I am currently studying directed acyclic graphs (DAGs). Recall the spectral radius $r(B)$ of a matrix $B$ is the largest absolute eigenvalue of $B$. The paper I am reading said the condition that $r(B) < 1$ is strong: although it is automatically satisfied when $B$ is a DAG, it is generally not true otherwise, and furthermore the projection is nontrival.
I am thinking what can we say about $B$ once we know $r(B)<1$. I know if $r(B) < 1$, it implies that the powers of $B$ converge to zero as the exponent increases. In other words, as $n$ approaches infinity, $B^n$ tends to the zero matrix. This property is related to the behavior of walks or paths in the graph. The convergence of $B^n$ to the zero matrix as $n$ approaches infinity implies that the "effect" of applying the adjacency matrix repeatedly diminishes.
But the above is about the matrix $B^n$, not $B$ itself. I am more interested in the matrix B itself, since we observe $B$ directly, not $B^n$.