Randomized shortest paths (RSP) net betweenness centrality
Randomized shortest paths (RSP) net betweenness centrality
is a variation of the simple RSP betweenness centrality specifically designed for directed networks [2]. Unlike standard RSP betweenness, which sums the total flow of random walkers along edges, the net variant considers the
net flow
, so that opposing flows along the same edge partially cancel each other. This can provide a more meaningful measure of node importance in directed networks where bidirectional flows exist.
The RSP net betweenness of node \(i\), denoted \(c_{\mathrm{RSPnet}}(i)\), is defined as
\begin{equation*}
c_{\mathrm{RSPnet}}(i) = \sum_{s=1}^{N}\sum_{t=1}^{N} \sum_{(i,j) \in \mathcal{L}} \left| \bar{η}_{ij}(s,t) - \bar{η}_{ji}(s,t) \right|
= \sum_{(i,j) \in \mathcal{L}} \bar{η}_{ij}^{\mathrm{net}},
\end{equation*}
where
\[
\bar{η}_{ij}^{\mathrm{net}} = e^T \big| N^{ij} - N^{ji} \big| e,
\quad
N^{ij} = w_{ij} \left( \frac{z_{si} z_{jt}}{z_{st}} - \frac{z_{ti} z_{jt}}{z_{tt}} \right),
\]
and \(z_{st}\) are the elements of the fundamental matrix
\[
Z = (I - W)^{-1} = \left( I - D^{-1}A \odot e^{-β C} \right)^{-1}.
\]
Here, \(A\) is the adjacency matrix, \(D\) is the diagonal matrix of row sums of \(A\), and \(C\) is the cost matrix, whose elements represent the traversal cost, or distance, between adjacent nodes.
In the limit \(β \to 0\), the RSP net betweenness converges to the
current flow betweenness
centrality, highlighting the equivalence between random-walk and electrical-flow interpretations of betweenness in this regime [2].