Studying the behaviour of a recursive formula with graph theory

Question

I have a recursive formula of the type : $$ \begin{equation} A_{r+1}(a)=\sum_{b \in X}A_{r}(b)\omega(a,b) \end{equation} $$ where $X$ is a set (vertices) and $\omega(a,b)$ is a COMPLEX weight such that : $$ \begin{equation} \sum_{b \in X}|\omega(a,b)|=1. \end{equation} $$ My work is studying the behaviour of $A_{k}(a))$ when $k$ goes to $+\infty$ and I wish to write $$ \begin{equation} A_{k}(r)=\pi(r)+O(2^{-kl}) \end{equation} $$ for a certain $l$. An idea consists in considering the matrice $M=(\omega(a,b))_{a,b \in X}$ and looking at the graph made by vertices. Do exist some conditions for the weights in order to apply ergodic theory (like a "Perron-Frobenius theorem")? Here, the graph I have is aperiodic, finite and irreductible, but as you understood, the weights are complex...