A singular perturbation approach for choosing the PageRank damping factor

We study the PageRank mass of principal components in a bow-tie web graph as a function of the damping factor $c$. It is known that the web graph can be divided into three principal components: SCC, IN, and OUT. The giant strongly connected component (SCC) contains a large group of pages having a hyperlink path connecting them. The pages in the IN (OUT) component have a path to (from) the SCC, but not back. Using a singular perturbation approach, we show that the PageRank share of the IN and SCC components remains high even for very large values of the damping factor, in spite of the fact that it drops to zero when $c$ tends to one. However, a detailed study of the OUT component reveals the presence of "dead ends" (small groups of pages linking only to each other) that receive an unfairly high ranking when $c$ is close to 1. We argue that this problem can be mitigated by choosing $c$ as small as ½.