Three further spreader-selection procedures in the VoteRank family. All three elect one node per round until every node is placed and return the election order as a score, 1 for the first elected down to \(1 / n\); ties go to the lowest node index. Direction and self-loops are ignored.
Usage
centrality_wvoterank(x, ...)
centrality_enrenew(x, enrenew_depth = 2, ...)
centrality_voterank_plus(x, voterank_lambda = 0.1, ...)Arguments
- x
Network input (matrix, igraph, network, cograph_network, tna object).
- ...
Additional arguments passed to
centrality.- enrenew_depth
Renewal radius \(l\) for
enrenew. Default 2.- voterank_lambda
Suppression factor \(\lambda\) for
voterank_plus. Default 0.1.
Details
wvoterank(Sun, Chen, He & Ch'ng 2019)VoteRank for weighted graphs: \(s_v = \sqrt{k_v \sum_{u \in N(v)} va_u w_{vu}}\). After an election the winner's ability is 0 and its neighbors lose \(1 / \langle w \rangle\), where \(\langle w \rangle\) is the average strength (the paper's Figure 1 pins strength, not degree). Uses edge weights; with unit weights it is VoteRank with a square-root score. Reproduces all sixty numbers of the paper's Figure 1.
enrenew(Guo, Yang, Guo, Pan & Chen 2020)Entropy-based selection: \(E_v = \sum_{u \in N(v)} -p_{uv} \ln p_{uv}\) with \(p_{uv} = k_u / \sum_{l \in N(v)} k_l\); after electing the largest \(E\), every entropy term flowing outward to depth \(d \le l\) is scaled by \(1 - 1 / (2^{d-1} \ln \langle k \rangle)\), with \(l\) =
enrenew_depth(default 2). Reproduces the paper's Figure 1. The authors' released code differs from the paper in several ways; the paper is implemented. Note the factor turns negative when \(\langle k \rangle < e\).voterank_plus(Liu, Li, Fang & Yao 2021)Initial ability \(\ln(1 + k_i / k_{\max})\), degree-proportional vote shares over unelected neighbors, score \(\sqrt{k_i \sum_j va_j w_{j \to i}}\), and after an election abilities are multiplied by \(\lambda\) one step away and \(\sqrt{\lambda}\) two steps away (
voterank_lambda, default 0.1). The article is closed access; the implementation matches the authors' released code exactly, including its 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
adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_wvoterank(adj)
#> A B C D E F
#> 0.6666667 0.3333333 1.0000000 0.8333333 0.5000000 0.1666667
centrality_enrenew(adj)
#> A B C D E F
#> 0.6666667 0.3333333 1.0000000 0.8333333 0.5000000 0.1666667
centrality_voterank_plus(adj)
#> A B C D E F
#> 0.6666667 0.3333333 1.0000000 0.8333333 0.5000000 0.1666667