A Bifurcation Model of Change in Group Structure

Abstract

A new topological approach is proposed for the analysis of change in group structure over time. Special force fields are defined operationally to explain changes in the structure, which are estimated by a method called DYNASCAL, given a set of longitudinal dissimilarity judgements such as longitudinal sociograms.These fields are presumably generated by interactions among individuals.For theoretical simplicity, these fields are supposed to be two-dimensional vector fields, which are described by a system of general nonlinear differential equations.Qualitative theories of the dynamical system such as those of singularities, bifurcations, and structural stability of the vector field are shown to be useful for describibg qualitative aspects of the force fields. Some difficulties in application of our method to empirical data are indicated and ways of overcoming them are discussed.

A Brief Survey of Asymmetric MDS and Some Open Problems

“…On the other hand, a DYNAmical System SCALing method (abbreviated as DYNASCAL) proposed by Chino and Nakagawa (1990), estimates the vector field at each point in time (which is not restricted to the gradient field of the scalar potentials), given a set of longitudinal asymmetric relational data matrices, such as longitudinal attraction data which were gathered by Newcomb (1961). In this method, the configuration of members of a dormitory at each time is estimated from each of the longitudinal asymmetric data matrices via MULTISCALE proposed by Ramsay (1982).…”

Vector Representation of Asymmetry in Multidimensional Scaling

“…Tobler (1976-77) considered the result of his analysis, that is, the coordinates of objects and the vectors of "flow", as a vector field, so that he calculated directly the rotation (divergence-free part) and the divergence (curl-free part) of the vector field. Chino and Nakagawa (1990) and Chino (1991) proposed the method for analyzing the longitudinal dissimilarity data. In these papers, the changes in group structure over the time were represented by two dimensional vector fields, which are described by a system of general nonlinear equations.…”

“…Yadohisa and Niki (1994) extend this model for three-way data, and more recently, dynamic representation and vector field representation of this asymmetry were proposed in Yadohisa et al (1998) and Yadohisa and Niki (1999), respectively. Chino and Nakagawa (1990), Chino (1991), andTobler (1976-77) are particularly concerned with a model proposed in our paper. In the former two papers, changes in group structure over time are represented by vector fields.…”

Predicting the Development of Friendship Ties Among A Group of Students Via Multidimensional Scaling: An Analysis of Nordlie’s Fraternity Data

“…Newcomb, 1961 have been done (Breiger, Boorman, & Arabie, 1975;Chino, 1984;Chino & Nakagawa, 1990;Nakao & Romney, 1993;Okada & Imaizumi, 1996;White, Boorman & Breiger, 1976). The data consists of friendship ties shown by attraction ranks among 17 students at a fraternity over 16 weeks.…”

“…The data consists of friendship ties shown by attraction ranks among 17 students at a fraternity over 16 weeks. Although some studies (Breiger, Boorman & Arabie, 1975;Chino, 1984;Chino & Nakagawa, 1990;Nakao & Romney, 1993) seem to pay attention not only to disclose the change of the process over time underlying the data but also to predict the process, their focus were mainly on disclosing the change of the process over time.…”