multiplications of graph adjacency matrix

Question

Suppose $A$ is a directed graph adjacency matrix. Is there any good interpration of the $(i,j)-$entry of the matrix $(A^{32}\cdot (A^T)^{32})$ ?

Answer 1

Start with equal edge weight, so $A$ is a boolean matrix. Think about what happens when you apply $A$ to the $k$th column of the identity, $\mathbf e_k$. Are the nonzeros in $A \mathbf e_k$ upstream or downstream of node $k$? What happens when you apply $A$ again, $A(A \mathbf e_k)$? What happens when there are two length-2 paths from node $i$ to node $k$?

Try the same experiment with $A^T$. You should now have an interpretation for the $k$th column of $A^{32} (A^T)^{32}$.