Graph Regularization Centrality

Graph regularization centrality (Dal Col and Petronetto 2023) is the reciprocal of the diagonal of the inverse regularized Laplacian, where $L$ is the weighted Laplacian of the undirected network and $\gamma$ is grc_gamma: $$GRC_i = \frac{1}{\left[(I + \gamma L)^{-1}\right]_{ii}}.$$ A larger score means that Laplacian smoothing retains less of a unit impulse placed at the node.

Usage

centrality_graph_regularization(x, grc_gamma = 1, ...)

Arguments

x: Network input accepted by centrality.
grc_gamma: Regularization strength, a finite nonnegative number. Default 1.
...: 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 measure is computed on the undirected network with loops removed. For a directed network the weights of opposite arcs are added, and with weighted = FALSE the simple undirected skeleton is used. Edge weights must be finite and nonnegative, and a zero weight is an absent edge. At grc_gamma = 0 every score is one, an isolated node always scores one, and within a component of $n$ nodes the scores lie between one and $n$. A negative or nonfinite grc_gamma, or a weight range beyond double precision, raises an error. The author software (Dal Col 2023) approximates the same filter with ten Chebyshev terms, so its values can differ from the exact inverse computed here.

References

Dal Col, A., & Petronetto, F. (2023). Graph regularization centrality. Physica A, 628, 129188. doi:10.1016/j.physa.2023.129188 .

Dal Col, A. (2023). GRC. Mendeley Data, version 1. doi:10.17632/ns63f5dj86.1 .

Examples

centrality_graph_regularization(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   2.144003   2.475617   2.454605   2.429099   2.112217   2.246126   1.659057 
#>   Evaluate     Create      Share 
#>   2.339699   2.300689   2.499953