  The accumulation of certain types of dark matter particles in neutron star
cores due to accretion over long timescales can lead to the formation of a mini
black hole. In this scenario, the neutron star is destabilized and implodes to
form a black hole without significantly increasing its mass. When this process
occurs in neutron stars in coalescing binaries, one or both stars might be
converted to a black hole before they merge. Thus, in the mass range of $\sim
\mbox{1--2}\, M_\odot,$ the Universe might contain three distinct populations
of compact binaries: one containing only neutron stars, the second population
of only black holes, and a third, mixed population consisting of a neutron star
and a black hole. However, it is unlikely to have a mixed population as the
various timescales allow for both neutron stars to remain or collapse within a
short timescale. In this paper, we explore the capability of future
gravitational-wave detector networks, including upgrades of Advanced LIGO and
Virgo, and new facilities such as the Cosmic Explorer and Einstein Telescope
(XG network), to discriminate between different populations by measuring the
effective tidal deformability of the binary, which is zero for binary black
holes but nonzero for binary neutron stars. Furthermore, we show that observing
the relative abundances of the different populations can be used to infer the
timescale for neutron stars to implode into black holes, and in turn, provide
constraints on the particle nature of dark matter. The XG network will infer
the implosion timescale to within an accuracy of 0.01 Gyr at 90% credible
interval and determine the dark matter mass and interaction cross section to
within a factor of 2 GeV and 10 cm$^{-2}$, respectively.
