  We study the evolution of structural disorder under cooling in supercooled
liquids, focusing on covalent networks. We introduce a model for the energy of
networks that incorporates weak non-covalent interactions. We show that at
low-temperature, these interactions considerably affect the network topology
near the rigidity transition that occurs as the coordination increases. As a
result, this transition becomes mean-field and does not present a line of
critical points previously argued for, the "rigidity window". Vibrational modes
are then not fractons, but instead are similar to the anomalous modes observed
in packings of particles near jamming. These results suggest an alternative
interpretation for the intermediate phase observed in chalcogenides.
