves an interplay of geometric constraints and hydrodynamic forcing mapped
to constant and fluctuating components. Loops require fluctuation in the
relative size of the flux between nodes, not just a temporal variation in the
flux at a given node. Hence, there is both a minimum and a maximum amount of
fluctuation relative to the constant-flux component where loops are supported.
