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On Mon, 16 Jan 2006, Joshua O'Madadhain wrote:

[...]
> I'm not sure this formulation is correct, for three reasons:
> (1) I don't see why it should matter whether any two 1s are in  
> adjacent rows/columns, since the ordering of the vertices is  
> presumably arbitrary.
> (2) I don't see how these conditions guarantee that the diameter is  
> <= 2.  (One easy way to do this is to square the adjacency matrix A,  
> and check to make sure that A_ij + (A^2)_ij > 0 for all i, j such  
> that i != j.)
> (3) How do these conditions guarantee that the resultant graph is  
> minimal?

Ooops.. my mistake: Writing this I was thinking of his example where n = 6 
and degree = 3. Thus, I've constructed a third sociomatrix as a case other 
than the two he has given. However, I agree with you, in general the 
situation is much more complicated.

--Moses

> I'd be happy to find that it's simply the case that my graph theory  
> is rusty, of course...
> 
> A final question for Juno: what is the difference metric that you're  
> using?
> 
> Good luck--
> 
> Joshua O'Madadhain
> 
> >
> > Incidentally in your example (n = 6) there is a third graph that  
> > you need
> > to consider and this is the following
> >
> > n 1 2 3 4 5 6
> > 1   1     1 1
> > 2 1   1 1
> > 3   1   1   1
> > 4   1 1   1
> > 5 1     1   1
> > 6 1   1   1
> >
> > --Moses
> >
> >
> > On Mon, 16 Jan 2006, Juno Blaauw wrote:
> >
> >> *****  To join INSNA, visit http://www.insna.org  *****
> >>
> >> Hello SOCNETTERS,
> >>
> >> My name is Juno Blaauw and I am a political research student at the
> >> University of Amsterdam. I am writing this e-mail on behalf of  
> >> Meindert
> >> Fennema, Jean Tillie and myself.
> >>
> >> Meindert Fennema and Jean Tillie are studying ethnic civic  
> >> communities.
> >> Lately they have been interested (among other things) in  
> >> introducing a
> >> normative definition of a civic community and in measuring the  
> >> difference
> >> between a given empirical civic community and this normative  
> >> definition. I
> >> have been working with them on this (thesis). We have been able to  
> >> formulate
> >> a, in our eyes, satisfactory definition of a normative civic  
> >> community,
> >> which we have called an 'old boys civic community', and a measure  
> >> for the
> >> above mentioned difference. The only problem is that our normative
> >> definition does not lead to one unique social network. In other  
> >> words: on a
> >> given number of nodes many social networks qualify as old boys civic
> >> communities.
> >>
> >> Before getting to the specific question we would like to ask you all,
> >> I willfirst give you our definitions. First of all our definition  
> >> of a
> >> civic
> >> community: "a civic community [...] consists of many voluntary  
> >> associations
> >> that are related to each other by way of overlapping membership and
> >> interlocking directorates' (Fennema, 2004, p. 433.)" The voluntary
> >> associations are formal organisations and the interlocking  
> >> directorates are
> >> people that are board members of two of these organisations. Thus,  
> >> a civic
> >> community can be defined as the set of formal organisations and  
> >> the ties, in
> >> the form of interlocking directorates, between them. A civic  
> >> community that is
> >> defined like this can be depicted as a graph were the nodes
> >> representorganisations and the lines interlocking directorates.
> >>
> >> Secondly, our definition of an old boys civic community: A civic  
> >> community
> >> is an old boys civic community if and only if (a) any two  
> >> organisations are
> >> either directly linked by an interlocking directorate or have a  
> >> maximum of
> >> two interlocking directorates between them and (b) each of the
> >> civiccommunity's organisations is adjacent to the same number of
> >> otherorganisations.
> >>
> >> Now, the question we would like to ask all of you is this: how can  
> >> we find
> >> out how many and which old boys civic communities there are on a
> >> givennumber of nodes? Note that we are only interested in minimal old
> >> boys civic
> >> communities, that is in old boys civic communities with a minimal  
> >> number of
> >> interlocking directorates (ties).
> >>
> >> To illustrate our problem: Here are two minimal old boys civic  
> >> communities
> >> on six nodes. Are these the only two? If so, how can we proof  
> >> that? If not,
> >> how can we find the other ones?
> >>
> >> n 1 2 3 4 5 6
> >> 1   1   1   1
> >> 2 1   1   1
> >> 3   1   1   1
> >> 4 1   1   1
> >> 5   1   1   1
> >> 6 1   1   1
> >>
> >> and
> >>
> >> n 1 2 3 4 5 6
> >> 1   1 1     1
> >> 2 1   1   1
> >> 3 1 1   1
> >> 4     1   1 1
> >> 5   1   1   1
> >> 6 1     1 1
> >>
> >> We have no idea whether or not this is actually an easy problem  
> >> that has
> >> already been solved by somebody else. We are just hoping that you  
> >> can inform
> >> us of anything you seem fit, given what we have told you so far.
> >>
> >>  Please let me know if you need any additional information. Thank  
> >> you in
> >> advance!
> 
> jmadden@ics.uci.edu...Obscurium Per Obscurius...www.ics.uci.edu/~jmadden
> Joshua O'Madadhain: Information Scientist, Musician, and Philosopher- 
> At-Tall
>    It's that moment of dawning comprehension that I live for--Bill  
> Watterson
>    My opinions are too rational and insightful to be those of any  
> organization.
> 
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