Universal Emergence of PageRank

The PageRank algorithm enables to rank the nodes of a network through a specific eigenvector of the Google matrix, using a damping parameter α ∈]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 PageRank vector at its emergence when α → 1. The whole network can be divided into a core part and a group of invariant subspaces. For α → 1 the 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 α → 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.