Product of Random Adjacency Matrices Based on an Underlying Fixed Graph

Question

Consider the adjacency matrix $$A$$ of a fixed undirected graph. So the $i,j$ element of $A$ is 1 if node $i$ is connected to node $j$ and it is zero, else. Define a random graph by deleting each edge of the original graph, independently with probability $p$. Define the adjacency matrix of such a random graph by $B$. Let $B(k)$ be independent copies of this random adjacency matrix. What can we say about the product $P(n)=B(1) B(2)...B(n)$ and how it scales with $n$. I expect this is known, but was not able to find anything in the literature. For example, can be quantify $(1/n)\log(||P(n)||)$ for an appropriate matrix norm?