centrality_spectralrank: SpectralRank with optional diagonal prior information

In cograph: Analysis and Visualization of Complex Networks

SpectralRank with optional diagonal prior information

Description

Xu et al.'s SpectralRank is the positive right eigenvector belonging to the largest real eigenvalue of the augmented adjacency B = \left(\begin{smallmatrix}A+P&\mathbf{1}\\ \mathbf{1}^T&0\end{smallmatrix}\right). A ground node connects bidirectionally to every original node with unit edge weight. The diagonal P is zero for ordinary SpectralRank; a nonnegative prior gives the paper's weighted SpectralRank family.

Usage

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

Arguments

x	Network input accepted by centrality.
sr_prior	Nonnegative finite scalar or one value per original node. Default zero selects SpectralRank; one selects a uniform unit prior.
...	Additional arguments to centrality.

Details

Raw scores are scaled by the maximum over ALL nodes, including the ground node, which is then omitted from the output. Its score is not redistributed. Consequently the largest returned score can be below one. Optional normalized = TRUE additionally divides by the maximum over original nodes, changing this source-defined scale.

The paper uses binary adjacency and outgoing neighbors: an edge i to j contributes j's score to i. The function preserves this orientation; transpose the graph to use incoming neighbors. Finite nonnegative edge weights extend the same matrix definition; they are interaction weights, separate from the diagonal-prior meaning of weighted SpectralRank. Unit ground edges stay fixed, so scaling original edge weights generally changes scores. For tiny asymmetric matrix weights, set directed = TRUE or use a directed igraph object because the shared parser otherwise uses approximate symmetry detection.

Loops are removed and remaining parallel edges sum after the generic simplify rule; unweighted remaining edges count once each. Zero weights are absent. Mode, path-weight inversion and cutoff are ignored. Named vector priors are matched to node names. Scalar priors broadcast; the ground prior is always zero. Supply externally computed degree, H-index or coreness scores as a vector to select those prior families.

All nodes, including isolates, receive positive spectral scores because of the ground links. Without edges or priors, each of n original nodes scores 1/\sqrt{n}. For an edgeless graph the paper's unshifted power iteration oscillates, although the Perron eigenvector is unique. This function explicitly uses that eigenvector definition, without claiming convergence of the published iteration. A singleton scores one; an empty graph returns no scores. Adding disconnected nodes generally changes other scores because all share the ground node.

For nonzero priors the implementation follows section III-A2's B=\widetilde A+P. Algorithm 1 constructs that matrix but its update line prints \widetilde A, omitting P; this inconsistency is retained in the verification audit. No author-software parity is claimed.

Dense eigendecomposition takes O(n cubed) time and O(n squared) memory. Extreme weight/prior ranges or unresolved positive eigenpairs raise errors. The score defines a spectral ranking, not a spreading probability or a general guarantee of predictive performance.

Value

Named numeric vector in input node order.

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/TCYB.2018.2861568")}.

Examples

centrality_spectralrank(igraph::make_ring(5))
centrality_spectralrank(igraph::make_star(5), sr_prior = 1)

cograph documentation built on Sept. 30, 2026, 5:08 p.m.