The VoteRank\(^{++ \)} method is an enhanced variant of the VoteRank algorithm designed to identify a set of influential nodes that are broadly distributed across a network [2]. Each node \(i\) is characterized by a tuple \((s_i, v_i)\), where \(s_i\) denotes its voting score and \(v_i\) its voting ability. Initially, these values are assigned as
\[
(s_i, v_i) = \left( 0, \log\left(1 + \frac{d_i}{d_{\max}}\right) \right),
\]
where \(d_i\) is the degree of node \(i\) and \(d_{\max}=\max_j{d_j}\) is the maximum degree in the network. The algorithm proceeds iteratively through the following steps:


  1. Vote: each node \(i\) casts votes to its neighbors according to \[ s_i = \sqrt{d_i \sum_{j=1}^{N} \left( \frac{a_{ji} d_i}{\sum_{l=1}^{N} a_{jl} d_l} v_j \right)}, \] where \(a_{ji}\) is the element of the adjacency matrix of the network.

  2. Select: the node \(k\) with the highest voting score \(s_k\) is selected as an influential node. This node is then excluded from subsequent voting rounds by setting its voting ability \(v_k = 0\).

  3. Update: the voting abilities of nodes that voted for node \(k\) are reduced to \(λ v_i\), where \(λ \in [0,1]\) is a suppressing factor. For nodes within two hops of \(k\), the voting ability is reduced to \(\sqrt{λ} v_i\). Following Liu et al. [2], the suppressing factor is typically set to \(λ = 0.1\).


Nodes with high VoteRank\(^{++}\) scores are thus identified as influential spreaders that are not only central but also spatially dispersed, ensuring broad network coverage.

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] Liu, P., Li, L., Fang, S., & Yao, Y. (2021). Identifying influential nodes in social networks: A voting approach. Chaos, Solitons & Fractals, 152, 111309. doi: 10.1016/j.chaos.2021.111309.