Dynamical importance by exact vertex deletion — centrality_dynamical

Restrepo, Ott & Hunt's node dynamical importance is the relative drop in adjacency spectral radius on removing that node: \(I_i = (\rho(A)-\rho(A_{-i}))/\rho(A)\) (equation 2). This function recomputes the spectral radius after every deletion. The paper's left/right eigenvector product (equation 5) is an approximation and can differ substantially on small networks; it is not used here.

Usage

centrality_dynamical_importance(x, ...)

Arguments

x: Network input accepted by centrality.
...: Additional arguments to centrality. The default normalized = FALSE preserves the published relative loss; TRUE additionally divides positive scores by their maximum.

Value

Named numeric vector in input node order.

Details

Supports directed or undirected nonnegative weighted networks. Self-loops are always removed, as in the paper's zero-diagonal definition. Edge weights, weighted and simplify follow the same adjacency conventions as centrality_diffusion_centrality. The measure is invariant to reversing all arcs and ignores mode and path-weight inversion. Disconnected graphs use the spectral radius of the whole graph.

When the original spectral radius is zero (including any directed acyclic graph), the ratio is undefined and all vertices receive NaN. Isolates in a graph with positive spectral radius receive zero. The empty graph returns an empty vector. Strong components are evaluated separately so acyclic parts contribute exactly zero, avoiding numerical eigenvalues of nilpotent blocks. Roundoff in the final ratio is clipped to zero or one.

Repeated eigendecomposition is costly. Select this measure explicitly or use include = "dynamical_importance"; it is held back from the default type = "all" tier.

References

Restrepo, J. G., Ott, E., & Hunt, B. R. (2006). Characterizing the Dynamical Importance of Network Nodes and Links. Physical Review Letters, 97, 094102. doi:10.1103/PhysRevLett.97.094102 .

Examples

centrality_dynamical_importance(igraph::make_full_graph(4))
#>         1         2         3         4 
#> 0.3333333 0.3333333 0.3333333 0.3333333