  We apply a recently developed measure of multiscale complexity to the
Gaussian model consisting of continuous spins with bilinear interactions for a
variety of interaction matrix structures. We find two universal behaviors of
the complexity profile. For systems with variables that are not frustrated, an
exponential decay of multiscale complexity in the disordered regime shows the
presence of small-scale fluctuations, and a logarithmically diverging profile
of fixed shape near the critical point describes the spectrum of collective
modes. For frustrated variables, oscillations in complexity indicate the
presence of global or local constraints. These observations show that the
multiscale complexity may be a useful tool for interpreting the underlying
structure of systems for which pair correlations can be measured.
