PageRank Computation, with Special Attention to Dangling Nodes

Abstract

We present a simple algorithm for computing the PageRank (stationary distribution) of the stochastic Google matrix G. The algorithm lumps all dangling nodes into a single node. We express lumping as a similarity transformation of G and show that the PageRank of the nondangling nodes can be computed separately from that of the dangling nodes. The algorithm applies the power method only to the smaller lumped matrix, but the convergence rate is the same as that of the power method applied to the full matrix G. The efficiency of the algorithm increases as the number of dangling nodes increases. We also extend the expression for PageRank and the algorithm to more general Google matrices that have several different dangling node vectors, when it is required to distinguish among different classes of dangling nodes. We also analyze the effect of the dangling node vector on the PageRank and show that the PageRank of the dangling nodes depends strongly on that of the nondangling nodes but not vice versa. Last we present a Jordan decomposition of the Google matrix for the (theoretical) extreme case when all Web pages are dangling nodes.