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.
Arguments
- g
An igraph object
- norm
True(Default) if the normalized majorization gap should be returned.
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] 0.7260459
majorization_gap(g, norm = FALSE) # number of reverse unit transformation
#> [1] 538