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*Mathematics *is the human activity of constructing axiomatic definitions of abstract patterns among unspecified or arbitrary elements and studying the properties of such patterns by deductive elaboration, using principles of logic. Any such abstract pattern, arising in such a context, may be said to define a class of mathematical objects, e.g., “differential equations,” “Markov chains,” “semigroups,” and “vector spaces.” If T is the axiomatic theory that defines a class M of mathematical objects, then any entity in M is said to be a T-model. Such models play a central role in the sciences. For example, Von Neumann and Morgenstern (1944) formulated the axiomatic theory of games of strategy, and this game theory (T) defines the class M of game models.

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As a science, sociology includes the use of such mathematical models. For most sociologists, however, this connection between mathematics and sociology is confined to problems of data analysis, employing *statistical models. *In other words, in this case the mathematical theory (T) is the *theory of statistics, *and the T-models deal with such things as *linear regression *and *statistical significance tests. *However, the linkage between mathematics and sociology goes well beyond simple uses of applied statistics and extends back to the mid-twentieth century.

## Origins of the Field

After World War II, as part of a more general zeitgeist involving the deepening and broadening of the interpenetration of mathematics and the social and behavioral sciences, some sociologists began to employ mathematical models in contexts different from traditional data analysis. Their point of view was a common one in the newly developing field of *mathematical social science. *The idea was to create more rigorous scientific theories than had hitherto existed in the social and behavioral sciences (Berger et al. 1962). Traditionally, for instance, sociological theories were strong in intuitive content but weak from a formal point of view. Assumptions and definitions were not clearly stipulated and distinguished from factual descriptions and inferences. In particular, there was rarely a formal deduction of a conclusion from specified premises. The new and preferred style was encapsulated in the phrase “constructing a mathematical model.” This means making specified assumptions about some mathematical objects and providing an empirical interpretation for the ideas. It also means deducing properties of the model and comparing these with relevant empirical data.

*Mathematical sociology *was part of this general movement in the social and behavioral sciences. Sociologists who contributed most to the development of mathematical sociology were greatly influenced by these wider developments, especially in disciplines with which sociology overlaps such as psychology, economics, and demography. Some of these wider developments will be briefly described because of their particular importance in this respect. Other examples of early work in mathematical social science may be found in the compendium edited by Lazarsfeld and Henry (1966).

Starting in the late 1940s, the mathematical biologist Anatol Rapoport developed a probabilistic approach to the characterization of large *social networks. *Starting from a baseline of a “random net” and then introducing “bias parameters,” Rapoport logically derived formulas connecting parameters such as density of contacts to important global network features, especially connectivity (Rapoport 1957). The logic of this approach is to compare these actual structural features with those that would hold if the network were generated by random connections. Then bias parameters that relate, for instance, to a tendency to transitivity (if a is connected to b and b to c, then a is connected to c) are shown to account for the way in which the real network differs from the random net, which functions as a baseline model.

In another early social networks development, mathematician Frank Harary and social psychologist Dorwin Cartwright collaborated in a discrete mathematical approach to social networks, featuring the theory of graphs—large parts of which were being created by Harary and his collaborators as they worked on social science problems. Graph theory is an axiomatic theory that defines models taking the form of an abstract pattern consisting of entities (nodes, points) in various pairwise relations (ties, edges, links). A graph model is employed to represent the network of connections among a set of acting units such as positive and negative sentiment relations among persons. From this starting point, Harary and Cartwright went on to prove the important and nonobvious structure theorem (Cartwright and Harary 1956). The theorem states that if a structure of interrelated positive and negative ties is balanced—illustrated by the psychological consistency of “my friend’s enemy is my enemy”—then it consists of two substructures, with positive ties within and negative ties between them. (There is a special case where one of the two substructures is empty.) Note that because the proof is based only on the formal object—the typical model in the specified class of models (which in this instance consists of *signed *graphs)—the theorem may be given various interpretations and associated differing empirical identifications that operationalize an interpretation.

In these two developments we have mathematical models bearing on the *analysis of structure. *Other early influential developments pertained to *process. *In the analysis of processes, two types of mathematical model are relevant: deterministic and stochastic.

An early example of the use of deterministic models is Herbert Simon’s (1952) mathematical formalization of a social systems theory set out by George Homans ([1950] 1992). Mechanisms describing interrelationships among the key variables of Homans’s theory—the intensity of interaction among group members, the level of friendliness, and the amounts of externally imposed and internally generated activity—are embedded in a system of differential equations. The system is then studied in its abstract form, leading to theorems about the dynamics and the implied equilibrium states.

The stochastic approach was strongly developed in mathematical learning theory (Bush and Mosteller 1955). The general probabilistic approach came to be known as stimulus sampling theory, in which the human being is viewed as sampling stimulus elements and connecting these to responses as a function of reinforcement contingencies. Intended for application to aggregate data acquired in experimental situations, these models enabled the derivation of predicted sequences of expected frequencies of a certain response under given conditions of reinforcement. Although the model is based on assumptions that refer to unobservable events (the stimulus sampling process), the derived predictions are in reference to observables (the over-time sequence of observed frequencies of a response) and thereby enable an empirical test of the theory based on the assumptions of the process that generates the observables. As in other such instances we will discuss, the model includes parameters pertaining to the underlying process (which usually appear as constants in derived equations). The values of these parameters are not usually known in the setting up of the model. To test the model, therefore, data need to be employed first to estimate the parameters. After parameter estimation, the data then can serve the function of testing the derived consequences, which can now be put in a definite numerical form corresponding to the particular values of the parameters. Hence, the construction of mathematical models of processes involves a sequence that includes such steps as deriving model predictions, estimating parameters, and testing for the goodness of fit between predicted and actual values of observables.

The field of mathematical sociology received selfconscious recognition with the work of James S. Coleman in the 1960s. Coleman came to sociology from an engineering background, studying with Paul Lazarsfeld at Columbia University in the 1950s. As an engineer, he thought about social processes in terms of differential equations, as had Simon and others. But how could one connect differential equations to the data of sociology? That was Coleman’s question. He noted that surveys reported results in the forms of proportions. Yet the proportion of people believing or doing something at a given time had to be correctly interpreted. First, it was not necessarily a stable proportion since it could change. So such proportions should be conceptualized as states of a probabilistic dynamic system, with a flow of probabilities over time that might indeed have some equilibrium state. Second, although each person held a belief or voted a certain way, the process by which these individual orientations came about was socially mediated—that is, we should understand the process by which the probability state changed over time as a network process in which individuals influence each other to change orientations. The results of these sorts of considerations were embodied in Coleman’s (1964) widely influential *Introduction to Mathematical Sociology. *The publication of this book marks the legitimation of mathematical sociology as a distinctive and important part of sociology. Coleman’s innovation was to show how processes in social networks could be analyzed in such a way as to come to grips with relevant sociological data, allowing empirical identification of abstractions, estimation of parameters, and calculations of the goodness of fit between model and data. Coleman’s interest in purposive/ rational action as the foundation for understanding social processes culminated in a major work on rational choice theory in sociology, including the use of the mathematics of general equilibrium theory (Coleman 1990).

## Scientific Realism and the Theory of Models

How should the models of mathematical sociology be viewed? Recent work in the philosophy of science and its implications for sociological inquiry in general and comparative historical sociological theories in particular and in the theory of models (Casti 1992a, 1992b; Land 2001) has converged to show the fundamental unity and continuity of the formal models of mathematical sociology with the statistical and verbal models of conventional sociology.

Gorski (2004) builds on recent developments in the branch of the philosophy of science known as *scientific realism, *the view that science seeks to reveal the underlying structures of the world that generate different outcomes under different conditions. He specifically uses an approach he calls *constructive realism, *which construes explanations as *linguistic representations *or causal models constructed out of theoretical terms. In the constructive realist model of explanation developed by Gorski, a scientific explanation is a semantic relation between causal models and causal processes or systems. A *causal model *is a simplified, linguistic representation of one or more real causal processes, which contribute to some set or type of outcomes. Thus, to explain something is to represent, and thereby make more comprehensible, the principal processes that produced it. A *theory *in this framework is a symbolic construct, stated in ordinary or mathematical language, that defines certain classes of objects and specifies their key properties. It is assumed that the objects refer to real entities in the world and the properties to actual qualities of these entities. In other words, a theory is a set of ontological assumptions that are used, explicitly or implicitly, in the construction of a causal model or models.

Gorski’s version of scientific realism is similar to, but also different in certain respects from, the philosophical approach taken by Fararo (1989), who frames a synthesis of scientific realism within the mathematical axiomatic method, thereby retaining the deductive element in sociological theorizing that Gorski largely abandons. The underlying difference is that one version of scientific realism is linked to historical interests—the explanation of particular events and trends—while the other is linked to a generalizing interest in the sense of explanation of abstractly stated empirical generalizations. (For a further discussion of the latter strategy see Fararo and Kosaka 2003, chap. 1.)

The scientific realist approach to sociological explanation can be compared with recent statements of the theory of models and its application in sociology (Casti 1992a, 1992b; Land 2001). We now sketch a formal representation for the theory of models to illustrate its compatibility with the constructive realist approach and to show that many classes of sociological models, from verbal to statistical to mathematical, can be accommodated within this formalism. The *theory of models *commences with the following general and encompassing definition:

Models are cognitive tools, namely linguistic devices, by which individuals order and organize experiences and observations.

Experiences/observations vary among individuals and can be organized in many different ways. Even if the observations are common and shared, as in the case of a single set of observations on specific social phenomena summarized in a data set, they can be organized in different ways. It follows that there can be many different models of the same experiences/observations. Hence, as Casti (1992b:380) notes, there can be many alternative realities—at least to the degree that individuals represent reality in models.

An implication of this generic definition of models is that virtually every functioning person in a society can be presumed to be a modeler. Beginning with the characterization of natural language as a tool for ordering and describing experiences (see, e.g., Whorf 1956), one can regard much of linguistic, popular, and material culture as providing the “tools” by which individuals, on a dayto-day basis, model their experiences (see Swidler 1986). And, of course, it is the objective of long traditions of ethnographic research in sociology and anthropology to record and study the structure of such “natural” models.

Another implication of this definition is that many of the verbal characterizations of social phenomena that we use in sociology are properly regarded and respected as models. Many of these *verbal models *have stimulated much research over many years and will continue to do so. As one of just two examples, Notestein’s (1945) verbally stated *demographic transition model *stimulated social demographers to focus their research attention on a host of historical and contemporary questions about trends in birth and death rates and their relationship to economic development and improvements in health and longevity. Similarly, the verbally stated *life-course model *that has been developed over the years by many sociologists and articulated succinctly by Elder (2000) has stimulated the work of researchers with many diverse substantive interests.

Taking models most generically as cognitive tools for ordering our experiences, formal or mathematical models can be defined as follows:

Formal or mathematical modelsencapsulate some slice of experiences/observations within the confines of the relationships constituting a formal system such as formal logic, mathematics, or statistics. (Casti 1992a:1)

Thus, a *formal sociological model *is a way of representing aspects of social phenomena within the framework of a formal apparatus that provides us with a means for exploring the properties of the social life mirrored in the model. Why construct formal sociological models? Why not just use verbally stated models? Basically, we construct formal models to assist in bringing a more clearly articulated order to our experiences and observations as well as to organize more complex theories of experiences and observations and to make more precise predictions about certain aspects of the social world (Lave and March 1975). Henceforth, we will drop the *formal *or *mathematical *adjective and simply use the term *models.*

Some notation will be useful. Consider a particular subset *S *of social life, and suppose that *S *can exist in a set of distinct *abstract states *Ω = {ω_{1}, ω_{2}, . . . }. The set Ω defines the *state space *of *S. *Whether or not the sociological observer can determine the state of *S *at a particular moment of study depends on the experiences, observations, or measurements (*observables*) at the sociologist’s disposal. An *observable *of *S *is a rule *f *associating a real number with each ω in the state space Ω. More formally, an observable is a map *f: *Ω α *R.*

For example, consider the abstract empirical generalization that as initially unacquainted members of task groups interact, a status hierarchy tends to emerge. The explanation, in a realist mode, employs a model in which the state space consists of the conjunction of underlying performance expectation states as to ability levels relevant to the task. These states are not observable in the flow of interaction—either to the members or to the observing sociologist. The observables are the acts of the individuals. Postulated expectation states and a coding of the acts together yield a particular interpretation of the general notion of a state space and one or more observables defined on it. Theoretical assumptions about the interaction process generate a trajectory of the group in state space along with derived predictions about the observables, thereby enabling empirical tests of the assumptions (Skvoretz and Fararo 1996).

Generally, for a full accounting of social life, many sociologists feel that we need an infinite number of observables *f*_{α}*: *Ω α *R, *where the subscript α ranges over a possibly uncountable index set. Thus, the complete slice of social life *S *is described by Ω and the possibly *infinite set of observables F *= {*f** _{a}*}. But it is impossible to deal with such a large set of observables, and it is not necessary to do so to build useful sociological models and/or theories. As Smith-Lovin (2000) has argued,

Social life is very complex. To be completely described, any historical event or current interactional situation requires a virtually infinite catalog of contextual, historically specific information to be conveyed . . . But the fact that social life is complex does not imply that we need complex descriptions of it . . . Indeed, I think that the most successful theories often focus on just one basic process, while most situations involve the simultaneous occurrences of many different processes. (P. 302)

In brief, in model construction, most of the possible observables in social life are thrown away and attention is focused on *a proper subset A *of *F. *It is, of course, the case that this means that our models may poorly capture the full complexity and nuances of social life. Certainly, this would be true of our example of the emergence of a status hierarchy in any group in a natural setting. This, again, reinforces the position that models are worth constructing and dealing with only if they assist in bringing a more clearly articulated order to experiences and observations and in making more precise predictions about certain aspects of the social world. Baseline, oversimplified models also can be criticized and improved.

We now can characterize a *sociological model S* *as an abstract state space Ω together with a finite set of observables *f** _{i }*: Ω α

*R, i*= 1, 2, . . . ,

*n.*Symbolically,

*S* *= {Ω, *f*_{1}, *f*_{2}, . . . , *f** _{n}*}.

To capture the scientific realist sense of a model, the *observables may be regarded as in two classes. *One class represents the *observable conditions *under which some process in state space occurs. In turn, these divide into the slow-changing conditions that can be represented as *constant parameters *in the time frame of the process and the more rapid changing conditions that can be called *inputs. *The other class represents the observable *outputs *of the process. These outputs—acts in the context of the emergent status hierarchy example—depend on the state and the inputs at the particular moment.

Note that, in that example, the outputs become inputs as the members observe acts. The complexity of the interaction process is thereby reflected in the complexity of the generative model.

A model that is *static, *that is, not dynamic, in this sense may be regarded as a special case of the general form given by

Change of state = F(state, input)

Outputs = G(state, input)

If the change of state is zero—as in static models—then the first of these expressions becomes

M* _{i }*(state,

*f*

_{1},

*f*

_{2}, . . . ,

*f*

*) = 0,*

_{n}*i*= 1, 2, . . . ,

*m,*

where the M_{i}(*f** _{j}*) are mathematical relationships expressing the dependence relations among the observables. This can be more compactly written as

M(*f*) = 0. [1]

Now suppose that the last *m *observables, *f*_{n-m}_{+1}, . . . , *f*_{m}*, *called *endogenous *(or determined within the system under consideration), are functions of the remaining observables, *f*_{1}, *f*_{2}, . . . , *f*_{n-m}*, *where the latter are termed *exogenous *(or determined outside the system under consideration). The endogenous terms of the static model correspond to the outputs of the dynamic model and the exogenous terms correspond to the inputs to the system with the given state description.

In other words, suppose we can define *m *functional relations, with some finite number *r *of numerical parameters, β_{1}, β_{2}, . . . , β_{r}*, *for determining values of the endogenous observables as a function of the exogenous observables. Then, if we introduce the notation β α (β_{1}, β_{2}, . . . , β* _{r}*)

to denote the vector of parameters and the notation

- α (
*f*_{1},*f*_{2}, . . . ,*f*) and_{n-m} - α (
*f**n-m+1**, f**n-m+2**,*. . . ,*f**n*)

to denote vectors of the exogenous and endogenous observables, henceforth *variables, *respectively, then the equations of state become

*y *= Φ_{β }(state, *x*). [2]

This last expression now is beginning to take on a form similar to the sociological models often seen in practice. In particular, suppose we define an additive vector

ε = (ε_{n-m+1}, ε_{2}, . . . , ε_{n})

of error terms (with the usual specifications on the error terms, namely, that the expected value of each ε_{i}*, E*(ε* _{i}*) = 0 with constant variance,

*E*(ε

*ε*

_{i}*) = σ*

_{i}^{2}

_{i}*, i*= 1, 2, . . . ,

*m*), one for each endogenous variable, to take explicitly into account the fact that there may be stochastic shocks to the equations of state due either to factors unaccounted for in our system model or to an intrinsic random element in the behavior of the endogenous variables. Then the equations of state, Equation 2, become

*y *= Φ_{β }(state, *x*) + ε. [3]

Equation 3 now is in a form such that many of the common formalisms used in model construction in sociology can be recognized as special cases.

To highlight this, we now treat the state variable as measured by one or more of the observables (functioning as indicators) rather than, as in the status hierarchy example, defined as an underlying unobservable element. Thus, a special case of Expression 3 applies in which the state variable is suppressed. Consider the following instances:

- If
*m*= 1, so that the vector*y*of endogenous variables contains a single element, then the equations of state, Equation 3, reduce to the form of a*conventional regression model.*If, in addition,*y*is a continuous variable, Equation 3 becomes a*classical regression model,*either linear or nonlinear, depending on the function form of Φ_{β}, whereas a dichotomous*y*and a logistic form for Φ_{β }yield a*logistic regression model*(Neter et al. 1996). Other specifications of measurement and functional formats yield other types of regression models, including the*multilevel*or*hierarchical models*for the analysis of contextual effects and growth models that have been developed and widely applied in recent years (see, e.g., Raudensbush and Bryk 2002). - In the case where
*y*contains*m*endogenous variables, Equation 3 is in a form similar to that of the reduced form of a*classical econometric/structural equation model.*If the functional form Φ_{β }incorporates recursive or nonrecursive dependences among the endogenous variables, then the equations of state, Equation 3, are in the structural-equation form of a classical econometric/ structural equation model (Christ 1966). If explicit measurement models taking into account random measurement errors of the exogenous variables and/or endogenous variables are specified in addition to structural equations linking latent variables, then the equations of state, Equation 3, take the form of contemporary*simultaneous-equation models*(*SEM*s, also termed*structural equation models with latent variables,*of which LISREL models are the most widely known (Bollen 1989; Hoyle 1995). - The characterization of formal models given in the foregoing and the notions of observables and equations of state also can be applied to many other types of modeling formalisms used in sociology. For instance, the equations of state can be given a
*dynamic formulation*by specifying them in*differential or stochastic differential equation form.*If the endogenous variables then are defined in terms of time to a transition of some type, the equations of state then may yield*hazard or event history regression models.*

In brief, the formalism represented by Equation 3 is capable of subsuming the logics of many approaches to the development of models in sociology, from verbal to statistical to mathematical and within the latter, both dynamic and static models. Thus, there is an intrinsic continuity between mathematical sociology and other parts of sociology.

Continuing with the parallels between constructive realism and the theory of models, we note that the terms *model *and *theory *are sometimes distinguished and sometimes used interchangeably. Usually, however, scientific theories are regarded as more general than scientific models:

A

theoryis afamily of related models,and amodelis a formal manifestation of a particular theory. (Casti 1992b:382)

A key characteristic of models, as noted in the foregoing, is that they are constructed relative to a given set of observables. Theories, on the other hand, are more generally applicable to numerous sets of observables. One example is the expectation states theory mentioned above, which is far more general than the particular model that was sketched (Wagner and Berger 2002). Another example, as mentioned earlier, is *game theory *(Fudenberg and Tirole 1991), which can be construed as a family of models of the behavior we observe when rational decision makers interact. The game theory family of models is more general than, say, a *game-theoretic model *of crime control policies and criminal decision making (de Mesquita and Cohen 1995). In this case, because game theory is a mathematical theory, the family of models is defined by a set of axioms, as noted earlier. As another example, but now pertaining to a nonmathematical type of theory, *functionalism *as a sociological theory can be regarded as a family of functionalist models of social structures and processes (see, e.g., Turner 1991). And functionalist sociological theory is more general than, say, a *functionalist model *of how organizations try to reduce uncertainty in their environments (Thompson 1967).

From sociological theories and theories in other social and biological sciences, sociologists develop ideas about what observations and measurements of social life should be made, generative mechanisms (Fararo 1989; Hedström and Swedberg 1998; Smith-Lovin 2000) with which to build and choose models, and hypotheses to be tested. But since sociological theories are typically stated in very general terms, they leave measurement instruments and functional forms of models to be specified. At this point, models become relevant. They develop linkages of theory to observations and data (Land 1971). Indeed, Skvoretz (1998) has forcefully argued that *theoretical models*— models that represent the generative mechanisms and processes embodied in sociological theory—are the missing or underdeveloped link in the discipline today.

Yet sociological theory sometimes does not provide complete guidelines for model building, which often leads to controversies about the adequacy of models constructed and applied in sociology. The following are two important issues of this kind:

- The endogenous-exogenous distinction
- Model completeness

Sometimes there is no disagreement that certain variables are appropriately specified as exogenous or determined outside a particular set of functional relationships. At other times, however, questions certainly can be raised, and disputes over the “endogenous/exogenous” distinction often are at the heart of disagreements over the “correct” model specification. Questions about “model completeness” also are often sources of dispute concerning the adequacy of sociological models. An analyst, in an effort to be complete, may extend the list of exogenous variables to a very large number. This can produce models with so many explanatory variables that the interpretation of results becomes difficult, if not impossible. On the other hand, some of the most damaging critiques of models are the claims that the modelers omitted variables (observables) that were centrally important to understanding the behavior of the observables employed. A dividing line on this issue also often depends on one’s theoretical versus empirical orientation toward modeling. Those who approach the construction of sociological models from a theoretical perspective tend to emphasize parsimony in model specification (e.g., Smith-Lovin 2000). Empirically oriented modelers, on the other hand, often feel that many explanatory variables are necessary for model completeness (e.g., in discussing the complications in sociological modeling due to multiple causation, Blalock (1984) stated that “upwards of 40 or 50 factors [may be] at work” (p. 40) in the determination of a social phenomenon).

In sum, there are remarkable parallels between the recent scientific realist approach to explanations in sociology and the theory of models. These parallels show the unity and continuity of models in sociology, from verbal to statistical to mathematical, and the commonality of uses of mathematical models in sociology with those in other scientific disciplines.

## Models of Social Processes

We now turn to a review of models of several types of social objects that are found in contemporary mathematical sociology. Consider first the *modeling of social processes. *To represent a social process, some sort of dynamic model is required. The basic ingredients in such a type of model are as follows.

Time domain: discrete or continuous

State space: discrete or continuous

Parameter space: discrete or continuous

Generator: deterministic or stochastic

Postulational basis: equations, transition rules

These ingredients are obvious in the case of physical theories, and sociologists employing system models (such as Parsons and Homans) were committed to the project of carrying this type of analysis into sociological theory. No clearer example of this exists than in Homans’s treatment of the social system in *The Human Group *(Homans [1950] 1992). So clearly did Homans try to model his discursive analysis of group phenomena on the setup and analysis of a system of differential equations that shortly after this book appeared it was formalized as noted in the foregoing by Herbert Simon (1952). Simon postulated the basic mechanisms described by Homans in terms of quantitative expressions in a differential equation (deterministic continuous-state, continuous-time) model with continuous parameters. Simon’s paper has been an exemplar for sociologists who formalize theories in terms of differential equations, for instance, Land’s (1970) formalization of the dynamics of Durkheim’s (1933) classical theory of the division of labor in society and Mayer’s (2002) class dynamics model of the collapse of Soviet communism.

As mentioned earlier, Coleman (1964), responsive to the needs of survey research with its discrete data summarized as proportions, developed a family of dynamic models that are stochastic processes in continuous-time with discrete states but continuous parameters. Each individual makes transitions from one discrete state to another—for instance, shifting candidates during an election campaign—and the group makes transitions among states representing the number of individuals in each of the discrete individual states (e.g., the number of people favoring a particular candidate at a particular time). With estimation of parameters by statistical methods, followed by tests for the goodness of fit of the model, this Coleman methodology extends to the social network context in which each individual’s transition is influenced by a composite flow of influence from other individuals to whom the person is connected in some social relationship. (For an introduction to this type of model, see Fararo ([1973] 1978, chap. 13) and for another type of dynamic model of social influence in a network context with numerous applications, see Friedkin (1998).)

Formally, the stochastic processes that Coleman invoked are *Markov chains. *Such Markov chains are directly analogous to deterministic processes and, in fact, are deterministic at the level of probabilities: Future probability distributions depend only on the present distribution and not on earlier ones. Other applications of Markov chains became common in the field of mathematical sociology. In particular, social mobility was the subject of a considerable number of mathematical modeling efforts in which Markov chains played a large role (Fararo [1973] 1978, chap. 16). More generally, stochastic processes have played an important role in theory and methodology in sociology (Tuma and Hannan 1984).

Of particular interest in sociology are two types of process models that relate to the concept of social structure. In one type, the structure is represented by a network or some other model object, and other phenomena are taken as “dependent variables.” The aim is to show how the outcome of a postulated process varies with parameters descriptive of the social structure, as represented in the structural model. This is illustrated by the research tradition involving the analysis of exchanges in networks (see Willer 1992). The shape of the network is the parameter, and the eventual distribution of resources among occupants of positions in the network is the dependent variable. Theoretical models of the process postulate how actors make and respond to offers, leading up to competed exchanges and so the eventual resource distribution.

In the other type of model, the structure is treated as emergent. The previously mentioned interaction process model involving the use of expectation states or *E-states *may serve to illustrate this (Skvoretz and Fararo 1996). The structure in this instance is local, that is, specific to a given group of actors and emergent in their situated interaction. The dynamic process involves the over-time construction of stable relationships among pairs of actors until equilibrium, when the postulated rules lead to a reproduction of the generated pattern of relationships. In this model, the state space consists of a set of logically possible forms of structure in the network sense. The emergent structure is a set of social relationships among group members, each defined in terms of stabilized E-states. The process is the trip through this space. Which trip is taken, in terms of which network states are visited, depends on the initial state, the parameters, and the specific realization of the stochastic process representation of the generator.

There is another mode of model specification and study that instantiates the dynamic type of theoretical model and that relates to the emergence of structure, namely *computer simulation. *Simulation models are generative of specific instances of “observables” from postulated rules that mimic a concatenation of social processes. These specific instances are pseudo-data outputs of the simulation, so that it is often necessary to employ statistical or graphical methods to reduce these “data” to an interpretable form in which the analyst can conclude that the process generates various types of outcomes as conditions vary.

Simulation models arise in at least *two different contexts. *In one context, a mathematical model is postulated or derived that involves nonlinear equations, for which analytical solutions are generally unavailable. The analyst may be able to derive qualitative results but may also turn to computer simulation to generate particular instances of the over-time behavior of the system of variables. The other context involves the postulation of rules of behavior of a collection of actors to generate the over-time consequences of their interactions. The dynamics of the system is given by the concatenation of simulated actions rather than by a system of equations. One important class of examples involves the problem of social order in social theory, in which theorists postulate rules of behavior in a repeated Prisoner’s Dilemma context and then simulate the over-time emergence of cooperation under various conditions represented in an “artificial world.” Examples of this type and other social simulation models may be found in the online *Journal of Artificial Societies and Social Simulation *(in particular, see Macy 1998).

## Models of Social Structure

Sociologists have employed at least *four different types of models in the analysis of structure in social life. *We may regard these as four representation principles under the following headings: structure as *network; *structure as *distribution; *structure as *grammar; *and structure as *game.*

In earlier sections we have made reference to the *network representation of structure. *The metaphor of a social system as a network, widely employed informally in sociology, was transformed into a mode of model building and analysis through a convergence of ideas and techniques from several disciplines. One such source was *sociometry *(Moreno 1934), involving the analysis of network diagrams indicating relationships among people in a small population. *Balance theory *was another source, where in this case positive and negative sentiments were represented in diagrams and analyzed in terms of the balance among the relationships. These ideas were absorbed into social network analysis via the formalization of the ideas in terms of *signed graph theory, *a branch of mathematics built to deal with structures of positive and negative sentiment relations (Harary, Norman, and Cartwright 1965). A third source was the analysis of *structures of kinship, *especially after the publication of an influential monograph by White (1963).

Sociometric models, balance-theoretic models, models of kinship structure, and other model-building efforts, such as those treating diffusion processes, converged by the late 1970s, and the term *social network paradigm *was used to describe this whole area of model building (Leinhardt 1977). Over time, it became common in sociology for measured properties of networks to be employed in the formulation and testing of empirical hypotheses about the behavior of actors. For instance, concepts such as status, centrality, and power have been defined in operational ways in terms of the network representation of structure. By the late twentieth century, social network analysis had become a mode of structural analysis with an extensive battery of formal techniques at its disposal (Scott 1991; Wasserman and Faust 1994) and with significant contributions to the analysis of substantive problems (e.g., Burt 1992).

Social network analysis has been regarded by most macrosociologists as not the sort of model required for the description of macrostructure. Sociologists often speak, in the latter context, of such entities as *occupational structure *or *income structure. *These terms refer to *distributions. *Blau (1977) proposed a systematic theory in which the key analytical properties of such distributions, in relation to rates of intergroup relations, provide one type of answer to the Durkheimian problem of the nature of the integration of a large complex social system. Blau employed the concepts of heterogeneity, inequality, and consolidation as such key parameters and formulated theorems relating them to the extent of intergroup relations, for example, rates of intermarriage.

A definite model that would represent such a macrostructure was not a part of this theory, but subsequently a mathematical model was developed (Skvoretz 1983). The model is based on the concept of a biased net, namely a network that departs from a random network in specified ways represented by bias parameters. All the key variables of Blau’s theory are formally linked to key parameters of the *biased net model*—in particular, the contact density and the connectivity of the network. The latter ideas had been important in the social networks tradition, particularly in the strand of work encapsulated in the notion of “the strength of weak ties” (Granovetter 1973). By synthesizing this “micro” strand of network analysis with the Blau-type macrosociology, this development exhibited one of the important theoretical benefits of formal model building: episodes of unification (Fararo [1973] 1978, chap. 4).

A third type of model of structure emerges out of the language analogy or metaphor. European linguists, after the foundational work of de Saussure ([1915] 1966), distinguished between language as a system and the particular utterances that occur in given occasions. American linguists, after the pioneering work of Chomsky (1957), treated language as an infinite system of legitimate possible utterances generated by a finite set of rules, its grammar. In the social sciences, *structuralism *has been a perspective based on the idea that in some sense, social and cultural systems should be treated with a languagelike model (Lévi-Strauss [1958] 1963). One implication of this idea is abstraction from time: The system exists as an infinite totality to be analyzed using algebraic or other formal tools.

However, the idea of a set of finite rules of *social structure as grammar *that generates a system of symbolically mediated interactions has been synthesized with information processing representations that had been developed in cognitive psychology (Newell and Simon 1972). The resulting model can be studied from two points of view. On the one hand, the finite rule basis and the institution stand to each as grammar and language: the analysis is in the spirit of structuralism. On the other hand, the finite rule basis can be used to analyze a system of symbolic interaction as it is generated locally and in real time. The system of finite rules is a generative mechanism, and the “outputs” are streams of coordinated social action by the socialized occupants of institutional positions. From this standpoint, the model is a special case of the general form of sociological model cited earlier, with the exception that the state and the observables are nonnumerical. This synthesis was motivated by the attempt to explicate the sociological concept of *institution *and to thereby provide a method for the formal analysis of institutional structures at various levels of the organization of action and interaction (Fararo and Skvoretz 1984). This type of model is one among a variety of those that have arisen out of sociological applications of techniques drawn from artificial intelligence, linguistics, and cognitive science (Bainbridge et al. 1994). One example is affect control theory, which employs a mathematical model grounded in grammatical analysis and control systems theory to analyze social interaction in symbolic interactionist terms (Heise 1979, 1989).

A fourth way in which sociologists have represented structure is again focused on the organization of action, treating a *structure of action as a game in the formal sense of the mathematical theory of games.*An utterance in a particular language is analogous to a play of a particular game. And the rules of the game play the role of the grammar. Given the rules of the game, a tree of all possible sequential unfoldings of the game is implied. However, the focus in game-theoretic analysis is on strategic interaction, so that a model of rational choice supplements the game model. The aim of the game-theoretic model-builder is to derive the consequences of rational choices made by each player, often with a view of showing how outcomes involve “perverse effects” (Boudon 1982). Thus, an alternative to the grammatical model of structure is the game model. The former emphasizes emergent order at the level of the tacit or implicit rules governing institutionalized social action. The latter emphasizes the way in which the structure, as represented by the game, produces predictable and often paradoxical effects from the conjunction of rational choices.

## Population Models

Mathematical modeling has been extensively developed and applied in the discipline of demography. Sociology overlaps substantially with demography because an essential feature of a society is its human population. The survival of a society’s population from birth and childhood through the adult years and its reproduction through fertility are the fundamental processes studied by demographers. Associated with these processes is an *age structure, *which is the fundamental structure studied by demographers. There is a long tradition of mathematical models of the survival or mortality and fertility processes in demography, with roots in actuarial science, biostatistics, epidemiology, mathematics, and statistics, fields with which it retains strong ties today (see, e.g., Jordan 1975; Keyfitz 1977, 1985; Lotka [1924] 1956; Smith and Keyfitz 1977). Through the population models used in mathematical demography, sociology thus has ties to these other scientific fields as well. Space limitations do not permit any detailed exposition of the life table, stable population, and reproductive models of demography (for recent reviews, see Land, Yang, and Yi 2005; Preston, Heuveline, and Guillot 2001; Schoen 1988).

As but one example of recent developments in population models, consider the measurement of the level of fertility in a human population. Although the demographic literature contains many measures of fertility, the period total fertility rate (TFR) is now used more often than any other indicator. The TFR is defined as the average number of births a woman would have if she were to live through her reproductive years (usually taken as ages 15 to 49) and bear children at each age at the rates observed in a particular year or period. The actual childbearing of cohorts of women is given by the completed or cohort fertility rate (CFR), which measures the average number of births 50-year-old women had during their past reproductive years. Formally, let *f** _{p}*(

*t,a*) denote the agespecific fertility rates for women aged

*a*at time

*t,*and let

*f*

*(*

_{c}*T,a*) represent the age-specific fertility rates at age

*a*for cohorts of women born at time

*T.*Then the

*period total fertility rate*for time

*t*is

TFR(*t*) = ∫*f*_{p}*(t,a)*d*a, *[4]

and the CFR for the cohort born at time *T *is

CFR(*T*) = ∫*f*_{c}*(T,a)*d*a. *[5]

In applications, the integrals are replaced by finite summations, and the sums are taken over the reproductive ages. Note also that the TFR can be made specific to the order of births (i.e., first, second, and so forth), but to simplify the notation, subscripts for the order of births are omitted.

The CFR measures the true reproductive experiences of a well-defined group of women. But it has the disadvantage of representing past experience, as women currently of age 50 did most of their childbearing two to three decades ago, when they were in their 20s and 30s. The advantage of the TFR is that it measures current fertility and therefore gives up-to-date information on levels and trends in fertility. The TFR also has a convention metric (births per woman) that nondemographers can readily understand.

However, the TFR has been widely subjected to criticism among demographers. Demographers interpret the conventional period TFR (TFR(*t*)) as the total number of births an average member of a hypothetical cohort would have for her whole life if this hypothetical cohort *exactly *(with no changes in quantum or level, tempo, or timing of births across the ages and in the shape of the fertility schedule) experienced the observed period age-specific fertility rates. This interpretation is equivalent to imagining that the observed period age-specific fertility rates are *constantly *extended sufficiently many years into the future (e.g., 35 years), so that a hypothetical cohort would have gone through the whole reproductive life span (e.g., from age 15 to 49) during this imagined extended period. This unadjusted conventional period TFR is the total number of births an average member of the hypothetical cohort would have for her whole life in such a static situation, in the absence of mortality throughout the reproductive ages. Note, however, that the assumption of no changes in tempo or timing of births inherent in the conventional period, TFR(*t*), is violated when the timing of fertility is changing. This violation results in well-known distortions of the conventional period TFR(*t*).

For this reason, Bongaarts and Feeney (1998) proposed an adjusted version of the period total fertility rate to minimize tempo effects—distortions in the observed TFR(*t*) due to changes in the tempo or timing of births. Specifically, based on the underlying assumption that the shape of the period age-specific fertility schedule does not change and on its implied assumption about equal changes in timing of births at all reproductive ages, Bongaarts and Feeney (1998) derived the following *quantum adjustment formula:*

TFR′(*t*) = TFR(*t*)/(1 – *r*(*t*)), [6]

where TFR′(t) is the adjusted order-specific TFR in year *t, *TFR(*t*) is the observed period order-specific TFR in year *t, *and *r(t) is the annual change in the order-specific period mean age at childbearing in year t. *The annual change, *r*(*t*), is defined as the difference in the mean age at childbearing of a particular birth order between two successive years. The unit of *r*(*t*) is “years old/per year.”

The Bongaarts-Feeney (B-F) quantum adjustment formula in Equation 6 looks like a relatively simple and straightforward adjustment to a long-standing fertility indicator used by demographers. But its development has led to substantial discussion, controversy, and additional analysis and applications to other topics in population models (see, e.g., Kim and Schoen 2000; Kohler and Philipov 2001; Van Imhoff and Keilman 2000; Zeng and Land 2001, 2002). This has contributed to a better understanding of how tempo changes in the age schedules of fertility and mortality can affect the fertility and mortality rates calculated by demographers and how these, in turn, can affect estimates of indices of these processes such as fertility rates and life expectancies from the corresponding population models.

## Conclusion

With origins in the mid-twentieth century, when a relatively few social scientists began to explore the idea of framing theories in mathematical form, mathematical sociology has grown considerably, as measured by the extent of journal literature in the field over time (Edling 2002). In addition to the texts already cited, a very accessible introduction is provided by Leik and Meeker (1975). Articlelength contributions involving mathematical models regularly appear in the field’s major comprehensive journals, the *American Sociological Review, The American Journal of Sociology, *and *Social Forces. *In addition, there are specialist journals that regularly publish contributions to mathematical sociology, including *The Journal of Mathematical Sociology, Sociological Methodology, Rationality and Society, *and *Social Networks. *In addition, in the United States, the American Sociological Association includes a mathematical sociology section that organizes a yearly set of sessions where new work is presented in its prepublication stage.

Another way of noting the penetration of mathematical sociology into the broader field is to take note of theoretical research programs in which theory and research are linked in part through theoretical models in mathematical form. A theoretical research program is a long-term multiperson program for developing a system of interrelated theories in relation to empirical research. For instance, the recent publication *New Directions of Contemporary Sociological Theory *(Berger and Zelditch 2002) contains introductions to more than a dozen such programs of the larger number that exist in the field. In short, mathematical sociology is a thriving part of the field of sociology that we expect will continue to produce fresh examples of the types of model building that we have sketched and that contribute to the advancement of our empirical knowledge of the social world.

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