The WVoteRank centrality is a modification of VoteRank that incorporates both the number of neighbors and the weight of each link [2]. Each node \(i\) is represented by the tuple \((s_i, v_i)\), where \(s_i\) is the voting score and \(v_i\) is the voting ability, initialized as \((s_i, v_i) = (0, 1)\) for all \(i \in \mathcal{N}\). The voting procedure iteratively performs the following steps:


  1. Vote: Each node votes for its neighbors based on its voting ability. The voting score of node \(i\) is updated as \begin{equation*} s_i = \sqrt{d_i \sum_{j=1}^{N} w_{ji} v_j}, \end{equation*} where \(w_{ji}\) is the weight of the link from node \(j\) to node \(i\) and \(d_i\) is the degree of node \(i\).

  2. Select: The node \(k\) with the highest voting score \(s_k\) is elected. Node \(k\) will not participate in subsequent voting turns, i.e., its voting ability is set to zero (\(v_k = 0\)).

  3. Update: The voting ability of nodes that voted for \(k\) is reduced to account for influence spread. For each neighbor \(i \in \mathcal{N}(k)\), the updated voting ability is \begin{equation*} v_i \leftarrow \max(0, v_i - f), \end{equation*} where \(f\) is typically set to the inverse of the average degree of the network, i.e. \(f=1/\langle d \rangle\).


WVoteRank prioritizes nodes that are both well-connected and linked through high-weight edges, ensuring that influential nodes are identified while accounting for the propagation of influence through their neighbors.

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] Sun, H. L., Chen, D. B., He, J. L., & Ch’ng, E. (2019). A voting approach to uncover multiple influential spreaders on weighted networks. Physica A: Statistical Mechanics and its Applications, 519, 303-312. doi: 10.1016/j.physa.2018.12.001.