centrality_graph_regularization: Graph regularization centrality

In cograph: Analysis and Visualization of Complex Networks

Graph regularization centrality

Description

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.

Usage

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

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.

Value

Named numeric vector in input node order.

References

Dal Col, A., & Petronetto, F. (2023). Graph regularization centrality. Physica A, 628, 129188. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.physa.2023.129188")}.

Dal Col, A. (2023). GRC. Mendeley Data, version 1. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.17632/ns63f5dj86.1")}.

Examples

centrality_graph_regularization(igraph::make_ring(4), grc_gamma = 0.5)

cograph documentation built on Sept. 30, 2026, 5:08 p.m.