Quantifying social asymmetric structures
A. Solanas1, Ll. Salafranca1, C. Riba1, V. Sierra2 and D. Leiva1
1Department of Behavioral methodology, University of Barcelona,
Barcelona, Spain
2Quatitative and Management Methods Department, ESADE-Ramon Llull University,
Barcelona, Spain
This paper describes a mathematical procedure to analyze dyadic interactions
among agents
in a social system. The proposed technique mainly consists of decomposing
asymmetric
data into their symmetrical and skew-symmetrical parts. The relevant data
that describe
agents interactions are organized in the form of a matrix, which
is generally asymmetric.
The asymmetric matrix is decomposed into two matrices, one of them corresponding
to its symmetrical part and the other to its skew-symmetrical part. A
quantification of skewsymmetry for a social system can be obtained by
dividing the norm of the skew-symmetrical matrix by the norm of the asymmetric
matrix. This calculation makes available to researchers a quantity related
to the dyadic reciprocity in the social system. Complementarily, an index
to quantify the symmetry can also be computed.
If an adequate kind of behavior is recorded, the index of skew-symmetry
must be related
to dominance measurements. Landaus index of dominance requires each
agent of a
dyad to be categorized as dominating or not dominating. This transformation
suppresses
quantitative information and does not take into account the numeric difference
between
two-way measurements in a dyad. The index of skew-symmetry is calculated
from interval,
ratio and absolute scales of measurement, and no previous categorization
is needed.
Regarding agents, the procedure enables researchers to identify those
whose behavior is
asymmetric with respect to all agents. It is also possible to derive symmetric
measurements
among agents and to use a multivariate statistical technique, such as
multidimensional
scaling, to extract latent dimensions which can help researchers to understand
the
underlying social structure.
The paper will include an explanation of the mathematical concerns of
the technique and
two examples, one of them related to dominance studies and the other focused
on dyadic
reciprocity. Both examples will illustrate the computational aspects and
the interpretation
of the results.
Paper presented
at Measuring Behavior 2005
, 5th International Conference on Methods and Techniques
in Behavioral Research, 30 August - 2 September 2005, Wageningen, The
Netherlands.
© 2005 Noldus
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