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In a forthcoming article in CMOT, Malcolm Alexander and I propose a
bipartite clustering coefficient in the context of interlocking
directorates. In a 1-mode graph, one method to measure clustering is the
ratio of the number of triangles in the graph to the number of
2-paths. Our bipartite coefficient is analogous: it is the ratio of the
number of "closed 4-paths" in the bipartite graph to the number of
3-paths, where a closed 4-path comprises 4 edges on 4-nodes. Of course,
this bipartite configuration translates as a multiple connection between
nodes in the 1-mode graphs derived from the bipartite graph.
In 1-mode networks of relatively low density with a fixed number of edges,
many triangles can only form at the expense of possibly disconnecting the
network. In other words, localized closure of 2-paths into triangles (high
clustering) may be at the cost of more global connectivity. Analogously
for a bipartite graph, if the bipartite clustering coefficient is high,
then there is closure of many 3-paths. So for two bipartite graphs with
similar numbers of edges, we expect the graph with the higher bipartite
clustering ratio will show lower levels of global connectivity. In our
study, we found much higher values than expected of bipartite clustering
(ie compared to various baseline distributions), indicating that shared
memberships of different company boards by pairs of directors was a major
feature of the data, with potential repercussions for connectivity across
the bipartite system.
>Giovanni Roberto Ruffini wrote:
>I apologize if this is a hopelessly vague question. I am having a rather
>hard time articulating it, and know that I have tried to do so with some
>of you unsuccessfully in the past.
>I am exploring the utility of the concept of clustering coefficients in
>analyzing the social connections of an ancient Egyptian village. But the
>connections I am working with are ones I have derived by running an
>affiliations function on a two-mode network, thus turning indirect
>connections (person->legal document->second person) into direct ones.
>I would like to use the (exceptionally high) clustering coefficient of
>this derived one-mode network to tell me something about the extent to
>which this village was ordered at the group level, by guild, by
>peer-group, etc. But it starts to occur to me that I cannot escape the
>distorting lense of the (now removed) texts linking person 1 to person 2.
>In other words, isn't the clustering coefficient in this case nothing but
>a measure of how much the names in each text overlap? So, in that sense,
>it tells us nothing about the society's structure itself, and everything
>about the clustering of the evidence for it.
>Am I understanding this correctly? Should I despair? Or is the clustering
>coefficient still an interesting number, even in light of this distorting
>problem? If so, how?
>I have looked at Watts _JAS_ 1999 fruitfully, although I am alarmed at
>the prospect of calling my Egyptians connected cavemen! :) What I think I
>need next is a way to be sure I'm understanding what I've read, and can
>put it in appropriately concrete (social, textual, methodological) terms.
>Thanks for your thoughts!
Dr Garry Robins
Department of Psychology
School of Behavioural Science
The University of Melbourne
Tel: 61 3 8344 6372
Fax: 61 3 9347 6618
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