Temporal variations in the soft X-ray (SXR) emission and the radio emission above the solar magnetic network of the quiet corona are investigated using Yohkoh SXR images with deep exposure and VLA observations in the centimeter radio range. The SXR data show several brightenings, with an extrapolated occurrence probability of one brightening per 3 seconds on the total solar surface. During the roughly 10 minutes of enhanced flux, total radiative losses of the observed plasma are around 1025 ergs per event. These events are more than an order of magnitude smaller than previously reported X-ray bright points or active region transient brightenings. For all of the four SXR events with simultaneous radio observations, a corresponding radio source correlating in space and time can be found. There are several similarities between solar flares and the SXR/radio events presented in this paper. (1) Variations in temperature and emission measure during the SXR enhancements are consistent with evaporation of cooler material from the transition region and the chromosphere. (2) The ratio of the total energies radiated in SXR and radio frequencies is similar to that observed in flares. (3) At least one radio event shows a degree of polarization as high as 35%. (4) In three out of four substructures the centimeter radio emission peaks several tens of seconds earlier than in the SXR emission. (5) The associated radio emission tends to be more structured and to have faster rise times. These events thus appear to be flare-like and are called network flares.