Calculating the PageRank Vector of a Scale-Free Web Network Growing by Preferential Attachment

We consider a scale-free model of the Web network that is evolving by a preferential attachment scheme and derive an explicit formula of its PageRank vector. Its $i^{th}$ element indicates the probability that a surfer resides at a related Web page $i$ in a stationary regime of an associated random walk. Considering the growth of the underlying directed Web graph, we apply specific linear preferential attachment schemes proposed by Samorodnitsky et al. (2016). To express the probability of a connection between two nodes of this Web graph, our derivation allows us to avoid the consideration of complicated paths with random lengths and to cover both self loops and multiple edges between nodes. In this way, our approach enhances existing analysis schemes. It provides a better insight on the PageRank of growing scale-free Web networks and supports the adaptation of the model to gathered network statistics.