Majorization gap
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)
g |
An igraph object |
norm |
|
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≥q k-1 \} | + | \{ i : i>k \land d_i≥q k \} |.
the majorization gap is then defined as
1/2 ∑_{k=1}^n \max\{d'_k - d_k,0\}
The higher the value, the further away is a graph to be a threshold graph.
Majorization gap of an undirected graph.
David Schoch
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.
library(igraph) g <- graph.star(5,'undirected') majorization_gap(g) #0 since star graphs are threshold graphs g <- sample_gnp(100,0.15) majorization_gap(g,norm=TRUE) #fraction of reverse unit transformation majorization_gap(g,norm=FALSE) #number of reverse unit transformation
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