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.
centrality_dynamical_importance(x, ...)Named numeric vector in input node order.
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.
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.
Restrepo, J. G., Ott, E., & Hunt, B. R. (2006). Characterizing the Dynamical Importance of Network Nodes and Links. Physical Review Letters, 97, 094102. tools:::Rd_expr_doi("10.1103/PhysRevLett.97.094102").
centrality_dynamical_importance(igraph::make_full_graph(4))
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