1.
Introduction
One of the best known methods in social network analysis and research in group dynamics is the use of so called socio matrices or Moreno matrices (Moreno 1934; Freeman 1989). A socio matrix represents relations between the different group members, e.g., positive and negative attitudes or the frequency of social interactions. Given a group of three members A,B and C for example, then a socio matrix of this group may be
|
|
A |
B |
C |
|
A |
1 |
1 |
1 |
|
B |
0 |
1 |
0 |
|
C |
0 |
0 |
0 |
which could represent the different attitudes: A likes
himself and the others, B does not like A and C but only himself and C does not
like anyone, including himself. Accordingly one may construct a socio matrix
that measures the interactions. In our little example such a matrix could be
|
|
A |
B |
C |
|
A |
0 |
4 |
5 |
|
B |
4 |
0 |
1 |
|
C |
5 |
1 |
0 |
the zeroes mean, of course, that one does not interact
with oneself.
Usually socio
matrices are used to describe only the structure of a group, for instance when analysing
the emotional relations between students in a class room. Because some years
ago we systematically investigated the dynamics of a social system in
dependency of its structure (Klüver 2000; Klüver and Schmidt 1999) we analysed
the dynamics of social groups in dependency of the structure that is
represented in its particular Moreno matrix. In other words, we took a certain
socio matrix as the basis for an artificial model and looked for the dynamics
of the according group; the dynamics of each model is generated by locally
operating rules of interaction.
Because we were
aware that often results obtained by experiments with computer models may be
only artifacts, due to particular characteristics of the model, we used different
models and compared their respective results. In addition we did some group
experiments with students at an academy at Dortmund and tried to predict the
outcome with our models. Finally, one of our students, Dominik Kalisch,
accomplished a systematic observation of children in a summer camp at the North
West of France for two weeks and compared his data with the prediction of one
of our models.
The three models
we used are (1) a cellular automaton (CA) with rules of interaction based on
the feelings of group members towards the other members, (2) a Kohonen feature
map (KFM), i.e., a self organising artificial neural net, and (3) an
interactive neural net (IN), that is a recurrent network that does not learn -
in contrast to the KFM.
2.1.
The Moreno CA
CA are discrete system consisting of artificial cells;
the cells are in particular states that change according to the rules of
interaction in dependency of the states of the cells in the
"neighbourhood" of a particular cell. As the cells of a CA are usually
defined as squares on a grid, the neighbourhood of a cell is mostly the von
Neumann neighbourhood, i.e. the four adjacent cells at the sides of the square,
or the Moore neighbourhood, that is the eight adjacent cells at the sides and
at the corners. CA are logically equivalent to universal Turing machines which
means that it is possible to model each type of complex systems with a suited
CA (cf. Wolfram 2002; Klüver 2000).
The rules of our
"Moreno CA" are roughly as follows: an artificial member, represented
by a cell, is in a positive or negative emotional state dependent on the
neighbours; he either likes them or
does not like them or he is indifferent towards them. The artificial member is
in a positive mood if he is surrounded by more people he likes than by people
he does not like or feels indifferent to; he is in other emotional states if
there are more people in the neighbourhood he does not like than those he likes
etc.
If a member is in
a positive emotional state, he will stay where he is. If he is not, then he
looks in the extended Moore neighbourhood - or even around the whole grid of
the CA - if there are members he likes better than his neighbours. If that is
the case he/she "moves", i.e., leaves his place and moves into the
direction of the members he likes better. If he/she cannot find such members,
he/she moves at random .
The
sociopsychological assumption behind these rules is of course that people tend
to gather with other people they like rather well or at least they do not
dislike. Because this assumption is a truism, known from everyday experience,
the rules of our CA seem to be quite realistic. Conversely, if it is not
possible for certain group members to associate themselves with members they
like, then they will feel unhappy and try to go away.
Experiments done with this CA demonstrated that the CA
nearly always generated point attractors very soon regardless with which
initial states the runs started. The length of the preperiods of the attractors
depended mainly on the size of the "looking area": if the group
members could observe all other group members then they tried to move into the
best neighbourhood the whole group could offer; accordingly many runs (usually
up to 15 time steps with a group of 30 members) were needed to obtain an
attractor, i.e., a group state where no member moved any longer. If on the
other hand each member only could analyse its Moore neighbourhood, then the
members had to be content with local optima and accepted suboptimal
neighbourhoods: they could not perceive that better neighbourhoods would be
possible with group members they could not "see".
Only in one case
the preperiods became rather long and sometimes no point attractor could be
reached during many runs. That was the case when the Moreno matrix was
deliberately constructed totally asymmetrical: if A likes B, then B does not
like A and this way for all members. It is intuitively plausible and our
CA-experiments confirmed it that in this case A will go after B but B will go
away from A and so on. Therefore no point attractor can be reached in this case
but only attractors with periods significantly larger than 1. The Moreno CA
seems to be a realistic model of the dynamics of real groups and capable of
predicting group dynamics.
2.2.
The Moreno KFM
A Kohonen feature map is an artificial neural net that
learns "non supervised", i.e., the learning processes of a KFM are
directed by the criterion to cluster neural groups according to their
similarity. A KFM learns roughly speaking by obtaining certain information
about "concepts" and clustering these concepts according to the
degree of equality in regard to the conceptual information. An illustrating
example of the operations of a KFM is the ordering of animal concepts according
to information like "flesh eating", " having feathers",
having four legs" and so on (Ritter and Kohonen 1989; Stoica 2000). One
may say that KFM learn according to a criterion of topological nearness.
In our
experiments we took the group members as "concepts" and the emotional
relations between them as information about the concepts. Therefore the KFM had
to cluster the members according to their mutual feelings, i.e., the KFM had to
construct subgroups or neighbourhoods of members in dependency of the
information represented in the respective Moreno matrix. We skip for the sake
of brevity the particular algorithms of a typical KFM; each text book on neural
nets deals with KFM too.
Our experiments
demonstrated that the KFM also obtained point attractors in practically all
cases although the KFM needed more runs than the CA. The reason for this is not
only due to the fact that the algorithms of a KFM are more complex than those
of a CA because a KFM is basically a learning system - in contrast to the CA -
but also that the KFM always uses the
whole Moreno matrix to obtain the global optimum for its task. Remember, the CA
in most experiments only uses that part of the Moreno matrix that is needed for
a Moore neighbourhood or an extended Moore neighbourhood (the 28 adjacent
cells). Because of this principal difference between the two models one can
expect that their respective results will not
always be equal. It is of course possible to restrict the operations of a
KFM too by letting it use only certain parts of the Moreno matrix but because
we experimented only with rather small groups (see below) we did not perform
such experiments.
2.3.
The Moreno IN
An interactive network (IN) is a recurrent network
that does not learn. It is often used for the modelling of logical relations
between different parts of a system, i.e., social classes, propositional parts
of a text and the like (Stoica 2000). The activation rules of an IN are the
well known linear activation rules of neural nets, i.e. the state of an
artificial neuron is a result of the "weighted" sum of the inputs the
neuron gets from other neurons. Usually all neurons in an IN are directly
connected with all others although that must not be the case. The
"quality" of these connections, i.e., the strength of the influence
the neurons exert on each other, is represented in a "weight matrix"
(wij): wij = 0.5 means that the neuron j gets an input by
neuron i that is the output of i multiplied by 0.5.
For our Moreno
experiments we defined the socio matrix as the weight matrix of an IN: 0.5 means
in this representation that member i has a neutral feeling in regard to member
j, measured on a 10-scale from 0 to 1. The states of the neurons again
represent the emotional states of the members. Our experiments with an IN do
not predict the ordering of members into subgroups according to their mutual
feelings as is the case with the CA and the KFM. IN-experiments simulate the
emotional outcome of a certain group structure, i.e., the result of IN-runs
with a particular Moreno matrix are the emotional states of the members in regard to the whole group. In
contrast to this, the emotional states as results of the CA- and
KFM-experiments are dependent on the particular subgroups the different members
have formed (Stoica 2000).
3.
Results of the comparisons of the models and of empirical observations
In a direct sense model comparison is possible only
between the CA results and those of the KFM- experiments because they simulate
the same process of the forming of subgroups. Despite the mentioned fact that
the CA only in some experiments used information about the whole group for each member the results of the
predictions of the two models were in most cases rather similar or even equal
The results significantly differed, as was to be expected, only when the whole
group was rather large, i.e., larger than 30 members, and the initial states of
the CA, i.e., the randomly generated position of the cells on the grid, made it
impossible for the cells to perceive other cells with better opportunities for
them.
In those cases
where the CA used the whole Moreno matrix for each member as the KFM always
does, the results were practically always equal or differed only in small
degrees. These experiments were always done with Moreno matrices generated at
random. Because of the similarities or even equality of these results, despite
the very different algorithms of the models and the different use of the Moreno
matrix, we may safely assume that the results of the particular models are not
simply artifacts.
More interesting still than a model comparison is of course the comparison of the prediction of the models with empirical observations. The first experiment was done, as we mentioned above, with a group of students. The students had to construct the Moreno matrix of their class group themselves, i.e., give the information about their emotional relations to the other students to one of us and then they were asked to move into another class room, choose a sitting place and select their respective neighbours. Before they did this we gave the Moreno matrix as input to the CA and to the KFM and let them predict the sitting order.
To the
astonishment of the students (and to our satisfaction) the KFM exactly
predicted the new sitting order and so did the CA. The CA only erred in regard
to the "absolute" position of the students in the room, i.e. if they
were seated in the first row of the room or another row, in the middle of the
room or at the windows etc. But that of course could not be predicted; the CA
just had to choose some absolute physical order. The KFM does that not but
takes into regard only the emotional relations, i.e., the KFM just forms
subgroups without predicting their distribution in the physical space. To put
it into a nutshell, obviously our models are valid in the sense that they are
able to predict the outcome of certain social processes (although of course
rather simple ones) and in particular they did it with the same success.
The IN, as was
mentioned above, predicts the feelings of group members in regard to the whole group. Accordingly we asked the students to
express their feelings in this respect. Again the model rather exactly
predicted the outcome which the students acknowledged and confirmed. This model
seems to be valid too.
The second
experiment in this context was done by observations of a group of children in a
summer camp during two weeks. Dominik Kalisch, our student, asked before the
camp all children about their mutual feeling towards the other children and in
addition about their willingness to interact with children they did not know:
He had to ask this because in the beginning the children who came from
different towns in Western Germany did know only the children from the same
town. The student then constructed a Moreno matrix of the whole group in the
usual manner. In the cases where child A did not know child B the degree of
willingness of child A to interact with strangers was inserted in the matrix at
position (A,B) and accordingly the degree of willingness of child B was
inserted in position (B,A).
The student then
observed the interactions of the children (if he could not observe himself he
asked the teachers in the camp to do this). At the end he had a partition of
the whole group into subgroups with respect to the frequency of interactions
between the children. Finally the student inserted the Moreno matrix in the
Moreno CA and compared the results of the CA-runs with his observations.
Despite the methodical problem that the children did not know all other
children in the beginning the results were nearly the same.
The student also
asked the children about the emotional feelings in regard to the whole group
and compared his data with the results of an IN. Again the results were very
similar - a fact that is astonishing because children usually are not able to
express their feeling in an exact, i.e. quantifiable manner. The IN - and the
CA - seem to be rather robust with respect to empirical fuzzyness.
Is there a reason for the equivalent results of models with different algorithms? In our opinion the explanation for this equivalence of the model is due to the fact that all the models are constructed in the same methodical and theoretical way: the basis contains always rules of interactions and empirical data taken "directly" from empirical experiences, like the CA-rules and emotional states mentioned above, and these data and rules are taken as the logical frame for the models. In this sense that has been called "Soft Computing" by the founder of fuzzy set theory Zadeh, the models are methodically equivalent; that is on the one hand the strength of those models (cf. Klüver 2000) and on the other hand the reason for their largely equivalent results. These considerations were confirmed by other experiments where we combined a hybrid CA, i.e. coupled with a genetic algorithm (GA), with a hybrid IN that was coupled with a GA too. The two hybrid systems mutually exchanged the results of their respective optimising processes and obtained this way better results than each system obtained alone (Stoica 2000; Klüver 2000). That was possible only because the two hybrid systems are methodically equivalent in the manner just described despite their very different algorithms.
To be sure, group processes like those discussed here
are rather simple and that is why we chose them to check our models - in
comparison to each other and to empirical observations. But theoretical
reflections on the characteristics of these models and enlarged experiments
(Klüver 2000 and 2002; Stoica 2000) give reason to assume that one can capture
much more complex social processes with models of this type. Further results in
this direction will be reported in due time.
The models we
discussed can be obtained from us: juergen.kluever@uni-essen.de
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