We investigate the capabilities of the ESA Gaia mission for detecting and characterizing short time-scale variability, from tens of seconds to a dozen hours. We assess the efficiency of the variogram analysis, for both detecting short time-scale variability and estimating the underlying characteristic time-scales from Gaia photometry, through extensive light-curve simulations for various periodic and transient short time-scale variable types. We show that, with this approach, we can detect fast periodic variability, with amplitudes down to a few millimagnitudes, as well as some M dwarf flares and supernovae explosions, with limited contamination from longer time-scale variables or constant sources. Time-scale estimates from the variogram give valuable information on the rapidity of the underlying variation, which could complement time-scale estimates from other methods, like Fourier-based periodograms, and be reinvested in preparation of ground-based photometric follow-up of short time-scale candidates evidenced by Gaia. The next step will be to find new short time-scale variable candidates from real Gaia data, and to further characterize them using all the Gaia information, including colour and spectrum.