  Using multiwavelength observations of radio afterglows, we confirm the
hypothesis that the flux density of gamma-ray bursts (GRBs) at a fixed
observing frequency is invariable when the distance of the GRBs increases,
which means the detection rate will be approximately independent of redshift.
We study this behavior theoretically and find that it can be well explained by
the standard forward shock model involving a thin shell expanding in either a
homogeneous interstellar medium (ISM) or a wind environment. We also found that
short GRBs and supernova-associated GRBs, which are at relatively smaller
distances, marginally match the flux-redshift relationship and they could be
outliers. We rule out the assumption that the medium density evolves with
redshift as $n\propto(1+z)^4$ from the current measurements of $n$ and $z$ for
short and long GRBs. In addition, the possible dependence of host flux on the
redshift is also investigated. We find that a similar redshift independence of
the flux exists for host galaxies as well, which implies that the detection
rate of radio hosts might also be independent of the redshift. It is also
hinted that most radio hosts have the spectral indices ranging from
$\beta_h\simeq-1$ to 2.5 in statistics. Finally, we predict the detection rates
of radio afterglows by the next-generation radio telescopes such as the
Five-hundred meter Aperture Spherical Telescope (FAST) and the Square Kilometer
Array (SKA).
