The nonradiative relaxation behavior of water molecules across the Ge-As-Se glass series has been studied by persistent IR spectral hole burning and picosecond IR saturation spectroscopy. The bending and stretching modes of the guest molecule are found to display relaxation times in the range between 100 ps and 400 ps. When the vibrational lifetimes of the H2O modes are compared for different chemical compositions of these ternary compounds, it is found that these times are not correlated with the chemical environment but instead with the glass network topology as given by the average number of covalent bonds per atom.