Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

Similarity breeds connection. This principle—the homophily principle—structures network ties of every type, including marriage, friendship, work, advice, support, information transfer, exchange, comembership, and other types of relationship. The result is that people's personal networks are homogeneous with regard to many sociodemographic, behavioral, and intrapersonal characteristics. Homophily limits people's social worlds in a way that has powerful implications for the information they receive, the attitudes they form, and the interactions they experience. Homophily in race and ethnicity creates the strongest divides in our personal environments, with age, religion, education, occupation, and gender following in roughly that order. Geographic propinquity, families, organizations, and isomorphic positions in social systems all create contexts in which homophilous relations form. Ties between nonsimilar individuals also dissolve at a higher rate, which sets the stage for the formation of niches (localize...

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Birds of a Feather: Homophily in Social Networks

The structure of scientific collaboration networks is investigated. Two scientists are considered connected if they have authored a paper together and explicit networks of such connections are constructed by using data drawn from a number of databases, including MEDLINE (biomedical research), the Los Alamos e-Print Archive (physics), and NCSTRL (computer science). I show that these collaboration networks form “small worlds,” in which randomly chosen pairs of scientists are typically separated by only a short path of intermediate acquaintances. I further give results for mean and distribution of numbers of collaborators of authors, demonstrate the presence of clustering in the networks, and highlight a number of apparent differences in the patterns of collaboration between the fields studied.

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The structure of scientific collaboration networks

A population is “hidden” when no sampling frame exists and public acknowledgment of membership in the population is potentially threatening. Accessing such populations is difficult because standard probability sampling methods produce low response rates and responses that lack candor. Existing procedures for sampling these populations, including snowball and other chain-referral samples, the key-informant approach, and targeted sampling, introduce well-documented biases into their samples. This paper introduces a new variant of chain-referral sampling, respondent-driven sampling, that employs a dual system of structured incentives to overcome some of the deficiencies of such samples. A theoretic analysis, drawing on both Markov-chain theory and the theory of biased networks, shows that this procedure can reduce the biases generally associated with chain-referral methods. The analysis includes a proof showing that even though sampling begins with an arbitrarily chosen set of initial subjects, as do most chain-referral samples, the composition of the ultimate sample is wholly independent of those initial subjects. The analysis also includes a theoretic specification of the conditions under which the procedure yields unbiased samples. Empirical results, based on surveys of 277 active drug injectors in Connecticut, support these conclusions. Finally, the conclusion discusses how respondent- driven sampling can improve both network sampling and ethnographic 44 investigation.

Respondent-driven sampling : A new approach to the study of hidden populations

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Random graphs with arbitrary degree distributions and their applications.

Introduction - James S Coleman and Thomas J Farraro PART ONE: THEMES OF ADVOCACY The Method of Decreasing Abstraction - Siegwart Lindenberg The Role of Models of Purposive Action in Sociology - Margaret Marini Rational Choice as a Principle - David Willer The Case of Elementary Theory The Attainment of Global Order in Heterogeneous Societies - Michael Hechter, Debra Friedman and Satoshi Kanazawa PART TWO: CRITICAL PERSPECTIVES Rationality and Emotion - Thomas J Scheff Homage to Norbert Elias Rationality and Robustness in Multilevel Systems - Michael Hannan Rational Choice Theory - Richard Munch A Critical Assessment of Its Explanatory Power PART THREE: RATIONAL CHOICE: PRO AND CON The Limits of Rational Choice Explanation - James Bonham Is Rational Choice Theory a Rational Choice Theory? - Peter Abell

Rational Choice Theory: Advocacy and Critique

This article presents a dynamic model for the formation of status orders in small task groups whose members may be differnetiated by a diffuse status characteristic. This theoretical strategy embodies " E-state structuralism," wich unites the social psychological formalism of expectation states theory with social network research. Using the development of status hierarchies via differential participation in group discussion, the formal model synthesizes (a) the conceptual structure of Fisek, Berger, and Norman, (b) Balkwell's formula for status effects on probabilities of directed participation, and (c) Fararo and Skvoretz's E-state structuralism model for the formation of dominance structures. After describing this synthesis, the article presents the model in axiomatic form and provides a simulation of its dynamic consequences.

Status and Participation in Task Groups: A Dynamic Network Model

A study of a biased friendship net

Presently the world is undergoing tremendous social, cultural and economic transformation. For sociologists, the challenge is arriving at a sound mapping of this tumultuous world stage. In this book, the contributing authors consider solidarity as a cognitive problem of basic science. They examine how solidarity is produced and reproduced, how it is related to social processes, and how such processes can be formalized and create conditions for productively studying their properties. Mathematical models and representations are presented by the authors as a coherent set of tools for understanding many social phenomena.

The Problem of Solidarity: Theories and Models

Even social and biological scientists familiar with the uses of mathematical models in their fields will find conceptually novel and sometimes mathematically forbidding elements in catastrophe models. One major difficulty is that these models use concepts that are more familiar to mathematicians and physicists than to behavioral scientists: potential functions and gradient systems, for instance. Starting from the basic elements of linear systems and their usual analysis (trajectories, equilibrium, stability of equilibria, comparative statics) it is possible to construct a conceptual and mathematical path from these familiar notions to more unfamiliar ideas and techniques dealing with structural stability and catastrophes. This is the aim: to present such a path from simple to complex, from old to new, from familiar to unfamiliar.



In a later section of the paper, the approaches of Thom and Zeeman to the scientific use of the catastrophe concept are presented. The conceptual framework employed in the paper is that of general systems theory and the basic notion within which all the remaining techniques and ideas find their place is that of a state-determined dynamic system. Apart from such an expository function, the paper also shows that the use of catastrophe concepts need not imply a special catastrophe model if analysis of any given dynamic model is extended to the problem of the implications for the events in state space of trajectories in parameter space. In this way, an effort is made to make this notion more accessible and useful in science.

Thomas J. Fararo

Papers

Rational Choice Theory: Advocacy and Critique

Status and Participation in Task Groups: A Dynamic Network Model

A study of a biased friendship net

The Problem of Solidarity: Theories and Models

An introduction to catastrophes