<p>I might be able to get part way there (although this is not my area of expertise). The answer may be:</p>
<pre><code>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) &lt;doi:10.1016/S0378-8733(03)00009-1&gt;,
Leicht E., Holme P., Newman M. (2006) &lt;doi:10.1103/PhysRevE.73.026120&gt;,
Zhou T. and Zhang Y (2009) &lt;doi:10.1140/epjb/e2009-00335-8&gt;, and
Fouss F., Pirotte A., Renders J., and Saerens M. (2007) &lt;doi:10.1109/TKDE.2007.46&gt;.
</code></pre>
<p>The <code>Leicht E., Holme P., Newman M. (2006) &lt;doi:10.1103/PhysRevE.73.026120&gt;</code> 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: <a href="https://arxiv.org/abs/physics/0510143" rel="nofollow noreferrer">https://arxiv.org/abs/physics/0510143</a></p>
<pre><code>if(!require(&quot;linkprediction&quot;) ){ 
    install.packages(&quot;linkprediction&quot;, dependencies=TRUE); 
    library(linkprediction }
if(requireNamespace(&quot;igraph&quot;)) {
  g &lt;- igraph::make_graph(~ A -- C:D:E -- B -- F -- G:H -- I)
}
# LHN
proxfun(g, method=&quot;lhn_global&quot;) $ returns matrix, possibly what your eq. described
round( proxfun(g, method=&quot;lhn_global&quot;), 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
</code></pre>
<p>On the other hand you may be looking for &quot;the Katz Index&quot; which has a citation from 1953. Also look at <a href="https://arxiv.org/pdf/2105.01931.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/2105.01931.pdf</a> as well as at: <a href="https://dial.uclouvain.be/memoire/ucl/en/object/thesis%3A12878/datastream/PDF_01/view" rel="nofollow noreferrer">https://dial.uclouvain.be/memoire/ucl/en/object/thesis%3A12878/datastream/PDF_01/view</a>  (which offers Matlab code for calculations.)</p>
How to compute the "Katz similarity" measure of regular equivalence described in Newman (2010)
<p>I would like to know if there is an R package which is able to compute the <em>regular equivalence</em> measure described in Newman (2010:218) as &quot;Katz <em>similarity</em>&quot;:</p>
<p><a href="https://i.stack.imgur.com/86oEb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/86oEb.png" alt="enter image description here" /></a></p>
<p>where σ is the similarity score, A is the adjacency matrix of the graph and δ is meant to increase the &quot;self-similarity&quot; 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:</p>
<blockquote>
<p>&quot;The Katz centrality of a vertex would then be simply
the sum of the Katz similarities of that vertex to all others.&quot; (Newman, 2010: 219)</p>
</blockquote>
<p>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.</p>