  We develop an estimator for the change point parameter for a dynamically
evolving graphical model, and also obtain its asymptotic distribution under
high dimensional scaling. To procure the latter result, we establish that the
proposed estimator exhibits an $O_p(\psi^{-2})$ rate of convergence, wherein
$\psi$ represents the jump size between the graphical model parameters before
and after the change point. Further, it retains sufficient adaptivity against
plug-in estimates of the graphical model parameters. We characterize the forms
of the asymptotic distribution under the both a vanishing and a non-vanishing
regime of the magnitude of the jump size. Specifically, in the former case it
corresponds to the argmax of a negative drift asymmetric two sided Brownian
motion, while in the latter case to the argmax of a negative drift asymmetric
two sided random walk, whose increments depend on the distribution of the
graphical model. Easy to implement algorithms are provided for estimating the
change point and their performance assessed on synthetic data. The proposed
methodology is further illustrated on RNA-sequenced microbiome data and their
changes between young and older individuals.
