ABSTRACT For two given graphs and , a graph is said to be weakly ‐saturated if is a spanning subgraph of which has no copy of as a subgraph and one can add all edges in to in some order so that a new copy of is created at each step. The weak saturation number is the minimum number of edges of a weakly ‐saturated graph. In this paper, we deal with the relation between and , where denotes the Erdős–Rényi random graph and denotes the complete graph on vertices. For any graph and constant , we prove that with high probability. Also, for some graphs , including complete graphs, complete bipartite graphs, and connected graphs with minimum degree or , it is shown that there exists an such that, for every with high probability.