Return Random Walk Gravity (RRWG) centrality
Return Random Walk Gravity
(RRWG) centrality combines the concepts of return random walks, effective distance, and the gravity model to assess node importance in networks [2]. For node \(i\), it is defined as
\[
c_{RRWG}(i) = \sum_{j \neq i} \frac{d_i d_j}{\left(D_{j|i}\right)^2},
\]
where \(d_i\) and \(d_j\) are the degrees of nodes \(i\) and \(j\), respectively, and \(D_{j|i}\) is the effective distance from node \(j\) to node \(i\), given by
\[
D_{j|i} = 1 - \log_2 \left( \max_{t \neq k} \left( p_{itj} \, p_{jki} \right) \right),
\]
with \(p_{itj}\) representing the probability of reaching node \(j\) from node \(i\) via a transition node \(t\), and \(p_{jki}\) representing the probability of returning from node \(j\) to node \(i\) via another transition node \(k\).
The RRWG centrality integrates three aspects: the gravity model captures the attractive power of a node based on its connectivity, the effective distance encodes both static and dynamic structural information of the network, and return random walks quantify a node's importance by accounting for the strength of indirect interactions with other nodes.