The improved WVoteRank centrality is a modification of WVoteRank that incorporates both 1-hop and 2-hop neighbors in the voting process [2]. Each node \(i\) is characterized by the tuple \((s_i, v_i)\), where \(s_i\) is the voting score and \(v_i\) is the voting ability. Initially, \((s_i, v_i) = (0,1)\) for all \(i \in \mathcal{N}\). The voting procedure iteratively performs the following steps:

  1. Vote: each node distributes votes to its neighbors. The voting score of node \(i\) is computed as \begin{equation*} s_i = \sqrt{ d_i \sum_{j=1}^{N} w_{ij} v_j + \sum_{j \in \mathcal{N}(i)} \sum_{k \in \mathcal{N}(j) \setminus \{i\}} w_{jk} v_k }, \end{equation*} where \(w_{ij}\) is the weight of the edge \((i,j)\), \(d_i\) is the degree of node \(i\) and \(\mathcal{N}(j)\) denotes the neighbors of node \(j\).
  2. Select: the node \(k\) with the highest voting score \(s_k\) is elected. This node is removed from subsequent voting rounds by setting its voting ability to zero, \(v_k = 0\).
  3. Update: the voting ability of nodes within the 2-hop neighborhood of the elected node is reduced. For each \(j \in \mathcal{N}^{( \leq 2)}(k)\), the updated voting ability is \begin{equation*} v_j \leftarrow \max(0, v_j - f_{kj}), \end{equation*} where \(f_{kj} = \frac{1}{\langle d \rangle d_{kj}}\), \(\langle d \rangle\) is the average degree of the network, and \(d_{kj}\) is the shortest-path distance from the elected node \(k\).
The improved WVoteRank centrality accounts for both direct and indirect neighbors, ensuring that the influence of selected nodes appropriately diminishes the voting power of surrounding nodes.

References

[1] Shvydun, S. (2025). Zoo of Centralities: Encyclopedia of Node Metrics in Complex Networks. arXiv: 2511.05122 https://doi.org/10.48550/arXiv.2511.05122
[2] Kumar, S., & Panda, A. (2022). Identifying influential nodes in weighted complex networks using an improved WVoteRank approach. Applied intelligence, 52(2), 1838-1852. doi: 10.1109/ACCESS.2017.2679038.