Lluís Salafranca

Quantifying social asymmetric structures.

Many social phenomena involve a set of dyadic relations among agents whose actions may be dependent. Although individualistic approaches have frequently been applied to analyze social processes, these are not generally concerned with dyadic relations, nor do they deal with dependency. This article describes a mathematical procedure for analyzing dyadic interactions in a social system. The proposed method consists mainly of decomposing asymmetric data into their symmetric and skew-symmetric parts. A quantification of skew symmetry for a social system can be obtained by dividing the norm of the skew-symmetric matrix by the norm of the asymmetric matrix. This calculation makes available to researchers a quantity related to the amount of dyadic reciprocity. With regard to 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 multivariate statistical techniques.

Testing reciprocity in social interactions: a comparison between the directional consistency and skew-symmetry statistics.

In the present article, we focus on two indices that quantify directionality and skew-symmetrical patterns in social interactions as measures of social reciprocity: the directional consistency (DC) and skew-symmetry indices. Although both indices enable researchers to describe social groups, most studies require statistical inferential tests. The main aims of the present study are first, to propose an overall statistical technique for testing null hypotheses regarding social reciprocity in behavioral studies, using the DC and skew-symmetry statistics (Φ) at group level; and second, to compare both statistics in order to allow researchers to choose the optimal measure depending on the conditions. In order to allow researchers to make statistical decisions, statistical significance for both statistics has been estimated by means of a Monte Carlo simulation. Furthermore, this study will enable researchers to choose the optimal observational conditions for carrying out their research, since the power of the statistical tests has been estimated.

P-SPACE: a program for simulating spatial behavior in small groups.

P-SPACE is a computer program that simulates spatial behavior in a small group of individuals. The program describes how interpersonal distances change through time as a result of changes in microlevel features, such as the minimization of local dissatisfaction. Agents are located in a two-dimensional lattice and can move some discrete space units at each discrete time unit within their neighborhood. A nonsymmetrical matrix of ideal distances between agents must be specified. Agents move in order to minimize their dissatisfaction, defined as a function of the discrepancy between possible future distances and ideal distances between agents. At each iteration, agents will move to those cells in their neighborhoods for which the function is minimized. Depending on the specific values in the ideal-distance matrix, different kinds of social dynamics can be simulated.

Measuring and making decisions for social reciprocity

Social reciprocity may explain certain emerging psychological processes likely to be founded on dyadic relations. Although indexes and statistics have been proposed to measure and make statistical decisions regarding social reciprocity in groups, these tools were generally developed to identify association patterns rather than to quantify the discrepancies between what each individual addresses to his or her partners and what is received from those partners in return. Additionally, social researchers’ interest extends beyond measuring groups at the global level because dyadic and individual measurements are also necessary for proper descriptions of social interactions. This study is concerned with a new statistic for measuring social reciprocity at the global level and with decomposing that statistic in order to identify which dyads and individuals account for a significant part of asymmetry in social interactions. In addition to a set of indexes, some exact analytical results are derived, and a way of making statistical decisions is proposed.

Bias and standard error for social reciprocity measurements.

The directional consistency and skew-symmetry statistics have been proposed as global measure of social reciprocity. Although both measures can be useful for quantifying social reciprocity, researchers need to know whether these estimators are biased in order properly to assess descriptive results. That is, if estimators are biased, researchers should compare actual values with expected values under the specified null hypothesis. Furthermore, standard errors are needed to enable suitable assessment of discrepancies between actual and expected values. This paper aims to derive some exact and approximate expressions in order to obtain bias and standard error values for both estimators for round-robin designs, although the results can also be extended to other reciprocal designs.