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. Landau’s 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 , 5^th International Conference on Methods and Techniques in Behavioral Research, 30 August - 2 September 2005, Wageningen, The Netherlands.

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