  If a hierarchy of interatomic interactions exists in a solid, low-frequency
modes can be found from viewing this solid as a mechanical network. In this
case, the low-frequency modes are determined by the network rigidity. We study
the low-frequency modes (rigid unit modes, or RUMs) in several important oxide
materials and discuss how the RUMs affect their properties. In SiO$_2$ glass,
the ability to support RUMs governs its relaxation in the wide range of
pressures and temperatures, giving rise to the non-trivial pressure window. It
also affects other properties, including crystallization, slow relaxation and
compressibility. At ambient pressure and low temperature, RUM flexibility is
related to the large-scale localized atomic motions. Whether these motions are
interpreted as independent two-level systems or collective density excitations,
the RUM flexibility determines whether and to what extent low-energy
excitations can exist in a given glass structure. Finally, we discuss the RUM
in, perhaps unexpectedly, cuprate superconductors, and its relevance for
superconductivity, including the d-wave symmetry of the order parameter and
other properties.
