Dal Col and Petronetto's graph regularization centrality is \(GRC_i = 1/[(I+\gamma L)^{-1}]_{ii}\), where L is the unnormalized weighted graph Laplacian. The ith column of this inverse minimizes \(\|s-e_i\|^2+\gamma s^T Ls\). A larger score indicates that smoothing retains less of a unit impulse at its source vertex. This implements the centrality with unit impulses; the author's separate signal option returns smoothed signal values and is not this centrality.
Arguments
- x
Network input accepted by
centrality.- grc_gamma
Finite nonnegative regularization strength, default one.
- ...
Additional arguments to
centrality.
Details
grc_gamma accepts any finite nonnegative number, default one.
At zero every score is one. Isolates also score one. Within a component
of n vertices scores lie between one and n, approaching n as gamma
grows without bound. Adding disconnected components does not change
existing raw scores. Edge weights and gamma act multiplicatively;
uniform weight scaling changes scores unless gamma is adjusted inversely.
Uses finite nonnegative edge weights when weighted = TRUE.
Zero weights are absent connections. Unweighted inputs use the simple
undirected skeleton. Loops are removed. For weighted directed inputs,
opposite arcs are added. The generic simplify argument combines
parallel edges first; remaining weighted parallel edges are added.
These projections are explicit cograph conventions for the published
undirected domain. Generic mode, shortest-path weight inversion
and cutoff do not affect the result.
The native dense spectral calculation separates each component's constant eigenvector and evaluates the remaining filter in log space. This supports extreme finite gamma and uniform weight scales without forming their product. Unresolvable weight ranges or positive spectral condition numbers above 1/(64 times machine epsilon) raise an error. Runtime is O(n cubed) and memory O(n squared) per component. Empty graphs return an empty vector.
The author software approximates the same filter with ten Chebyshev
terms. This function evaluates the defining inverse to numerical
precision; default author-software values need not coincide. Optional
normalized = TRUE divides scores by their global maximum.
References
Dal Col, A., & Petronetto, F. (2023). Graph regularization centrality. Physica A, 628, 129188. doi:10.1016/j.physa.2023.129188 . Author implementation: GRC, Mendeley Data, version 1. doi:10.17632/ns63f5dj86.1 .
Examples
centrality_graph_regularization(igraph::make_ring(4), grc_gamma = 0.5)
#> 1 2 3 4
#> 1.714286 1.714286 1.714286 1.714286