How to compute the "Katz similarity" measure of regular equivalence described in Newman (2010)

Question

I would like to know if there is an R package which is able to compute the regular equivalence measure described in Newman (2010:218) as "Katz similarity":

where σ is the similarity score, A is the adjacency matrix of the graph and δ is meant to increase the "self-similarity" of diagonal elements. So far I was not able to find any specific function computing this score in R. Since the textbook also explicitly states that:

"The Katz centrality of a vertex would then be simply the sum of the Katz similarities of that vertex to all others." (Newman, 2010: 219)

I was thinking that maybe there is a way to derive the similarity score from the Katz centrality measure, but I could not find a proper way to disentangle each score in the centrality.

I'm wondering if the citation is to M. E. J. Newman. "Networks: An introduction." Oxford University Press, 2010. If so, you deserve a sharp handslap on both wrists because you should not be making us do searching for an item that you cite with insufficient detail. — IRTFM, Aug 22 '21 at 21:50
@IRTFM That is indeed the correct reference, sorry for not making it clear — Dani Mohammed, Aug 23 '21 at 12:08

IRTFM · Accepted Answer · 2021-08-22T22:02:30.310

I might be able to get part way there (although this is not my area of expertise). The answer may be:

Package ‘linkprediction’
October 19, 2018
Title Link Prediction Methods
Version 1.0-0
Description Implementations of most of the existing proximity-based methods of
link prediction in graphs. Among the 20 implemented methods are e.g.:
Adamic L. and Adar E. (2003) <doi:10.1016/S0378-8733(03)00009-1>,
Leicht E., Holme P., Newman M. (2006) <doi:10.1103/PhysRevE.73.026120>,
Zhou T. and Zhang Y (2009) <doi:10.1140/epjb/e2009-00335-8>, and
Fouss F., Pirotte A., Renders J., and Saerens M. (2007) <doi:10.1109/TKDE.2007.46>.

The Leicht E., Holme P., Newman M. (2006) <doi:10.1103/PhysRevE.73.026120> citation is behind a paywall, but there is a preprint version of it that makes me think the citation you mention is a later description of the same thing: https://arxiv.org/abs/physics/0510143

if(!require("linkprediction") ){ 
    install.packages("linkprediction", dependencies=TRUE); 
    library(linkprediction }
if(requireNamespace("igraph")) {
  g <- igraph::make_graph(~ A -- C:D:E -- B -- F -- G:H -- I)
}
# LHN
proxfun(g, method="lhn_global") $ returns matrix, possibly what your eq. described
round( proxfun(g, method="lhn_global"), 5)
        1       2       3       4       5       6       7       8       9
1 0.12648 0.04052 0.04052 0.04052 0.01199 0.00329 0.00101 0.00101 0.00038
2 0.04052 0.27352 0.02352 0.02352 0.03162 0.00867 0.00266 0.00266 0.00101
3 0.04052 0.02352 0.27352 0.02352 0.03162 0.00867 0.00266 0.00266 0.00101
4 0.04052 0.02352 0.02352 0.27352 0.03162 0.00867 0.00266 0.00266 0.00101
5 0.01199 0.03162 0.03162 0.03162 0.07439 0.02039 0.00625 0.00625 0.00237
6 0.00329 0.00867 0.00867 0.00867 0.02039 0.12603 0.03862 0.03862 0.01465
7 0.00101 0.00266 0.00266 0.00266 0.00625 0.03862 0.27152 0.02152 0.05556
8 0.00101 0.00266 0.00266 0.00266 0.00625 0.03862 0.02152 0.27152 0.05556
9 0.00038 0.00101 0.00101 0.00101 0.00237 0.01465 0.05556 0.05556 0.27107

On the other hand you may be looking for "the Katz Index" which has a citation from 1953. Also look at https://arxiv.org/pdf/2105.01931.pdf as well as at: https://dial.uclouvain.be/memoire/ucl/en/object/thesis%3A12878/datastream/PDF_01/view (which offers Matlab code for calculations.)

I had a look at the Leicht E., Holme P., Newman M. (2006) paper and it is indeed the same measure that was used in the Networks textbook. Therefore the linkprediction package does the job, great find thanks! — Dani Mohammed, Aug 23 '21 at 12:12

How to compute the "Katz similarity" measure of regular equivalence described in Newman (2010)

1 Answers1