Rank centrality
Rank centrality
is a spectral ranking algorithm for directed graphs that aggregates pairwise comparisons to derive a global ranking of nodes [2]. The method constructs a random walk on the comparison graph, where at each step the transition probability from node \( i \) to node \( j \) is proportional to the empirical probability that \( i \) beats \( j \) in pairwise comparisons. The transition probabilities are normalized by the maximum out-degree to ensure that the resulting matrix is stochastic. Formally, the random walk is described by an \( N \times N \) time-independent transition matrix \( P \) with elements
\begin{equation*}
p_{ij} =
\begin{cases}
\dfrac{A_{ij}}{d_{\max}}, & \text{if } i \neq j, \\[6pt]
1 - \dfrac{1}{d_{\max}} \sum_{k \neq i} A_{ik}, & \text{if } i = j,
\end{cases}
\end{equation*}
where \( A_{ij} = \dfrac{a_{ij}}{a_{ij} + a_{ji}} \) denotes the empirical probability that node \( i \) is preferred over node \( j \), and \( d_{\max} \) is the maximum out-degree among all nodes. Intuitively, the random walk tends to remain longer on nodes that perform well across comparisons. The rank centrality of a node corresponds to the stationary distribution \( \boldsymbol{π} \) satisfying \( \boldsymbol{π} = \boldsymbol{π} P \), which represents the relative importance of each node in the network as determined by the random walk.