  Inspired by works on the Anderson model on sparse graphs, we devise a method
to analyze the localization properties of sparse systems that may be solved
using cavity theory. We apply this method to study the properties of the
eigenvectors of the master operator of the sparse Barrat-M\'ezard trap model,
with an emphasis on the extended phase. As probes for localization, we consider
the inverse participation ratio and the correlation volume, both dependent on
the distribution of the diagonal elements of the resolvent. Our results reveal
a rich and non-trivial behavior of the estimators across the spectrum of
relaxation rates and an interplay between entropic and activation mechanisms of
relaxation that give rise to localized modes embedded in the bulk of extended
states. We characterize this route to localization and find it to be distinct
from the paradigmatic Anderson model or standard random matrix systems.
