Carter's Research Project Page

The Structure of Large-Scale Interpersonal Networks

Structural Measurement and Comparison

Strategic Behavior in Social Contexts

Social Structural Impacts on Social Influence, Judgment, and Choice

The Structure of Large-Scale Interpersonal Networks

Distance and Large-Scale Network Structure

A large body of literature in the social sciences has established a strong, inverse relationship between distance in socio-physical space and extent of interpersonal interaction. At the macro level (i.e., scales of apx 1km and greater) prior work suggests the dominance of physical distance as a predictor of social ties. Unfortunately, however, past research in this area has generally failed both at parameterizing the distance/tie probability relationship, and at exploring the implications of this relationship for network structure. In recent work, I have attempted to address these questions by fitting hierarchical Bayesian models to various distance/tie frequency data sets, and then by sampling from the posterior predictive distributions to estimate structural properties conditional on population geometry. This research suggests (among other things) that small-world properties arise "naturally" from the distance/tie probability relationship (Watts's hypothetical rewired lattice can be seen as a gross stylization of this empirical model), and that the form of the relationship in question is a power law. Extrapolation of model uncertainty to larger populations confirms the initial impression that physical distance accounts for the overwhelming majority of uncertainty at large scales, and demonstrates that this effect does not depend on population density (though it does depend on geometry). Application of the tie probability models to interregional relations (see below) indicates non-trivial effects of population geometry on the aggregate influence relations between regions; this can be applied to such questions as the evolution of urban areas over time. Work on these problems has been supported by grants from the NSF and CASOS, and results have been presented at the 2000 US-Japan Mathematical Sociology conference, the 1999 and 2001 ASA meetings, and the 2001 CASOS conference. Several CASOS working papers are available, and some foundational work is contained within a paper with Kathleen Carley which is currently under review; a related piece recently appeared in Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers, published in 2003 by the National Academies Press. (See also my dissertation page.)

Measures of Network Macrostructure

The vast majority of classical social network analysis was constructed to deal with small networks of less than 50-100 individuals. As we move beyond this range, to the consideration of structures among whole populations of actors, conventional methods begin to break down (both conceptually and computationally). At the same time, however, conventional alternatives (which generally consist simply of creating new "small-scale" networks of relations among macro-entities) are not sufficiently well-linked to the micro-structures they are supposed to express. To break this barrier, Kathleen Carley and I have proposed an approach to the measurement of network structure which is based on properties of aggregate tie-flow across regions in socio-physical space; the resulting measures can be expressed in terms of low-level entities, but are both amenable to sampling and computationally feasible to estimate. I have also applied this approach to the study of inter-regional relations, where the tie-flow model can be used as a foundation to bridge classical network analysis and the conventional "small networks of macro-entities" previously used in macrostructural analysis. Work on these problems has been presented at the 1999 ASA and 2001 CASOS conferences, and a paper on the subject is currently under review. (See also my dissertation page.)

Structural Measurement and Comparison

Canonical Labeling of Graphs to Facilitate Comparison

One fundamental open problem in the area of social network analysis is the appropriate comparison of unlabeled (or partially labeled) social structures. Given, for instance, a collection of kinship networks taken from different societies, how may one express the relative distances between these networks? Is it possible to utilize some general metric of distance to classify such structures? Can one identify a common, "central" structure from which all members of the set are minimally deviant in some sense? To answer these sorts of questions, Kathleen Carley and myself have been investigating the use of canonical labeling heuristics to uniquely identify social structures and to thereby permit the use of "classic" techniques such as Hamming distance analysis and the identification of the central graph among a set of structures. (Extensions of the methodology to visualization and to the cluster analysis of graph sets are also being investigated.) The resulting work has most recently been presented at the 1997 ASA conference, at the Carnegie Mellon University Network Seminar Series, and at the 1998 Sunbelt Network Conference.

Dealing with Error in Network Data

A great deal of research in the field of social network analysis (particularly the pathbreaking work of Bernard, Killworth, Sailer (BKS), and others), has focused on the extent to which informant reports can be used as proxies for behavioral data. A number of studies by BKS established the presence of extremely high error rates in self-report data; while important research by Burt and Bittner, Romney and Faust, and others indicated that these reports still contained valuable information, their use as proxy data continues to be problematic. (Similar results from David Krackhardt's CSS research have established that the problem extends to reports of third party observers as well.) Given that networkers are often interested in behavioral networks, dealing with the problem of error (as well as the related problem of missing data) depends on our constructing viable models to account for the uncertainty introduced by informant inaccuracy. As a step in this direction, I have developed a simple family of hierarchical Bayesian models which can cope with observer-dependent, asymmetric error rates in network data. In the spirit of related work by Romney and Batchelder, these models work by integrating multiple informant reports to get estimates of "the truth," with each informant's report being weighted by his or her estimated error rates. Given reasonable prior constraints on accuracy scores, monte carlo studies show excellent convergence of posterior estimates to the criterion ("true") structure, and to the underlying error rates. These studies also show considerable robustness on the part of the model to deviations from a number of assumptions, and indicate substantial gains over more conventional approaches (such as the LAS or central graph). This work was presented at the 2000 ASA meeting, and a paper containing the above results was published in Social Networks (2003). Subsequent elaborations of this work have been presented at the 2003 Sunbelt network conference, and the 2003 NAACSOS meeting.

The Interaction of Size and Density with Graph Level Indices

Graph level indices such as centralization, reciprocity, and hierarchy play an important role in structural theory. While many such measures have been defined, their properties are often poorly understood. Adding to this difficulty is the problem of size and density: by constraining these graph properties, one radically alters the distribution of graph level measures. Brigham Anderson, Kathleen Carley, and myself have done work which seeks to better characterize such behaviors of GLIs, and have developed a series of measures to characterize the GLI distributions across conditions. Using monte carlo simulation methods, we have further constructed a set of hypothesis tests which may be used to compare empirical GLIs both against theoretical models and against each other. Work on this problem has been published in a 1999 Social Networks article.

Graphical Representations of Life History Data

Life history data poses a number of challenges for would-be analysts. How can one integrate events in multiple life domains within a single analysis? How does one quantify the life-context of a particular event? How can life histories be compared? Joy Pixley and I have sought to answer some of these questions by means of a graphical approach. Applying the well-known formalism of the interval graph (widely used in operations research, computer science, and molecular biology) to life history data, we construct networks which represent individual life histories. These "life history graphs" can then be analyzed using techniques adapted from network analysis to uncover structural features of interest to life course researchers. Work on this problem will be presented at the 2001 ASA meeting; we have currently authored two working papers on the subject, one of which is currently under review.

Multivariate Methods for the Analysis of Graph Sets

While classical social network analysis has focused primarily on the study of positions within individual networks, the analysis of structural populations -- sets of graphs -- is an increasingly important area of study. Problems such as identifying underlying dimensional structure, uncovering archetypal substructures, constructing structural taxonomies, and finding shared tendencies between sets of structures surface in applications as diverse as the study of organizational structure, textual analysis, team performance, group structure/dynamics, and structural evolution. Given the large body of literature on multivariate data analysis which addresses very similar problems for attributional data, it is reasonable to inquire as to whether extant multivariate methods might be applied to relational data. In ongoing research, Kathleen Carley and myself have addressed this question, and have demonstrated a set of procedures which allow for the application of traditional multivariate methods to graph sets under arbitrary labeling assumptions. These procedures may be applied both to distance and covariance-based approaches, thereby generalizing both the metric distance methods of Banks and Carley and the network regression of Krackhardt. Work on this problem has been presented at the 2000 Sunbelt network and CASOS conferences, and has led to an ICES research report. A paper on this topic is currently under review.

Structural Complexity

The past decade has witnessed an explosion of interest in the notion of "complexity" throughout the sciences. Unfortunately, however, attempts to study complexity within the social sciences have been crippled by a lack of agreement on what constitutes "complexity," and of how complexity would be measured were such agreement to emerge. Is there a way to construct sensible, formal definitions of complexity within the social sciences? If so, are social systems actually complex (as is often claimed), or are they marked by simplicity? Focusing on graphical representations of social structure (i.e., networks), I have pursued the complexity question on two fronts. First, I have attempted to establish an axiomatic basis for the definition of structural complexity, and to assess the extent to which currently available complexity measures satisfy these axioms. Second, I have used one particular complexity measure (an adaptation of Lemple and Ziv's algorithmic complexity index) to examine a large number of social network data sets. With respect to the first, it can be shown that a number of "reasonable" assumptions regarding structural complexity lead to a family of measures which are degenerate; thus, relaxations are necessary in order to obtain non-trivial definitions. Using a more limited set of assumptions, it can be shown that the role complexity of Everett satisfies the axioms, though this is not unique. (Another measure, based on the number of distinct induced subgraphs, satisfies these same axioms while exhibiting somewhat different behavior.) The algorithmic complexity, which satisfies the core axioms, is also deeply connected with a range of important structural concepts (most prominently structural equivalence). Since this cannot be measured directly, the second stream of work employs a proxy index (known to have similar properties) to compare the apparent algorithmic complexities of empirical network structures to those of a random baseline. Surprisingly, empirical social networks appear to be nearly as complex as random structures of equivalent size and density, suggesting that it is these two factors which account for the bulk of the structure within this data. These results, while preliminary, suggest that there is a risk of "over-interpreting" apparent patterns in network data, where baseline effects from size and density have not been adequately accounted for. Work on this problem was presented at the 1999 Sunbelt network conference. A 2000 article in the Journal of Mathematical Sociology gave the results of the axiomatization, and the related empirical findings were published in a recent (2001) paper in Social Networks.

Strategic Behavior in Social Contexts

Communication and Cooperation

As it is trivial to show, cheap talk cannot affect the outcome of a Prisoner's Dilemma game played by rational actors when both payoffs and rationality are common knowledge. What, however, of adaptive agents? John Miller, David Rode, and myself have been exploring the coevolution of communication and cooperation among adaptive agents in a non-iterated Prisoner's Dilemma context. Modeling agents as finite state automata, we allow actors to develop their own symbolic languages, and find that these languages permit groups of actors to establish intervals of cooperative behavior. These "epochs" of cooperation are the result of a process by which actors develop special conversation sequences (or "secret handshakes") which allow cooperators to identify one another. Eventually, evolutionary pressures break down the handshake, permitting mimics to undermine group cooperation, and the system ultimately returns to its original state. Analysis of the model under a wide range of parameters reveals a strong relationship between actors' complexity and the amount of cooperation which can be sustained; populations of exceedingly complex actors cooperate far more often, and for longer, than populations of simple actors. These results have been verified across a variety of adaptation mechanisms, and have proven robust to changes in these assumptions. A working paper on this topic is available via the Santa Fe Institute Working Paper Series. A revised version appeared in the Journal of Economic Behavior and Organization (2002).

Empirical Play in the Hot Potato Game

Imagine being offered a bottle. Inside the bottle is a genie, who will grant you immense wealth should you accept the bottle and pass it on to another (willing) recipient; but there is a catch. Should you be unable or unwilling to promptly find another willing to take the bottle, the angered genie will cause you to die a horrible death. Worse yet, the 1001st person to accept the bottle will in doing so free the genie, who will then (being in a very foul mood) promptly visit the same horrible death on the unfortunate holder in place of the proffered reward. (You know all of this because it's written in ten languages on an indelible, glowing Surgeon General's warning label on the side of the bottle; anyone to whom you might wish to trade the bottle will clearly be aware of the terms.) Say that there are 1000 trades left. Do you take the bottle? This classic decision problem is the basis for what David Rode and I call the "hot potato good": an exchange good which becomes a bad after a finite number of trades (and/or after a finite trading period). Although we have shown that rational actors will not accept hot potato goods under a variety of circumstances, experiments we have conducted indicate that real people behave very differently. Analyzing various predictors of willingness to accept a hot potato good, we have found that the overall pattern of subject behaviors is inconsistent not only with a rational actor model, but also with various boundedly rational models of strategic behavior. Surprisingly, subjects' willingness to accept a hot potato good appears tied to a Benthamite strategy of "rational altruism" (a la the work of Andreoni and Miller), wherein the total value of the potential chain of trades is the primary determinant of individual action. This pattern of apparently altruistic behavior is not mirrored by subjects' self-reported decision making processes, suggesting the potential for covert or possibly unintended altruism on the part of some subjects. A paper with basic results from this project is currently under review, and will be presented at the 2001 ASA meeting.

Social-Structural Impacts on Social Influence, Judgment, and Choice

Belief Panics

Belief panics (transient events characterized by the spontaneous emergence and subsequent dissolution of socially contiguous "regions" of minority opinion within a larger population) have been documented by ethnographers such as Jeffrey Victor and Richard Hicks, but formal treatments have thusfar been lacking. To construct a framework in which to model the panic phenomenon, I have elaborated a simple model of social influence which treats actors as socially embedded, naive Bayesians who exchange categorical signals regarding their present belief states. Extensive analytical treatment of this model in the two-person case, as well as virtual experimentation across larger networks, reveals a strong tendency towards local convergence and global polarization. Further investigation has considered large populations in which actors are exposed to environmental signals, and in which saliency conditions/interaction rules are systematically varied. Current findings from this work seem to suggest that structural variables such as the presence of clustering do not affect the emergence of panic, although network density is a powerful predictor across all models; choice of saliency/interaction model appears to have significant and non-trivial effects on panic behavior. Work on this topic has been presented at the 1997 conference on Computational and Mathematical Organization Theory, and at the 1998 ASA conference. An initial paper developing the belief feedback model was published in Computational and Mathematical Organization Theory in 1998, and an ICES research report with subsequent results is also available.

Deliberation, Conformity, Rationality, and Influence

Traditional and normative theories of deliberative decision making have argued that the social processes of deliberation lead to increased cognitive sophistication, improved decision making, greater equity in outcomes, etc. Theories of social influence and conformity, by contrast, suggest that deliberation may lead to imitation of others' beliefs, circulation of redundant information, amplification of the opinions of particular alters, and a rapid convergence to stable minority/majority dichotomies. To what extent is it possible to reconcile these claims? Which outcomes best reflect the deliberative process, and what factors determine those outcomes? How does social structure both shape the evolution of opinion and itself change within a deliberative context? To help examine these questions, Peter Muhlberger and myself have designed a series of experiments to examine social network and cognitive phenomena in an ongoing political discussion forum. Via a series of web-based studies, we have also piloted a set of measures (based on ranking tasks) to assess violations of certain axioms of rational choice within the context of political decision making. By examining the extent to which deliberation does or does not facilitate the development of well-formed preferences, we hope to further concretize the link between deliberation, influence, and rational behavior in group settings. Work on this problem has been presented most recently at the 1998 Midwest Political Science Association conference.

Emergent Cultural Stratification

Life in a modern, structurally differentiated society entails the flow of individuals through a series of institutional tiers (such as jobs, schools, prisons, etc.), each of which contains some number of formal or informal organizations. In addition to this demographic process, however, actors are confronted with cultural expectations defining competent performance in social interaction. Given the existence of such processes, and given the basic social psychological mechanisms of imitation, trial and error learning, and homophily, what can we say about the emergence of stratification in society? Following Bourdieu's insights on the role of culture as a stratifying medium, and taking insights from Goffman's analysis of ritual social interaction, Thomas Fararo and myself have developed a simple computational model in which social stratification both within and across generations emerges from simple demographic, psychological, and interactive processes. Utilizing automata models, we allow actors to develop their own languages of ritual interaction, and permit them to explore new strategies within a cooperative game. Without invoking assortative mating, institutionalized restrictions on population flow, social capital, or other obvious promoters of stratification, we nonetheless find powerful forces of constraint which determine actors' trajectories through the life course. Investigation across model parameters suggests that these forces are unaffected by actor complexity or linguistic flexibility, and that it is macrostructure which has the strongest impact on the strength of stratification in the system. Results from this work are included in a 1999 paper in the Journal of Mathematical Sociology (see below), and additional material is available via an ICES research report.

Generative Structuralism

Given the dynamic, multi-level nature both of social theory and of empirical research into the social world, how can we hope to identify a unifying paradigm for sociology? Is it possible for such a framework to be compatible with and to assist formal theoretical efforts? In order to address questions such as these, Thomas Fararo (in collaboration with John Skvoretz and others) has constructed a developing approach to social theory known as Generative Structuralism. Recently, Fararo and myself have worked to extend the initial framework of GS to deal with problems of multi-level dynamics in complex systems, and to integrate three major traditions of sociological research. This extended version of GS has been illustrated with a simple model of cultural stratification which draws on work by Bourdieu, Goffman, White, and Fararo and Skvoretz. A paper on this topic was published in 1999 in the Journal of Mathematical Sociology.