Re: [igraph] mismatched closeness values in R

[Ju-Sung (Jay) Lee]

> On Fri, Dec 20, 2013 at 1:05 PM, Ju-Sung Lee <address@hidden> wrote:
> > The results igraph R's closeness() and centralization.closeness()$res are
> > different.  It appears closeness() uses the sum of shortest path lengths
> > whereas centralization.closeness() uses the average (before the 
> > inversion).
> > Was this intentional?
>
> I guess so. Centralization means maximization imho, and if you invert 
> the average distance, then you would need to minimize. But I agree I can 
> be confusing.

If by maximizing centralization, you mean how we use centrality, then I 
agree: higher value should indicate more of the centrality.  However, both 
functions accomplish this:  inversion of sum of shortest paths vs 
inversion of average of shortest paths.  When I wrote earlier, I left out 
that closeness() also performs an inversion (but on the sum of shortest 
paths). So, I'm not sure I agree with your second clause: "if you invert 
the average distance, then you would need to minimize". Also, the outputs 
of the two functions are merely off by a multiplicative factor of (n-1) 
and correlate 100%.  It's just that I expected the outputs to be identical 
but they're off by a factor of (n-1).

> > Also, the documentation is confusing.  It states the function employs 
> > the average length of shortest paths (which is what 
> > centralization.closeness() uses) but the formula does not reflect this 
> > (but closeness() is consistent with formula).
>
> If _does_ employ the average length of shortest paths. And then it
> inverts it, AFAIR.

The centralization.closeness() does that, but according to my 
calculations, closeness() employs inversion of *sum* of shortest path 
lengths.  Also, in various reports of the closeness centrality formula 
across the 'net, people seem to report both versions of the formula. 
Here's how you can confirm that the output are off by a factor of (n-1) 
(i.e., the averaging factor):

> centralization.closeness(graph.star(5,mode="undirected"))$res
[1] 1.0000000 0.5714286 0.5714286 0.5714286 0.5714286
> closeness(graph.star(5,mode="undirected"))
[1] 0.2500000 0.1428571 0.1428571 0.1428571 0.1428571

Also, when I said 'formula does not reflect this', I meant the formula as 
written in the help documentation does not reflect the averaging. From the 
igraph documentation:

"The closeness centrality of a vertex is defined by the inverse of the 
average length of the shortest paths to/from all the other vertices in the 
graph:

1/sum( d(v,i), i != v)"

If the formula was to reflect the text above it, it should be written as:
"(n-1)/sum (d(v,i), i != v)"

> > This was found under igraph_0.6.5-2 (I haven't upgraded my R yet so I 
> > don't know if this was fixed in igraph_0.6.6 but the README doesn't 
> > report any of this).
>
> Which README is this?

Sorry, I mean the NEWS file in the igraph_0.6.6 source package.  I also 
just managed to load igraph_0.6.6 and the closeness output disparity is 
still there.  FYI, for some reason update.packages('igraph') didn't do 
anything so I had to exit my R session and re-install the package; I don't 
know whether this is an issue with igraph, R, or both, but other people 
seem to have had this problem (with other packages).

Jay