Eigenvectors of a perturbed reducible stochastic matrix

Question

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix

$$\tilde{Q}\,=\,(1-\alpha)Q+\alpha J$$

Though $Q$ is reducible meaning it has a eigenspace of dimension more than $1$ corresponding to the Perron root, $\tilde{Q}$ is irreducible guaranteeing a unique Perron vector. So this transition from reducibility of $Q$ to irreducibility of $\tilde{Q}$ is a function of $\alpha$. I was curious if we can study the Perron vector as a function of $\alpha$. Has there been any study in literature on this?

PS: I know that this is referred to as the teleportation probability in computer science (for example, the famous Pagerank algorithm uses this). Is there any other application area where this is used frequently?

Answer 1

So this transition from reducibility of Q to irreducibility of $\tilde Q$ is a function of α.

It is easy to show that for any $\alpha > 0$ $\tilde Q$ will be irreducible. See Lemma 3.1 from A note on perturbations of stochastic matrices by Bertram and Wolfgang).

Applying the Theorem 3.2 from the same note, we obtain that Perron vector $P(\alpha)$ is a analytic (thus smooth, and continous) function.

(Actually, it should be a comment. But, unfortunately, I cannot add any comments)