Dynamical Importance

Dynamical importance (Restrepo et al. 2006) is the relative drop in the spectral radius $\rho$ of the adjacency matrix when the node is removed: $$I_i = \frac{\rho(A) - \rho(A_{-i})}{\rho(A)}.$$ The spectral radius is recomputed after each deletion, in place of the eigenvector approximation of the paper (eq. 5).

Usage

centrality_dynamical_importance(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality. The measure uses weighted (default TRUE) and normalized (default FALSE).

Value

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

Details

$A$ holds the edge weights, or ones with weighted = FALSE, and self-loops are removed. Negative or non-finite weights raise an error. The scores lie between 0 and 1 and are unchanged when every arc is reversed, so mode has no effect. A network with spectral radius 0, such as any directed acyclic network, gives NaN for every node with a cograph_undefined_measure warning. An isolated node in a network with positive spectral radius scores 0.

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(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.12106302 0.09396942 0.19308255 0.16302319 0.03448233 0.07603592 0.02203123 
#>   Evaluate     Create      Share 
#> 0.09419196 0.20502778 0.23434006