Adaptive omni-channel gravity centrality (AOGC)
The
adaptive omni-channel gravity centrality
(AOGC), originally termed the Gravity Centrality method based on an Adaptive Truncation radius and Omni-channel paths, is a gravity-based centrality measure that combines an adaptive truncation radius with omni-channel path analysis to identify influential nodes in complex networks [2]. For a node \(i \in \mathcal{N}\), the AOGC score is defined as
\[
c_{AOGC}(i) = \sum_{j \in \mathcal{N}^{(\leq r)}(i)} c_{ij} \frac{m_i m_j}{LD_{ij}^2},
\]
where
\[
c_{ij} = \exp\Bigg(\frac{k_s(i) - k_s(j)}{\max(k_s) - \min(k_s)}\Bigg)
\]
is the attraction coefficient based on the \(k\)-shell centrality of nodes \(i\) and \(j\), \(\mathcal{N}^{(\leq r)}(i)\) denote the set of nodes within \(r\)-hop neighborhood of node \(i\), \(m_i\) is the mass of node \(i\) defined by the neighborhood structure-based centrality (NSC) [2] and \(LD_{ij}\) is the
looseness distance
between nodes \(i\) and \(j\):
\[
LD_{ij} = \frac{1}{\sum_{l=1}^{r} σ^l (A^l)_{ij}},
\]
where \(σ \in (0,1)\) is a free parameter controlling the weight of paths of different lengths, \(A\) is the adjacency matrix, and \(r\) is the truncation radius.
The adaptive truncation radius \(r\) is determined based on the average shortest path length \(\langle d \rangle\) of the network as
\[
r =
\begin{cases}
3, & \text{if } \langle d \rangle \leq 3, \\
\lfloor \langle d \rangle \rfloor, & \text{if } 3 < \langle d \rangle \leq \Theta, \\
\Theta + \lfloor \ln(\langle d \rangle - \Theta + 1) \rfloor, & \text{if } \langle d \rangle > \Theta,
\end{cases}
\]
where \(\Theta\) is a threshold parameter, set to 6.
Nodes with high AOGC values are those that have large mass (high NSC), are well-connected to other influential nodes, and are reachable via multiple strong omni-channel paths, making them particularly important for spreading processes and maintaining network cohesion.