Convergence of recursive formula in TextRank
Your process is a Markov chain, so I suggest you try applying the theory of Markov chains. There's a rich and powerful theory that has been built up over time, which might be applicable here.

In particular, if we let $S$ denote the column vector $S = (S(V_1),S(V_2),\dots,S(V_n))$, then there is a matrix $n\times n$ $M$ such that each iteration maps $S$ to $MS$.  If the underlying graph is strongly connected, it looks like this Markov chain is irreducible, ergodic, aperiodic, and all its states are recurrent.  From this you should be able to deduce that it has a stationary distribution $\pi$ and that the stationary distribution is the unique limiting distribution of the chain, hence the algorithm eventually converges.  You should check all of this yourself carefully; I haven't checked any of this.

The rate of convergence will be dependent on the magnitude of the second-largest eigenvalue of the Markov chain.

Look at the analysis of PageRank for a detailed, worked example applying these concepts to a similar algorithm.  This analysis can be found in many textbooks and algorithms courses.