We conclude this book by speculating a bit about the future of social network methodology. The following comments include observations about gaps in current network methods and “hot” trends that we think are likely to continue. We also include some wishful thinking about the directions in which we would like to see network methodology develop.

Statistical Models

We believe that statistical models will be a major focus for continued development and expansion of network methods. Clearly scientific understanding is advanced when we can test propositions about network properties rather than simply relying on descriptive statements. Great steps have been made in statistical models for dyads (including p1 and its relatives for valued relations, multiple relations, and for networks including actor attributes). We expect that further development of Markov graph models, logistic regressions, and so on will make statistical models more useful. Such models avoid the assumptions of dyadic independence, and thus promise to be more “realistic” than models of social networks that assume dyadic independence.

These future developments make use of very important research by Frank and Strauss (1986) and Strauss and Ikeda (1990) on Markov random graphs. Specifically, one can postulate statistical models for social networks which do not assume dyads are independent; in fact, the dependence structure of these models can be quite complicated. However, fitting them exactly is quite tedious computationally, unless one relies on the approximations described by Strauss and Ikeda, which allow one to calculate approximate maximum likelihood estimates of model parameters using logistic regression. These models, because of their generality and realism, have tremendous potential, which has yet to be realized.

This chapter presents the terminology and concepts of graph theory, and describes basic matrix operations that are used in social network analysis. Both graph theory and matrix operations have served as the foundations of many concepts in the analysis of social networks (Hage and Harary 1983; Harary, Norman, and Cartwright 1965). In this chapter, the notation presented in Chapter 3 is used, and more concepts and ideas from graph theory are described and illustrated with examples. The topics covered in this chapter are important for the methods discussed in the remaining chapters of the book, but they are especially important in Chapter 5 (Centrality, Prestige, and Related Actor and Group Measures), Chapter 6 (Structural Balance, Clusterability, and Transitivity), Chapter 7 (Cohesive Subgroups), and Chapter 8 (Affiliations, Co-memberships, and Overlapping Subgroups).

We start this chapter with a discussion of some reasons why graph theory and graph theoretic concepts are important for social network analysis. We then define a graph for representing a nondirectional relation. We begin with simple concepts, and progressively build on these to achieve more complicated, and more interesting, graph theoretic concepts. We then define and discuss directed graphs, for representing directional relations. Again, we begin with simple directed graph concepts and build to more complicated ideas. Following this, we discuss signed and valued graphs. We then define and discuss hypergraphs, which are used to represent affiliation networks.

Many methods for the description of network structural properties are concerned with the dual notions of social position and social role. In social network terms these translate into procedures for analyzing actors' structural similarities and patterns of relations in multirelational networks. These methods, which have been referred to as positional, role, or relational approaches, are the topic of Part IV. Although these methods are mathematically and formally diverse, they share a common goal of representing patterns in complex social network data in simplified form to reveal subsets of actors who are similarly embedded in networks of relations and to describe the associations among relations in multirelational networks.

The diversity of methods and potential complexity of mathematics has influenced our organization of topics in the following chapters. We begin this chapter with an overview of the theoretical and historical background for network role and positional analysis. We then discuss the basics of positional analysis. These basics will occupy Chapter 9 and the first part of Chapter 10. Chapters 9 and 10 discuss how to perform basic positional analysis using measures based on the mathematical notion of structural equivalence. In Chapters 11 and 12 we take up more advanced approaches to the notions of role and position and explore alternative formal definitions of these concepts. These chapters are concerned with the algebraic analysis of role systems using relational algebras (Chapter 11) and more general definitions of equivalence (Chapter 12).

This chapter discusses characteristics of social network data, with an emphasis on how to collect such data sets. We categorize network data in a variety of ways, and illustrate these categories with examples. We also describe the data sets that we use throughout the book. As noted in Chapter 1, the most important difference between social network data and standard social and behavioral science data is that network data include measurements on the relationships between social entities. Most of the standard data collection procedures known to every social scientist are appropriate for collecting network data (if properly applied), but there are a few techniques that are specific to the investigation of social networks. We highlight these similarities and differences in this chapter.

Introduction: What Are Network Data?

Social network data consist of at least one structural variable measured on a set of actors. The substantive concerns and theories motivating a specific network study usually determine which variables to measure, and often which techniques are most appropriate for their measurement. For example, if one is studying economic transactions between countries, one cannot (easily) rely on observational techniques; one would probably use archival records to obtain information on such transactions. On the other hand, friendships among people are most likely studied using questionnaires or interviews, rather than using archival or historical records. In addition, the nature of the study determines whether the entire set of actors can be surveyed or whether a sample of the actors must be taken.

Our plan in sequential analysis is to compare the likelihood of the occurrence of a sequence for two groups. For example, do fathers smile as often in response to their infant's smiles as mothers do? Obviously, to answer such questions we will have to estimate parameters of sequential connection from our samples and will want to believe that these estimates are, in some senses, “stable”. What is meant by “stable?”

Stationarity is a concern with the stability of our parameters of sequential connection over time. Homogeneity is a concern with the stability of our parameters of sequential connection across subjects. Recall that we only estimate these parameters from our sample, so we need tests of statistical significance to decide whether, given our data, the parameters are likely to differ, say from the first to the second half of the interaction.

The reader should consider the assessment of stationarity as a decision that is not automatically made by one significance test. Unfortunately, this has come to be the way stationarity is usually viewed – the test for stationarity is seen only as a precaution that must be taken for our analyses to be valid. We suggest two things: First, the decision about stationarity need not be automatic. It can depend on the researcher's judgment to a much greater degree than has been presumed in the past. Such a decision is much like deciding on an adequate reliability index for a measure; we suggest that the spirit of Cronbach et al.'s (1972) generalizability conceptions can be applied to a discussion of stationarity.

The Arundale Programs

Arundale (1982) wrote two useful programs for Markov analysis, called SAMPLE and TEST. The program SAMPLE takes as input a sequence of as many as 3,000 observations of up to 15 different discrete states. It outputs transition frequency and transition probability matrices for the input sequence. SAMPLE also tests the stationarity of the sequence (user specifies the number of chunks to break the original data set into) and it also tests order. The program TEST takes as input a set of transition frequency matrices and compares pairs of these matrix sets by specifying one set as the expected values and one set as the observed values. Both Pearson and likelihood ratio statistics are computed. It is possible to average a set of the input matrices before testing, or to test a group of matrices for homogeneity. The Arundale programs are also now available for an IBM-compatible personal computer.

Example

The data in this chapter come from a study, conducted by Krokoff, Gottman, and Roy (1988), of 120 married couples in a random sample study of blue- and white-collar couples who were either happily or unhappily married. Fifty-two of the home audiotapes of these couples attempting to resolve an existing marital issue were coded using the Couples Interaction Scoring System (see Gottman, 1979a). In the present context we examine only the affect portion of these codes, of which there are six: (1) husband positive (H+); (2) husband neutral (HO); (3) husband negative (H–); (4) wife positive (W+); wife neutral (WO); and, (6) wife negative (W–).

This book was initially inspired by Gene Sackett's conference on observational methods, held at Lake Wilderness, Washington, in 1976. In many ways that conference was a call for statistical approaches to the detection of interaction sequences in the observation of social behavior. This book is an integration of the methods needed for this task.

Many of the methods were in place prior to 1976, but were not well known. Fortunately, statistical workers have recently shown interest in the problem, and new methods have been developed. In fact, at the time of this writing there may be a bewildering array of procedures for the researcher.

What we present here is an integration of current techniques with suggested guidelines and rules of thumb. We will learn about the power and limitations of these techniques only by using them. Our goal is to make the techniques accessible, and in fact even easy to use.

The book is a sequel to an introductory volume by Roger Bakeman and John Gottman, also published by Cambridge University Press. Beginners should start with that book.

We wish to acknowledge the help of our teachers, particularly James Ringland and Stanley Wasserman. Professor Wasserman was very generous with his time in agreeing to review a draft of this manuscript. This work would not have been possible without the support and release time provided by Research Scientist Development Award K00257MH and a semester's sabbatical leave in 1984 to the first author.

Many applications of sequence analysis demand an overall procedure in which the same set of analyses is done for each case (subject, dyad, family, group) in our study. Most of our book is designed with such an application in mind. For example, we will usually want to know, for all the married couples in the study, which one “best” order of the Markov chain to pick, and, in general, how many segments are meaningful for dividing conversations in a stationarity analysis. This is the usual practice, because results would be difficult to report a case at a time.

However, in part to illustrate the flexibility of the methods we have mentioned, we will present a worked everyday example of the analysis of one couple using log-linear methods. We selected a case that has limitations in the amount of data available and that illustrates how much artfulness is needed to select a best model.

This chapter is best thought of as: (1) a worked example of log-linear methods; and as (2) an approach to hypothesis generating with sequential methods.

A Six-Step Procedure

Remember that the goal is to find the simplest model smaller than the fully saturated model that fits, that is, produces nonsignificant chi square. For example, if a Markov (1) (first-order Markov) model produces a significant chi square, this is evidence of the inadequacy of the model. The example will also illustrate how to cope with practical problems that can emerge from a model-fitting procedure.

This chapter covers the history of sequential analysis, limited primarily to those applications that have bearing on the observation of social behavior in psychology. Readers should not worry too much about undefined terms. We will begin defining terms in Chapter 3.

Initially, interest in the mathematics of sequential analysis had to do with a concern about how to characterize change. In a seminal paper in 1952, George Miller introduced Markov processes into psychology by noting that probabilistic methods had proved themselves in sensory psychology and test construction, both of which rely heavily on the independence of observations and on invariance over time. He wrote,

However, for areas of psychology, such as learning, that are essentially concerned with change

It is important to note that Miller was writing in the postwar context of the work of Norbert Wiener and his student Claude Shannon, and that the work of both was concerned with change and dependent data.

This section will introduce the words commonly used in sequential analysis. Usually, there is a set of coding categories: call them A1, A2, A3,…,AS. For example, in Ekman and Friesen's Facial Action Coding System (FACS), A1 might be the brows drawn down and together and A2 might be a nose wrinkle.

The general data situation is illustrated in Table 3.1. Each row represents a code, and each column represents a time interval. The time interval is presumed to be sufficiently small so that coding decisions can be reliably made. Time is used here in a general sense; for most of the techniques discussed in this book all that will matter, to preserve order, is that time 1 precedes time 2, and so on.

One advantage of displaying the data as they are in Table 3.1 is that every code can be represented as a binary time series. Another advantage is that we can add up all codes of a certain type and create a non-binary time series. For example, with the FACS we can add up all codes that indicate brow activity, or all codes that are predictors of sadness. An example of this is given by Tronick, Als, and Brazelton (1977), who graphed the sum of all the behaviors that represented the mother's involvement with her baby and subtracted the sum of all the behaviors that represented the mother's disengagement from the interaction. A similar time series was generated for the baby.

Since the late 1970s, a number of papers have appeared that advance the statistics of sequential analysis. This chapter briefly introduces the reader tp some of them.

Introduction

Gottman (1979a) defined the concept of dominance in marital interaction as asymmetry in predictability of behavior from one member of a dyad to the other. In a dyad, for example, if a husband's behavior at some point in time is better predicted from the wife's behavior at previous points in time than the reverse (i.e., predicting the wife's behavior from the husband's past behavior), then the wife is said to be dominant. Gottman and Ringland (1981) expanded on this notion in the context of mother-infant interaction; they also added the modification that controls for auto-correlation. In this modification, the wife's past behavior needs to add information in predicting the husband's behavior over and above the information provided by the husband's past behavior. The use of “dominance” may be further qualified by the content of the behavior itself; for example, in playful mother-infant interaction the fact that the mother's behavior is highly predicted by the infant's behavior but not conversely may reflect the infant's limited developmental social-cognitive abilities and the mother's responsiveness, and not the infant's dominance.

The focus of studies in dyadic social interaction has recently shifted from testing for such lagged dominance toward formally “discovering” some methodology for modeling the nature and structure of this dependence.

This chapter is essentially an advertisement for the sequential and time-series analysis of observational data. We would go so far as to make the statement: Anyone who has collected data over time and ignores time is missing an opportunity. We would like to back up this statement with several examples and with a discussion of conceptual issues.

The substantive theoretical discussion in this chapter concerns the analysis of marital and family interaction. Perhaps the most exciting area in which sequential analysis has been applied is social interaction. Sequential analysis, we will see, has supplied a great many analytic tools for researchers studying social interaction. It is not a mere data analysis option. It is a whole new way of thinking about social processes. The dimension of time is so central to conceptualizing social interaction that its use will lead us to think of interaction itself as temporal form.

We will begin with a few examples. The first is metacommunication, a concept introduced in what has become known as the “doublebind” paper (Bateson et. al., 1957). A metacommunication qualifies or comments on communication. It can be a nonverbal act that says “all that follows is really play,” or it can be a statement that comments on the process of communication, such as, “you're interrupting me,” or “that's not what we were discussing.”

Originally, information theory was devised in response to a need to conceptualize the amount of information in a message that might be transmitted through a noisy channel. In this event errors would clearly take place in transmitting the message, and they would be more or less serious depending on the amount of redundancy in the language system used. In English, for example, if we received the word “Q_ICK” we could guess with complete confidence that the missing letter was “U.” We can make these guesses because languages have a temporal structure.What does this temporal structure mean? It means simply that we gain information in prediction by knowing the past.

There was also a great deal of concern during World War II about breaking secret codes that the Axis powers used to transmit information. Conceptually the code-breaking problem is very similar to the problems we have in analyzing a sequence of codes. We inductively seek to determine the sequential structures that are present in the data.

Shannon's Approximations to English

In his 1949 paper, Claude Shannon generated a series of approximations to English. He used a 27-symbol alphabet: 26 letters and a space. The first approximation assumed that all symbols were independent and equally probable; these obviously not very good assumptions produced

which looks very little like a typical sample of English text. Shannon's next approximation was to account for asymmetries in the frequencies of the symbols.

It is not enough to be able to say that the order of sequential dependency is two or three, or even that there is some temporal structure in the data. In most of our research we have an experimental design and want to know if specific sequential patterns vary with our experimental factors. For example, suppose we have studied younger and older children and observed interactions of pairs of children either with their best friends or with strangers. We may want to know if friends are more likely than strangers to clarify their message in response to a clarification request. This is the kind of question sequential analysis is designed for.

Log-linear analysis is the generalization of analysis of variance to contingency tables in which the entries in each cell are counts, sampled under some distributional assumption. The typical case of interest to us is one in which we have some experimental design (which we call the contextual design) and in each cell of this design we have a timetable. The timetable has been constructed to be of a particular order based on our order and stationarity tests. It has also been based on pooling data across subjects (dyads, families, groups, etc.) based on our homogeneity tests. We now want to know how particular cells in the timetable that are of theoretical interest vary as a function of main effects and interaction effects in the contextual design. Such questions can be answered with log-linear models.