PureRank assigns an importance score to every node of an arbitrary weighted directed network without any free parameter. It decomposes the network into strongly connected components, labels nodes as recurrent, transient, or dangling, solves a Katz-parameter optimization within each class to best approximate Bonacich eigenvector centrality, and aggregates the local vectors by the recursive-definition-of-importance principle. The learner sees why classical eigenvector and Seeley centrality fail on networks that are not strongly connected (loss of uniqueness and completeness), how strongly-connected-component structure restores both, and how PureRank differs from PageRank's damping-factor approach while admitting a random-surfer interpretation and parallel, incremental computation.
PureRank is a parameter-free centrality measure for arbitrary directed networks. It first decomposes the network into strongly connected components, classifying every node as recurrent, transient, or…