PureRank: A parameter-free recursive importance measure for network nodes

This study develops PureRank, a parameter-free importance measure for network nodes based on the recursive definition of importance (RDI). For any directed network, PureRank uniquely determines an importance score vector without user-specified parameters. PureRank can thus provide a neutral reference for parameter-dependent importance measures. PureRank is constructed in three steps: (i) nodes are classified into {\it recurrent}, {\it transient}, and {\it dangling} classes via strongly connected component decomposition; (ii) for each class, the local importance vector is obtained by choosing the parameters of the Katz equation on the class-restricted subnetwork according to the RDI principle; and (iii) the local importance vectors are aggregated into the PureRank vector. This modular design supports parallel and incremental computation while retaining a unified random-surfer interpretation. Numerical experiments on three SNAP networks show that PageRank has a computational advantage over PureRank except when the damping factor $d$ is close to one, and that the similarity of PageRank to PureRank depends on $d$ and the node classification. In the fully recurrent network, similarity increases monotonically with $d$ and reaches Kendall's $\tau_b=0.966$ and Pearson correlation coefficient $=1.000$ at $d=0.999$, whereas in the two transient-dominated networks, similarity varies nonmonotonically with $d$. PureRank is extended to multi-attribute networks.