Universal emergence of PageRank

The PageRank algorithm enables us to rank the nodes of a network through a specific eigenvector of the Google matrix, using a damping parameter alpha a [setmembership]]0, 1[. Using extensive numerical simulations of large web networks, with a special accent on British University networks, we determine numerically and analytically the universal features of the PageRank vector at its emergence when alpha arrow right 1. The whole network can be divided into a core part and a group of invariant subspaces. For alpha arrow right 1, PageRank converges to a universal power-law distribution on the invariant subspaces whose size distribution also follows a universal power law. The convergence of PageRank at alpha arrow right 1 is controlled by eigenvalues of the core part of the Google matrix, which are extremely close to unity, leading to large relaxation times as, for example, in spin glasses.