  In this paper, we consider mixtures of two Mallows models for top-$k$
rankings, both with the same location parameter but with different scale
parameters, i.e., a mixture of concentric Mallows models. This situation arises
when we have a heterogeneous population of voters formed by two homogeneous
populations, one of which is a subpopulation of expert voters while the other
includes the non-expert voters. We propose efficient sampling algorithms for
Mallows top-$k$ rankings. We show the identifiability of both components, and
the learnability of their respective parameters in this setting by, first,
bounding the sample complexity for the Borda algorithm with top-$k$ rankings
and second, proposing polynomial time algorithm for the separation of the
rankings in each component. Finally, since the rank aggregation will suffer
from a large amount of noise introduced by the non-expert voters, we adapt the
Borda algorithm to be able to recover the ground truth consensus ranking which
is especially consistent with the expert rankings.
