  Identifiability concerns finding which unknown parameters of a model can be
quantified from given input-output data. Many linear ODE models, used in
systems biology and pharmacokinetics, are unidentifiable, which means that
parameters can take on an infinite number of values and yet yield the same
input-output data. We use commutative algebra and graph theory to study a
particular class of unidentifiable models and find conditions to obtain
identifiable scaling reparametrizations of these models. Our main result is
that the existence of an identifiable scaling reparametrization is equivalent
to the existence of a scaling reparametrization by monomial functions. We also
provide partial results beginning to classify graphs which possess an
identifiable scaling reparametrization.
