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Thanks, Ian!

What I'm trying to do is NOT:

* to compare separate centrality indices to each other either to
benchmark them or to assess which centrality index performs better
than some other
* or locate a super node (according to separate indices or all of them
lumped together).

What I'm trying to do is:

1. To RANK all network nodes according to each separate centrality
index, obtaining K rankings of nodes (where K is the number of
centrality indices with regards to which I'm measuring the importance
of nodes). Of course, each ranking has a super/top node but I'm not
interested in it. I'm interested in the relative positions, in which
each node is placed in the order for every centrality ranking among
the K ones.
2. To extract a "meta-ranking" from the above K rankings as if each
nodal ranking was a "ranked choice/vote" that (metaphorically) that
each centrality index was casting in a preferential ballot. Of course,
the individual "vote" of each centrality index would have one or more
super/top nodes according to the importance attributed by this
centrality index. However, in preferential ballots this is not the way
that winners are counted. In Social Choice theory, there are many
different ways which count the winning outcomes always as rankings
(see for example
In my case, since each centrality index produces a ranking, I am
trying to find a winning meta-ranking among all the individual

Why would one want to do this?

Because by computing "distances" between the overall meta-ranking and
a partial subset of rankings (corresponding to a smaller subfamily of
centrality indices), one might say which ones among the centrality
indices (actually among a potentially large number of them that
nowadays can be easily computed) really matters and which ones are in
fact redundant (and it doesn't make any sense that one would bother to
include them in one's network computations, always for a particular
social network). To give a hypothetical example: Why should one bother
to compute both, let's say, degree, closeness, betweenness, PageRank
and eigenvector centralities, when the meta-ranking wouldn't change
considerably if, let's say, eigenvector or closeness centrality was
omitted? Of course, an obvious objection might be that this method is
only a posteriori useful, but think of the case that one would
subsequently need to process a number of computationally-demanding
statistical models, which would be groundlessly loaded by including a
number of redundant (in the above sense) centrality indices.

That was supposed to be my 0.02 bitcoins!

On Tue, Apr 3, 2018 at 1:34 PM, McCulloh, Ian A.
<[log in to unmask]> wrote:
> Moses,
> Thanks for sharing and maintaining a blog.
> I often find differences in centrality more interesting. One common approach
> I use is to plot two measures against each other, such as degree and
> betweenness. Centrality measures tend to be correlated (on my phone or I’d
> share refs, but there are pubs on this by me, Valente, Contractor). So, when
> you see a node stand out that is high in betweenness and low in degree, you
> have evidence of structural holes.
> I think of betweenness, closeness, degree, much like median, mean, and mode.
> All three (six actually) are measures of center. All tell you something a
> bit different. The differences are interesting (eg income median vs mean).
> The comparisons are really fascinating.
> So, I’d encourage you to think what do centrality patterns tell us about the
> structure than to locate a super node. Just my $0.02.
> Ian
> Ian McCulloh, PhD
> Johns Hopkins University
> 240-506-3417
> From: Moses Boudourides <[log in to unmask]>
> Date: Sunday, Apr 01, 2018, 5:20 PM
> To: [log in to unmask] <[log in to unmask]>
> Subject: [SOCNET] Cumulative Rank Aggregation of a Family of Network
> Centrality Indices
> *****  To join INSNA, visit  *****
> Hello everybody,
> You might be interested in the following brief report "Cumulative Rank
> Aggregation of a Family of Network Centrality Indices" that  I've just
> entered in my Medium blog:
> A growing number of centrality indices are used today in social
> network analysis. The purpose of using all these network centrality
> measures is that through them one might be able to identify the most
> important nodes according to a variety of structural criteria (like
> nodal degree, closeness, betweenness, eigenvector, PageRank etc.).
> Moreover, computations (in Python, R etc. or standalone applications)
> may very easily derive the tables of various centrality indices of
> network nodes. Therefore, knowing a good deal of network nodal
> centralities, the crucial question would be how to make sense for all
> such indices in a illuminating way that would account for the
> structural features that an empirical network exhibits. What I am
> proposing here is a methodology for a cumulative ranking of network
> nodes according to the scores that each node possesses, not on a
> single centrality measure, but on a whole group (a family) of
> centrality measures.
> Any remarks, corrections, comments, suggestions etc are more than welcomed.
> Best,
> --Moses
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