title: normalized local closeness centrality in R igraph::estimate_closeness()
body: <p>I am trying to calculate a normalized local closeness centrality. But setting the parameter <code>normalized = T</code> for <code>igraph::estimate_closeness()</code> only multiplies the results with (N-1). Is it possible to define <em>N_i</em> for vertex <em>i</em> individually by it's neighborhood, that is determined by the cut-off parameter (and of course by the graph itself)? </p>
  
  <p>The mini example demonstrates, that setting the parameter <code>normalized=T</code> does not work for this purpose. It normalizes with one and the same <em>(N-1)</em> for all vertices.</p>
  
  <p>Thanks for help.</p>
  
  <pre><code>set.seed(1210)
  require('igraph')
  g &lt;- random.graph.game(20,3/10)
  g &lt;- set.edge.attribute(g, "weight",  value= 1)
  cnt &lt;- estimate_closeness(g, cutoff = 3, normalized = T );cnt
  cnf &lt;- estimate_closeness(g, cutoff = 3, normalized = F );cnf
  
  #print results
  cbind(cnf * (length(V(g))-1), cnt)
  sum(abs(cnf * (length(V(g))-1) - cnt))
  
  #for visualization
  V(g)$name &lt;- paste("v", 1:length(V(g)), sep="")#letters[1:length(V(g))]
  set.seed(2)
  lay &lt;- layout.auto(g)
  
  ## set plotting parameters
  vs &lt;- 15# vertex.size
  ec &lt;- gray(0.8)#edge.color
  elx &lt;- 2# edge.label.cex
  elc &lt;- "black"#,edge.label.color
  vlc &lt;- 2#vertex.label.cex
  ew &lt;- 2#edge.width
  hd &lt;- paste(rep(" ",0), collapse="")
  cm &lt;- 3
  vc &lt;- "orange"#gray(0.8)#palette("default")#"grey"#vertex.color
  
  windows(width = 18, height=6)
  par(mfrow=c(1,3))
  
  plot(g, edge.label=paste(hd, round(E(g)$weight,1)), layout=lay, vertex.size=vs, edge.color=ec, edge.label.cex=elx, vertex.label.cex=vlc, edge.width=ew, edge.label.color=elc, vertex.color=vc)
  title("(a) Weights", cex.main = cm)
  
  V(g)$name &lt;- paste(round(cnt,3))
  plot(g, layout=lay, vertex.size=vs, edge.color=ec, edge.label.cex=elx, vertex.label.cex=vlc, edge.width=ew, edge.label.color=elc, vertex.color=vc)
  title("(b) local closeness normalized", cex.main = cm)
  
  V(g)$name &lt;- paste(round(cnf,3))
  plot(g, layout=lay, vertex.size=vs, edge.color=ec, edge.label.cex=elx, vertex.label.cex=vlc, edge.width=ew, edge.label.color=elc, vertex.color=vc)
  title("(c) local closeness", cex.main = cm)
  </code></pre>