  We examine the relationship between source position stability and
astrophysical properties of radio-loud quasars making up the International
Celestial Reference Frame. Understanding this relationship is important for
improving quasar selection and analysis strategies, and therefore reference
frame stability. We construct light curves for 95 of the most frequently
observed ICRF2 quasars at both the 2.3 and 8.4 GHz geodetic VLBI observing
bands. Because the appearance of new quasar components corresponds to an
increase in quasar flux density, these light curves alert us to potential
changes in source structure before they appear in VLBI images. We test how
source position stability depends on three astrophysical parameters: (1) Flux
density variability at X-band; (2) Time lag between flares in S and X-bands;
(3) Spectral index rms, defined as the variability in the ratio between S and
X-band flux densities. We find that small time lags between S and X-band light
curves, and low spectral index variability, are good indicators of position
stability. On the other hand, there is no strong dependence of source position
stability on flux density variability in a single frequency band. These
findings can be understood by interpreting the time lag between S and X-band
light curves as a measure of the size of the source structure. Monitoring of
source flux density at multiple frequencies therefore appears to provide a
useful probe of quasar structure on scales important to geodesy. We show how
multi-frequency flux density monitoring may allow the dependence on frequency
of the relative core positions along the jet to be elucidated. Knowledge of the
position-frequency relation has important implications for current and future
geodetic VLBI programs, as well as the alignment between the radio and optical
celestial reference frames. (Abridged)
