I am reading a paper about the TextRank algorithm in keyword extraction and they mention the recursive formula:$$ \displaystyle S(V_{i}) = (1 - d) + d \ast \sum_{j \in In(V_{i})} \frac{1}{|Out(V_{j})|} \; S(V_{j})  $$where $ In(V_{i}) $ are the vertices that point to $ V_{i} $ and $ Out(V_{i}) $ are the vertices that $ V_{i} $ points to. The paper claimed that this algorithm will converge for any arbitrarily small positive threshold and any arbitrary initial values by testing with experiments. However, I am attempting to prove it mathematically that this algorithm will always converge regardless of the choice of the threshold as well as the initial values, but haven't succeeded at this point. My approach is to prove that if $ E_{ij} $ is the error at vertex $ i $ at the $ j $ iteration, then the sequence $ E_{i1}, E_{i2}, \dots, E_{in} $ is a decreasing sequence. Intuitively, it means that the error rate approaches $ 0. $Any hint or suggestion is appreciated.
Convergence of recursive formula in TextRank
algorithm analysis;numerical algorithms