  A weighted $d-$complex is a simplicial complex of dimension $d$ in which each
face is assigned a real-valued weight. We derive three key results here
concerning persistence diagrams and minimal spanning acycles (MSAs) of such
complexes. First, we establish an equivalence between the MSA face-weights and
\emph{death times} in the persistence diagram. Next, we show a novel stability
result for the MSA face-weights which, due to our first result, also holds true
for the death and birth times, separately. Our final result concerns a
perturbation of a mean-field model of randomly weighted $d-$complexes. The
$d-$face weights here are perturbation of some i.i.d. distribution while all
the lower-dimensional faces have a weight of $0$. If the perturbations decay
sufficiently quickly, we show that suitably scaled extremal nearest
face-weights, face-weights of the $d-$MSA, and the associated death times
converge to an inhomogeneous Poisson point process. This result completely
characterizes the extremal points of persistence diagrams and MSAs. The point
process convergence and the asymptotic equivalence of three point processes are
new for any weighted random complex model, including even the non-perturbed
case. Lastly, as a consequence of our stability result, we show that Frieze's
$\zeta(3)$ limit for random minimal spanning trees and the recent extension to
random MSAs by Hino and Kanazawa also hold in suitable noisy settings.
