What is the explanation of infinite summation of random walk matrix?

Question

I'm reading about Katz index, I wonder what is the explanation about Katz index in intuitive meaning ? How about I replace a random walk matrix, i.e., use probability instead of adjacency indicator. Is there realistic meaning about matrix $S$ defined as follows: $$ S = \sum_{i=1}^{\infty}\alpha^iP^i $$ where $\alpha\in (0,1)$ and $P=D^{-1}A$, $A$ is the adjacency matrix in Katz index, and $ D$ is the diagonal matrix with degree of $A$.