  We consider the problem of distinguishing classical (Erd\H{o}s-R\'{e}nyi)
percolation from explosive (Achlioptas) percolation, under noise. A statistical
model of percolation is constructed allowing for the birth and death of edges
as well as the presence of noise in the observations. This graph-valued
stochastic process is composed of a latent and an observed non-stationary
process, where the observed graph process is corrupted by Type I and Type II
errors. This produces a hidden Markov graph model. We show that for certain
choices of parameters controlling the noise, the classical (ER) percolation is
visually indistinguishable from the explosive (Achlioptas) percolation model.
In this setting, we compare two different criteria for discriminating between
these two percolation models, based on a quantile difference (QD) of the first
component's size and on the maximal size of the second largest component. We
show through data simulations that this second criterion outperforms the QD of
the first component's size, in terms of discriminatory power. The maximal size
of the second component therefore provides a useful statistic for
distinguishing between the ER and Achlioptas models of percolation, under
physically motivated conditions for the birth and death of edges, and under
noise. The potential application of the proposed criteria for percolation
detection in clinical neuroscience is also discussed.
