I have a square matrix $A_{n \times n}$ whose elements are either 0 or 1. The matrix $A$ changes in response to different events (the elements always being 0 or 1). After a series of changes, it finally converges to a particular matrix i.e. subsequent events do not alter $A$ significantly.
How could I determine the converged form of $A$ given that I know all the intermediate forms (say, $A_1, A_2, ..., A_n$)? Of course, if $A_n$ is the final matrix, then it would give the converged form. But, what if the convergence has already been achieved at $A_k$ where $k < n$?
I was exploring Frobenius norm a bit, and thought of using: $\lVert A_{i+1}$ - $A_i \rVert_F \leq \epsilon$. However, that does not account for any changes later in the sequence.
What would be the suggested approach in this case?