Suppose there is a random walk process on a simple strongly connected directed graph $G=(V, E)$. The random walk follows a transmission matrix in the form $M=AD^{-1}$, where A is the adjacency matrix and $D$ is a diagonal matrix whose diagonal elements are exactly the out-degree ($d^+(v)$) of the vertices.
From the Perron-Frobenius theorem, it is easy to see that there is a unique strictly positive stationary distribution $\pi=\{x_v\}_{v\in V}$. How to prove that for every $v\in V, x_v\geq n^{-(n+1)}$? (Here, |V|=n).
I was thinking about using contradiction to prove this: suppose there exists $v\in V$ such that $x_v<n^{-(n+1)}$. Then utilize the equation $$x_v=\sum_{(u,v)\in E}\frac{x_u}{d^+(u)}$$ to find a contradiction. But I do not know how to proceed next.