SpectralRank Centrality

SpectralRank (Xu et al. 2019) is the positive eigenvector of the largest eigenvalue of the adjacency matrix augmented by a ground node, which is linked in both directions to every node with unit weight: $$B = \left(\begin{smallmatrix} A + P & \mathbf{1} \\ \mathbf{1}^T & 0 \end{smallmatrix}\right).$$ The diagonal prior $P$ is zero for ordinary SpectralRank, and a nonnegative prior gives the weighted SpectralRank family of the paper.

Usage

centrality_spectralrank(x, sr_prior = 0, ...)

Arguments

x: Network input accepted by centrality.
sr_prior: Diagonal prior $P$, a nonnegative scalar or one value per node. A named vector is matched to the node names. Default 0, which gives ordinary SpectralRank.
...: Further arguments to centrality. The measure uses weighted (use edge weights, default TRUE).

Value

A named numeric vector with one score per node, in input node order.

Details

The eigenvector is divided by its maximum over all nodes including the ground node, which is then dropped, so the largest returned score can be below one. normalized = TRUE further divides by the maximum over the original nodes. An arc from $i$ to $j$ contributes the score of $j$ to $i$. Edge weights must be finite and nonnegative, weighted = FALSE gives every edge weight one, and loops are removed. Every node, isolated nodes included, scores positive, and without edges or priors each of $n$ nodes scores $1/\sqrt{n}$. The update line of Algorithm 1 in the paper omits $P$, and the implementation follows section III-A2 with $B = \tilde{A} + P$. An invalid sr_prior or an unresolved eigenvector raises an error.

References

Xu, S., Wang, P., Zhang, C.-X. and Lu, J. (2019). Spectral Learning Algorithm Reveals Propagation Capability of Complex Networks. IEEE Transactions on Cybernetics, 49(12), 4253-4261. doi:10.1109/TCYB.2018.2861568 .

Examples

centrality_spectralrank(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.3383201  0.4405105  0.3341652  0.3555215  0.2984056  0.3518547  0.3347994 
#>   Evaluate     Create      Share 
#>  0.3590902  0.3730458  0.3900942