Two-way random walk betweenness (2RW) centrality
Two-way random walk betweenness
(2RW) centrality is a variant of betweenness centrality based on two-way random walks [2]. For a pair of nodes \(i\) and \(j\), the probability of reaching \(j\) from \(i\) through a transition node \(t\) is defined as
\[
p_{itj} = \frac{a_{it} a_{tj}}{d_i d_t},
\]
where \(a_{ij}\) is the adjacency matrix entry and \(d_i\) is the degree of node \(i\).
The most likely two-way random walk between nodes \(i\) and \(j\) passes through a pair of transition nodes \((t^*, k^*)\) that maximizes the probability
\[
(t^*, k^*) = \arg\max_{t,k} \, p_{itj} \, p_{jki}.
\]
The 2RW betweenness centrality of a node counts how often it appears as one of these optimal transition nodes \((t^*\) or \(k^*)\) for all pairs of nodes \((i,j)\) in the network. Nodes that frequently serve as high-probability intermediate steps are considered more central.