Three spreader-selection procedures in the VoteRank family that elect
one node per round. WVoteRank (Sun et al. 2019) scores a node by
\(s_v = \sqrt{k_v \sum_{u \in N(v)} va_u w_{vu}}\)
and lowers the ability of the neighbors of a winner by
\(1 / \langle w \rangle\), the inverse of the average strength.
EnRenew (Guo et al. 2020) elects the largest neighbor entropy
\(E_v = -\sum_{u \in N(v)} p_{uv} \ln p_{uv}\),
\(p_{uv} = k_u / \sum_{l \in N(v)} k_l\),
and scales the entropy terms within enrenew_depth steps by
\(1 - 1 / (2^{d-1} \ln \langle k \rangle)\).
VoteRank++ (Liu et al. 2021) starts from ability
\(\ln(1 + k_i / k_{\max})\), splits votes in
proportion to degree, and multiplies abilities by \(\lambda\)
one step from a winner and by \(\sqrt{\lambda}\) two
steps away.
Usage
centrality_wvoterank(x, ...)
centrality_enrenew(x, enrenew_depth = 2, ...)
centrality_voterank_plus(x, voterank_lambda = 0.1, ...)Arguments
- x
Network input accepted by
centrality.- ...
Further arguments to
centrality. WVoteRank usesweighted(use edge weights, defaultTRUE).- enrenew_depth
Renewal radius \(l\) for EnRenew (default 2).
- voterank_lambda
Suppression factor \(\lambda\) for VoteRank++ (default 0.1).
Details
Elections continue until every node is placed. The first elected scores 1 and the last \(1/n\), so scores lie in \((0, 1]\), and ties go to the lowest node index. Direction and self-loops are ignored. WVoteRank uses edge weights, and with unit weights it is VoteRank with a square-root score. The other two ignore weights. The renewal factor of EnRenew is negative when \(\langle k \rangle < e\). EnRenew follows the article where the released code of the authors differs from it. VoteRank++ follows the released code of the authors, including the exclusion of elected nodes from the vote-share denominator.
References
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, 519, 303-312.
Guo, C., Yang, L., Guo, X., Pan, J., & Chen, X. (2020). Influential nodes identification in complex networks via information entropy. Entropy, 22(2), 242.
Liu, P., Li, L., Fang, S., & Yao, Y. (2021). Identifying influential nodes in social networks: A voting approach. Chaos, Solitons & Fractals, 152, 111309.
Examples
centrality_wvoterank(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.7 0.6 1.0 0.9 0.5 0.8 0.4
#> Evaluate Create Share
#> 0.3 0.2 0.1
centrality_enrenew(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.4 0.8 1.0 0.6 0.2 0.3 0.1
#> Evaluate Create Share
#> 0.7 0.9 0.5
centrality_voterank_plus(regulation_net)
#> Explore Plan Monitor Adapt Reflect Discuss Synthesize
#> 0.4 0.6 1.0 0.9 0.5 0.7 0.3
#> Evaluate Create Share
#> 0.2 0.8 0.1