Showing papers in "Journal of Mathematical Sociology in 1971"

Dynamic models of segregation

Abstract: Some segregation results from the practices of organizations, some from specialized communication systems, some from correlation with a variable that is non‐random; and some results from the interplay of individual choices. This is an abstract study of the interactive dynamics of discriminatory individual choices. One model is a simulation in which individual members of two recognizable groups distribute themselves in neighborhoods defined by reference to their own locations. A second model is analytic and deals with compartmented space. A final section applies the analytics to ‘neighborhood tipping.’ The systemic effects are found to be overwhelming: there is no simple correspondence of individual incentive to collective results. Exaggerated separation and patterning result from the dynamics of movement. Inferences about individual motives can usually not be drawn from aggregate patterns. Some unexpected phenomena, like density and vacancy, are generated. A general theory of ‘tipping’ begins to emerge.

Structural equivalence of individuals in social networks

Abstract: The aim of this paper is to understand the interrelations among relations within concrete social groups. Social structure is sought, not ideal types, although the latter are relevant to interrelations among relations. From a detailed social network, patterns of global relations can be extracted, within which classes of equivalently positioned individuals are delineated. The global patterns are derived algebraically through a ‘functorial’ mapping of the original pattern. Such a mapping (essentially a generalized homomorphism) allows systematically for concatenation of effects through the network. The notion of functorial mapping is of central importance in the ‘theory of categories,’ a branch of modern algebra with numerous applications to algebra, topology, logic. The paper contains analyses of two social networks, exemplifying this approach.

A note on the measurement of racial integration of schools by means of informational concepts

Abstract: This note is concerned with the measurement of racial integration of schools in a way that permits a simple aggregation of the measure to sets of schools such as school districts. [The procedure can be applied to institutions other than schools, but we prefer a more specific terminology.] There are several standard procedures for measuring integration, but the dissimilarity index appears to be more popular than any other index.2 This index is based on a comparison of the number of white students in each school, measured as a fraction of the total number of white students in the city, and the analogous nonwhite proportion of the same school. The index is defined as one-half of the sum over all schools of the absolute differences of these proportions. The value of the index is zero when the racial compositions of all schools are identical and it takes larger and larger positive values (up to a maximum of 1) when the racial compositions are more different; also, it has a simple interpretation in terms of minimum shifts which are needed in order to obtain identical racial compositions in all schools. However, the use of absolute differences makes the dissimilarity index less suitable when one wants to aggregate schools to school districts.3 The objective of this note is to show that there are some simple measures derived from information theory which are superior in this respect, and to illustrate their use by means of data on the elementary schools of the Chicago public school system for the years 1963–1969.

The checkerboard model of social interaction

Abstract: The checkerboard model is a computer simulation of social interaction among members of two groups. The checkerboard represents a social field on which two groups of checkers move on the board on the basis of positive, neutral or negative attitudes toward one another assigned to them. The resulting pattern of positions of the pieces represents the social structure. The theoretical basis for the checkerboard model is explained and the rules for operating the model are outlined. This is followed by illustrative runs named Crossroads, Mutual Suspicion, Segregation, Social Climber, Social Worker, Boy‐Girl, Couples and Husband‐Wives, showing intermediate and final positions on the board for each. It is concluded that the checkerboard model is capable of demonstrating the intimate connection between attitudes of group members toward their own group and toward others to a continuous social interactional process and to the resulting social structure.

Semi‐Markov processes and mobility†

Abstract: A stochastic model of migration, occupational and vertical mobility, based on the theory of Semi‐Markov processes, is presented and important features of these processes derived. The model is a generalization of the Markov process in which the probability of leaving a state can depend in any arbitrary way on the length of time the state has been occupied (duration‐of‐stay) and on the next state entered (pushes and pulls). For mobility processes it thus captures McGinnis’ ‘axiom of cumulative inertia.’ Several distributions with cumulative inertia are presented and the relationship between the Semi‐Markov model and the Mover‐Stayer model explored. A method of including age effects is described. The model is shown to have applications to many other social processes, in addition to mobility, which have duration‐of‐stay effects.

A finite model of mobility

Abstract: The Markov chains with stationary transition probabilities have not proved satisfactory as a model of human mobility. A modification of this simple model is the ‘duration specific’ chain incorporating the axiom of cumulative inertia: the longer a person has been in a state the less likely he is to leave it. Such a process is a Markov chain with a denumerably infinite number of states, specifying both location and duration of time in the location. Here we suggest that a finite upper bound be placed on duration, thus making the process into a finite state Markov chain. Analytic representations of the equilibrium distribution of the process are obtained under two conditions: (a) the maximum duration is an absorbing state, for all locations; and (b) the maximum duration is non‐absorbing. In the former case the chain is absorbing, in the latter it is regular.

Some exhaustible Poisson process models of divorce by marriage cohort

Abstract: A one‐parameter exhaustible Poisson process model is formulated to represent the cumulative divorce trajectory of marriage cohorts. On the basis of recently published data of nine‐year cumulative records of all one‐year United States marriage cohorts, 1949–1958, it is found that the one‐parameter model does not provide an empirically adequate representation of cohort divorce trajectories. Therefore, two modifications of the basic model are explored. First, it is assumed that the longer the couples remain married the smaller is the probability of their becoming divorced (a cumulative inertia modification). Second, it is assumed that the cohort can be divided into two groups one of which is subject to risk of divorce while the other is not (a mover‐stayer modification). It is concluded that the latter model provides a better approximation to the empirically observed divorce trajectories. Given availability of disaggregated observations of divorces for marriage cohorts, the model could be used as a mathemati...

Should social choice be based on binary comparisons

Abstract: This paper argues that social choice from among more than two feasible alternatives should not be based on social choice from two‐alternative subsets. It considers in some detail the case where one alternative ties or beats every other alternative on the basis of simple majorities, and raises the question of whether such an alternative should be chosen. A condition of ‘stochastic unanimity’, introduced in this context, is shown to be incompatible with the simple majority rule when it can apply. This new condition plus a consideration of ties leads into a brief discussion of the use of individual expected utility in social choice theory.

Clustering in N dimensions by use of a system of forces

Abstract: This paper describes a model and associated computer program for carrying out clustering of elements in a space of a predetermined number of dimensions. In addition to clustering of elements, the model may be used for multi‐dimensional scale analysis and for construction of sociograms. The model takes as input data a set of ‘affinities’ between elements, the inverse of which may be considered as psychological or sociological distances. It then moves elements in a Euclidean n‐space toward the point at which the geometric distance is equal to the given psychological or social ‘distance,’ as if a set of attractive and repulsive forces was acting upon each element from other elements. The second major portion of the paper consists of ten examples, which are analyzed by means of the accompanying computer program.

Demiarcs: An atomistic approach to relational systems and group dynamics

Abstract: In the course of extensive conversations during the past decade with Dorwin Cartwright, the concept of demiarcs was conceived, developed, explored, and applied to a great variety of relational systems. The present communication is not intended as definitive, but rather as introductory and exploratory. There is still much room for clarification, empirical justification, and application to a great variety of relational systems including the study of group dynamics, innovation, and the dissemination of information.

A formalization of measurement scale forms

Abstract: The general measurement scale hierarchy developed by S. S. Stevens has had widespread use and discussion, but has suffered from various criticisms, many of which may be traced to the informal descriptions and minimal mathematical formulations used in the original definitions and claims. In this paper, a mathematical model of measurement and specific scale forms is developed, using the minimum mathematical structure necessary to define each scale type. Measurement is defined, and a series of restrictions are applied to measurements to yield a precisely specified hierarchy of measurement scale forms including nominal, ordinal, interval, and ratio scales, as well as several others. Equivalences are mathematically denned for each scale form, and theorems are presented specifying the sets of transformations under which equivalence is preserved, thus yielding the scale form invariances discussed by Stevens. The hierarchy of measurement scale forms is summarized in a pictorial ‘spectrum’ diagram. The development...

The accuracy of alternative stochastic models of participation in group discussion

Abstract: Four related models of participation in group discussion are presented and compared in the accuracy with which they predict proportional participation, mean run length, mean recurrence time and the variances of runs and recurrences. For some quantities, such as proportional participation and mean run length, the most restricted model does virtually as well as the least restricted model; for other quantities it does not. None of the models does a good job of predicting variances of runs or recurrences.