Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node connected to its four neighbors, with the top row connected to the bottom, and the right column connected to the left. Suppose that, within each unit of time, each edge "breaks" for a randomly placed time interval lasting $\delta \lt 1$; perhaps $\delta \ll 1$ could be assumed. The edge-failures are independent of one another. Let $G_n(t)$ be the graph at time $t$, with, in general, several edges missing.

Q. I would like to know the expected duration $T$ that $G_n(t)$ will remain connected for all $t < T$, i.e., the wait time to the first disconnection of one or more nodes from the remainder.

Although disconnection is likely to be dominated by the probability of single-node isolation, it seems difficult to capture all the different ways that $G_n$ could become disconnected. Below, the three shaded nodes are disconnected by $8$ missing edges.

One could of course ask the same question for grid graphs based on $\mathbb{Z}^d$ rather than $\mathbb{Z}^2$ as above. It seems likely this has been studied for network robustness. Thanks!