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Dear Socnetters

I used photos of 7 young women in my very old empirical study of
structural balance in signed graphs.

Tn that study 45 female students were informed that the women in the
pictures marked A,B,C,D,E,F,G  work together (actually, I found the photos
of anonymous persons in a weekly newspaper for women; the readers were
invited to assess for fun psychological traits of people from their
photos)

Each student obtained the set of 7 cards with the photos and a form in
which a graphical scale looking more or less like this

|_---__--__-__|__+__++__+++_|

was printed next to each of 21 pairs of symbols AB, AC,…., FG (the pairs
were listed in a random order rather than lexicographically). The
subject’s task for each pair was to pick the respective two pictures from
the set, look at them  and guess the sign (but putting a tick on the left
or right segment  of the scale), of an emotional tie that must have arisen
between  two women when they had to meet face-to-face at work.

Having  ignored the intensity of liking/disliking, I obtained the set of
45 7-point complete signed graphs. My aim was to test if there is a
tendency for  structural balance in the data.  Concretely, I wanted to
check if the mean number of negative triangles in the empirical
structures was significantly lower than in randomly generated structures
(the probability of + in any pair being equal to ˝  and under independent
assignment of signs to pairs).

At the second step of  my study I asked each subject to guess the sign of
her relationship with women A….G, if she had to join the group as its 8th
member H. My aim was to test if the degree of imbalance in the 7-point
structure depends on the degree of imbalance in triangles containing H (as
the ‘focal person’ as Heider would say). This problem was inspired by a
theorem by Claude Flament (the author of ‘'Applications of Graph Theory to
Group Structure'’, 1963). He proved that any n-point complete signed graph
is balanced  (that is, all triangles are positive) iff  all triangles
containing one node are positive.

I wonder if any similar study has ever been done. My own research remains
unpublished. It was reported only in a section of  my Ph.D thesis (1982,
in Polish). I lost my interest in empirical applications of signed graphs,
when I plunged totally into mathematical theory of signed graphs (see my
papers in Mathematiques et Sciences Humaines 1976, in French,  and in the
J. of Graph Theory, 1980).

With best regards