Calculates the (normalized) majorization gap of an undirected graph. The majorization gap indicates how far the degree sequence of a graph is from a degree sequence of a threshold_graph.

majorization_gap(g, norm = TRUE)

Arguments

g

An igraph object

norm

True (Default) if the normalized majorization gap should be returned.

Value

Majorization gap of an undirected graph.

Details

The distance is measured by the number of reverse unit transformations necessary to turn the degree sequence into a threshold sequence. First, the corrected conjugated degree sequence d' is calculated from the degree sequence d as follows: $$d'_k= |\{ i : i<k \land d_i\geq k-1 \} | + | \{ i : i>k \land d_i\geq k \} |.$$ the majorization gap is then defined as $$1/2 \sum_{k=1}^n \max\{d'_k - d_k,0\}$$ The higher the value, the further away is a graph to be a threshold graph.

References

Schoch, D., Valente, T. W. and Brandes, U., 2017. Correlations among centrality indices and a class of uniquely ranked graphs. Social Networks 50, 46–54.

Arikati, S.R. and Peled, U.N., 1994. Degree sequences and majorization. Linear Algebra and its Applications, 199, 179-211.

Examples

library(igraph) g <- graph.star(5,'undirected') majorization_gap(g) #0 since star graphs are threshold graphs
#> [1] 0
g <- sample_gnp(100,0.15) majorization_gap(g,norm=TRUE) #fraction of reverse unit transformation
#> [1] 0.7274
majorization_gap(g,norm=FALSE) #number of reverse unit transformation
#> [1] 547