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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!

Juno Blaauw.

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