DirichletRank
DirichletRank
is a variant of PageRank designed to address the “zero-one gap” problem inherent in the classical PageRank algorithm [2]. In PageRank, a random surfer moves to one of a node’s outgoing links with probability \(α\), or jumps to a random node with probability \(1-α\). For nodes with no outgoing links (sink nodes), the surfer cannot follow a link, so they must jump to a random node with probability 1. This creates a large difference in transition behavior between a sink node and a node with even a single outgoing link, leading to the so-called “zero-one gap” in PageRank probabilities. DirichletRank overcomes this issue by using a Bayesian estimation with a Dirichlet prior to compute smoother and more realistic transition probabilities. The DirichletRank score of the nodes, denoted \(c_{DR}\), is obtained by solving the eigenvector equation
\[
c_{DR} = \tilde{M} \, c_{DR},
\]
where
\[
\tilde{M} = \operatorname{diag}(1-ω_1, \dots, 1-ω_N) \, D^{-1}A + \frac{\operatorname{diag}(ω_1, \dots, ω_N)}{N} \, u u^T,
\]
with
\[
ω_i = \frac{μ}{μ + \sum_{j=1}^{N} a_{ij}},
\quad μ = 20,
\quad u \text{ is an \(N \times 1\) all-one vector}.
\]
Here, \(ω_i\) represents the random jumping probability for node \(i\). As defined, the more outgoing links a node has, the less likely a surfer is to jump randomly, and the more likely they are to follow one of its outgoing links.