Threshold for weak saturation stability

We study the weak $K_s$-saturation number of the Erd\H{o}s–R'{e}nyi random graph $\text{mathbbmsl}G(n,p)$, denoted by $$\mathrm{w}\mathrm{s}\mathrm{a}\mathrm{t}$(\mathbbmsl{G}(n, p), K_s)$,where$K_s$isthecompletegraphon$s$ vertices. Kor'{a}ndi and Sudakov in 2017 proved that the weak $K_s$-saturation number of $K_n$ is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Kor'{a}ndi and Sudakov. A general upper bound for $\mathrm{w}\mathrm{s}\mathrm{a}\mathrm{t}(\text{mathbbmsl}G(n,p),K_s)$ is also provided.