Mathematical sociology

From Wikipedia, the free encyclopedia

Mathematical sociology is the area of sociology that uses mathematics to construct social theories. Mathematical sociology aims to take sociological theory, which is strong in intuitive content but weak from a formal point of view, and to express it in formal terms. The benefits of this approach include increased clarity and the ability to use mathematics to derive implications of a theory that cannot be arrived at intuitively. In mathematical sociology, the preferred style is encapsulated in the phrase "constructing a mathematical model." This means making specified assumptions about some social phenomenon, expressing them in formal mathematics, and providing an empirical interpretation for the ideas. It also means deducing properties of the model and comparing these with relevant empirical data. Social network analysis is the best-known contribution of this subfield to sociology as a whole and to the scientific community at large. The models typically used in mathematical sociology allow sociologists to understand how predictable local interactions are and they are often able to elicit global patterns of social structure.

History

Starting in the early 1940s, Nicolas Rashevsky, and subsequently in the late 1940s, Anatol Rapoport and others, developed a relational and probabilistic approach to the characterization of large social networks in which the nodes are persons and the links are acquaintanceship. During the late 1940s, formulas were derived that connected local parameters such as closure of contacts – if A is linked to both B and C, then there is a greater than chance probability that B and C are linked to each other – to the global network property of connectivity.

Moreover, acquaintanceship is a positive tie, but what about negative ties such as animosity among persons? To tackle this problem, graph theory, which is the mathematical study of abstract representations of networks of points and lines, can be extended to include these two types of links and thereby to create models that represent both positive and negative sentiment relations, which are represented as signed graphs. A signed graph is called balanced if the product of the signs of all relations in every cycle (links in every graph cycle) is positive. Through formalization by mathematician Frank Harary this work produced the fundamental theorem of this theory. It says that if a network of interrelated positive and negative ties is balanced, e.g. as illustrated by the psychological principle that "my friend's enemy is my enemy", then it consists of two subnetworks such that each has positive ties among its nodes and there are only negative ties between nodes in distinct subnetworks. The imagery here is of a social system that splits into two cliques. There is, however, a special case where one of the two subnetworks is empty, which might occur in very small networks. In another model, ties have relative strengths. 'Acquaintanceship' can be viewed as a 'weak' tie and 'friendship' is represented as a strong tie. Like its uniform cousin discussed above, there is a concept of closure, called strong triadic closure. A graph satisfies strong triadic closure If A is strongly connected to B, and B is strongly connected to C, then A and C must have a tie (either weak or strong). 

n these two developments we have mathematical models bearing upon the analysis of structure. Other early influential developments in mathematical sociology pertained to process. For instance, in 1952 Herbert A. Simon produced a mathematical formalization of a published theory of social groups by constructing a model consisting of a deterministic system of differential equations. A formal study of the system led to theorems about the dynamics and the implied equilibrium states of any group. 

The emergence of mathematical models in the social sciences was part of the zeitgeist in the 1940s and 1950s in which a variety of new interdisciplinary scientific innovations occurred, such as information theory, game theory, cybernetics and mathematical model building in the social and behavioral sciences.

Further developments

In 1954, a critical expository analysis of Rashevsky's social behavior models was written by sociologist James S. Coleman. Rashevsky's models and as well as the model constructed by Simon raise a question: how can one connect such theoretical models to the data of sociology, which often take the form of surveys in which the results are expressed in the form of proportions of people believing or doing something. This suggests deriving the equations from assumptions about the chances of an individual changing state in a small interval of time, a procedure well known in the mathematics of stochastic processes. 

Coleman embodied this idea in his 1964 book Introduction to Mathematical Sociology, which showed how stochastic processes in social networks could be analyzed in such a way as to enable testing of the constructed model by comparison with the relevant data. The same idea can and has been applied to processes of change in social relations, an active research theme in the study of social networks, illustrated by an empirical study appearing in the journal Science.

In other work, Coleman employed mathematical ideas drawn from economics, such as general equilibrium theory, to argue that general social theory should begin with a concept of purposive action and, for analytical reasons, approximate such action by the use of rational choice models (Coleman, 1990). This argument is similar to viewpoints expressed by other sociologists in their efforts to use rational choice theory in sociological analysis although such efforts have met with substantive and philosophical criticisms.

Meanwhile, structural analysis of the type indicated earlier received a further extension to social networks based on institutionalized social relations, notably those of kinship. The linkage of mathematics and sociology here involved abstract algebra, in particular, group theory. This, in turn, led to a focus on a data-analytical version of homomorphic reduction of a complex social network (which along with many other techniques is presented in Wasserman and Faust 1994). 

In regard to Rapoport's random and biased net theory, his 1961 study of a large sociogram, co-authored with Horvath turned out to become a very influential paper.  There was early evidence of this influence. In 1964, Thomas Fararo and a co-author analyzed another large friendship sociogram using a biased net model. Later in the 1960s, Stanley Milgram described the small world problem and undertook a field experiment dealing with it. A highly fertile idea was suggested and applied by Mark Granovetter in which he drew upon Rapoport's 1961 paper to suggest and apply a distinction between weak and strong ties. The key idea was that there was "strength" in weak ties. 

Some programs of research in sociology employ experimental methods to study social interaction processes. Joseph Berger and his colleagues initiated such a program in which the central idea is the use of the theoretical concept "expectation state" to construct theoretical models to explain interpersonal processes, e.g., those linking external status in society to differential influence in local group decision-making. Much of this theoretical work is linked to mathematical model building, especially after the late 1970s adoption of a graph theoretic representation of social information processing, as Berger (2000) describes in looking back upon the development of his program of research. In 1962 he and his collaborators explained model building by reference to the goal of the model builder, which could be explication of a concept in a theory, representation of a single recurrent social process, or a broad theory based on a theoretical construct, such as, respectively, the concept of balance in psychological and social structures, the process of conformity in an experimental situation, and stimulus sampling theory.

The generations of mathematical sociologists that followed Rapoport, Simon, Harary, Coleman, White and Berger, including those entering the field in the 1960s such as Thomas Fararo, Philip Bonacich, and Tom Mayer, among others, drew upon their work in a variety of ways.

Present research

Mathematical sociology remains a small subfield within the discipline, but it has succeeded in spawning a number of other subfields which share its goals of formally modeling social life. The foremost of these fields is social network analysis, which has become among the fastest growing areas of sociology in the 21st century. The other major development in the field is the rise of computational sociology, which expands the mathematical toolkit with the use of computer simulations, artificial intelligence and advanced statistical methods. The latter subfield also makes use of the vast new data sets on social activity generated by social interaction on the internet.

One important indicator of the significance of mathematical sociology is that the general interest journals in the field, including such central journals as The American Journal of Sociology and The American Sociological Review, have published mathematical models that became influential in the field at large. 

More recent trends in mathematical sociology are evident in contributions to The Journal of Mathematical Sociology (JMS). Several trends stand out: the further development of formal theories that explain experimental data dealing with small group processes, the continuing interest in structural balance as a major mathematical and theoretical idea, the interpenetration of mathematical models oriented to theory and innovative quantitative techniques relating to methodology, the use of computer simulations to study problems in social complexity, interest in micro–macro linkage and the problem of emergence, and ever-increasing research on networks of social relations. 

Thus, topics from the earliest days, like balance and network models, continue to be of contemporary interest. The formal techniques employed remain many of the standard and well-known methods of mathematics: differential equations, stochastic processes and game theory. Newer tools like agent-based models used in computer simulation studies are prominently represented. Perennial substantive problems still drive research: social diffusion, social influence, social status origins and consequences, segregation, cooperation, collective action, power, and much more.

Research programs

Many of the developments in mathematical sociology, including formal theory, have exhibited notable decades-long advances that began with path-setting contributions by leading mathematical sociologists and formal theorists. This provides another way of taking note of recent contributions but with an emphasis on continuity with early work through the use of the idea of “research program,” which is a coherent series of theoretical and empirical studies based on some fundamental principle or approach. There are more than a few of these programs and what follows is no more than a brief capsule description of leading exemplars of this idea in which there is an emphasis on the originating leadership in each program and its further development over decades. 

• Rational Choice Theory and James S. Coleman: After his 1964 pioneering Introduction to Mathematical Sociology, Coleman continued to make contributions to social theory and mathematical model building and his 1990 volume, Foundations of Social Theory was the major theoretical work of a career that spanned the period from 1950s to 1990s and included many other research-based contributions.. The Foundation book combined accessible examples of how rational choice theory could function in the analysis of such sociological topics as authority, trust, social capital and the norms (in particular, their emergence). In this way, the book showed how rational choice theory could provide an effective basis for making the transition from micro to macro levels of sociological explanation. An important feature of the book is its use of mathematical ideas in generalizing the rational choice model to include interpersonal sentiment relations as modifiers of outcomes and doing so such that the generalized theory captures the original more self-oriented theory as a special case, as point emphasized in a later analysis of the theory. The rationality presupposition of the theory led to debates among sociological theorists. Nevertheless, many sociologists drew upon Coleman’s formulation of a general template for micro-macro transition to gain leverage on the continuation of topics central to his and the discipline's explanatory focus on a variety of macrosocial phenomena in which rational choice simplified the micro level in the interest of combining individual actions to account for macro outcomes of social processes.
• Structuralism (Formal) and Harrison C. White: In the decades since his earliest contributions, Harrison White has led the field in putting social structural analysis on a mathematical and empirical basis, including the 1970 publication of Chains of Opportunity: System Models of Mobility in Organizations which set out and applied to data a vacancy chain model for mobility in and across organizations. His very influential other work includes the operational concepts of blockmodel and structural equivalence which start from a body of social relational data to produce analytical results using these procedures and concepts. These ideas and methods were developed in collaboration with his former students François Lorraine, Ronald Breiger, and Scott Boorman. These three are among the more than 30 students who earned their doctorates under White in the period 1963-1986.  The theory and application of blockmodels has been set out in detail in a recent monograph.. White's later contributions include a structuralist approach to markets and, in 1992, a general theoretical framework, later appearing in a revised edition
• Expectation states theory and Joseph Berger: Under Berger’s intellectual and organizational leadership, Expectation States Theory branched out into a large number of specific programs of research on specific problems, each treated in terms of the master concept of expectation states. He and his colleague and frequent collaborator Morris Zelditch Jr not only produced work of their own but created a doctoral program at Stanford University that led to an enormous outpouring of research by notable former students, including Murray Webster, David Wagner, and Hamit Fisek. Collaboration with mathematician Robert Z. Norman led to the use of mathematical graph theory as a way of representing and analyzing social information processing in self-other(s) interactions. Berger and Zelditch also advanced work in formal theorizing and mathematical model building as early as 1962 with a collaborative expository analysis of types of models. Berger and Zelditch stimulated advances in other theoretical research programs by providing outlets for the publication of new work, culminating in a 2002 edited volume that includes a chapter that presents an authoritative overview of Expectation states theory as a program of cumulative research dealing with group processes. 
• Formalization in Theoretical Sociology and Thomas J. Fararo: Many of this sociologist’s contributions have been devoted to bringing mathematical thinking into greater contact with sociological theory. He organized a symposium attended by sociological theorists in which formal theorists delivered papers that were subsequently published in 2000. Through collaborations with students and colleagues his own theoretical research program dealt with such topics as macrostructural theory and E-state structuralism (both with former student John Skvoretz), subjective images of stratification (with former student Kenji Kosaka), tripartite structural analysis (with colleague Patrick Doreian) and computational sociology (with colleague Norman P. Hummon). Two of his books are extended treatments of his approach to theoretical sociology.
• Social Network Analysis and Linton C. Freeman: In the early 1960s Freeman directed a sophisticated empirical study of community power structure. In 1978 he established the journal Social Networks. It rapidly became a major outlet for original research papers that used mathematical techniques to analyze network data. The journal also publishes conceptual and theoretical contributions, including his paper “Centrality in Social Networks: Conceptual Clarification.” The paper has been cited more than 13,000 times. In turn, the mathematical concept defined in that paper led to further elaborations of the ideas, to experimental tests, and to numerous applications in empirical studies. He is the author of a study of the history and sociology of the field of social network analysis.
• Quantitative Methodology and Kenneth C. Land: Kenneth Land has been on the frontier of quantitative methodology in sociology as well as formal theoretical model building. The influential yearly volume Sociological Methodology has been one of Land’s favorite outlets for the publication of papers that often lie in the intersection of quantitative methodology and mathematical sociology. Two of his theoretical papers appeared early in this journal: “Mathematical Formalization of Durkheim's Theory of Division of Labor” (1970) and “Formal Theory” (1971). His decades-long research program includes contributions relating to numerous special topics and methods, including social statistics, social indicators, stochastic processes, mathematical criminology, demography and social forecasting. Thus Land brings to these fields the skills of a statistician, a mathematician and a sociologist, combined. 
• Affect Control Theory and David R. Heise: In 1979, Heise published a groundbreaking formal and empirical study in the tradition of interpretive sociology, especially symbolic interactionism,Understanding Events: Affect and the Construction of Social Action. It was the origination of a research program that has included his further theoretical and empirical studies and those of other sociologists, such as Lynn Smith-Lovin, Dawn Robinson and Neil MacKinnon. Definition of the situation and self-other definitions are two of the leading concepts in affect control theory. The formalism used by Heise and other contributors uses a validated form of measurement and a cybernetic control mechanism in which immediate feelings and compared with fundamental sentiments in such a way as to generate an effort to bring immediate feelings in a situation into correspondence with sentiments. In the simplest models, each person in an interactive pair, is represented in terms of one side of a role relationship in which fundamental sentiments associated with each role guide the process of immediate interaction. A higher level of the control process can be activated in which the definition of the situation is transformed. This research program comprises several of the key chapters in a 2006 volume of contributions to control systems theory (in the sense of Powers 1975) in sociology.
• "Distributive Justice Theory" and Guillermina Jasso: Since 1980, Jasso has treated problems of distributive justice with an original theory that uses mathematical methods. She has elaborated upon and applied this theory to a wide range of social phenomena. Her most general mathematical apparatus – with the theory of distributive justice as a special case -- deals with any subjective comparison between some actual state and some reference level for it, e.g., a comparison of an actual reward with an expected reward. In her justice theory, she starts with a very simple premise, the justice evaluation function (the natural logarithm of the ratio of actual to just reward) and then derives numerous empirically testable implications.
• Collaborative research and John Skvoretz. A major feature of modern science is collaborative research in which the distinctive skills of the participants combine to produce original research. Skvoretz, in addition to this other contributions, has been a frequent collaborator in a variety of theoretical research programs, often using mathematical expertise as well as skills in experimental design, statistical data analysis and simulation methods. Some examples are:
• Collaborative work on theoretical, statistical and mathematical problems in biased net theory.
• Collaborative contributions to Expectation States Theory.
• Collaborative contributions to Elementary Theory.
• Collaboration with Bruce Mayhew in a structuralist research program. From the early 1970s, Skvoretz has been one of the most prolific of contributors to the advance of mathematical sociology.

The above discussion could be expanded to include many other programs and individuals including European sociologists such as Peter Abell and the late Raymond Boudon.

Awards in mathematical sociology

The Mathematical Sociology section of The American Sociological Association in 2002 initiated awards for contributions to the field, including The James S. Coleman Distinguished Career Achievement Award. (Coleman had died in 1995 before the section had been established.) Given every other year, the awardees include some of those just listed in terms of their career-long research programs:

• 2018: Ronald Breiger, University of Arizona
• 2017: Lynn Smith-Lovin, Duke University.

• 2014: Philip Bonacich, University of California, Los Angeles.
• 2012: John Skvoretz, University of South Florida.
• 2010: David R. Heise, Indiana University.
• 2008: Scott Boorman, Yale University.
• 2006: Linton Freeman, University of California, Irvine.
• 2004: Thomas Fararo, University of Pittsburgh.
• 2002: Harrison White, Columbia University.

The section's other categories of awards and their recipients are listed at ASA Section on Mathematical Sociology

Texts and journals

Mathematical sociology textbooks cover a variety of models, usually explaining the required mathematical background before discussing important work in the literature (Fararo 1973, Leik and Meeker 1975, Bonacich and Lu 2012). An earlier text by Otomar Bartos (1967) is still of relevance. Of wider scope and mathematical sophistication is the text by Rapoport (1983). A very reader-friendly and imaginative introduction to explanatory thinking leading to models is Lave and March (1975, reprinted 1993). The Journal of Mathematical Sociology (started in 1971) has been open to papers covering a broad spectrum of topics employing a variety of types of mathematics, especially through frequent special issues. Other journals in sociology who publish papers with substantial use of mathematics are Computational and Mathematical Organization Theory, Journal of social structure, Journal of Artificial Societies and Social Simulation.

 

Articles in Social Networks, a journal devoted to social structural analysis, very often employ mathematical models and related structural data analyses. In addition – importantly indicating the penetration of mathematical model building into sociological research – the major comprehensive journals in sociology, especially The American Journal of Sociology and The American Sociological Review, regularly publish articles featuring mathematical formulations.

Social network

From Wikipedia, the free encyclopedia

A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics. 

Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s. Social network analysis is now one of the major paradigms in contemporary sociology, and is also employed in a number of other social and formal sciences. Together with other complex networks, it forms part of the nascent field of network science.

Overview

Evolution graph of a social network: Barabási model.

 

The social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies. The term is used to describe a social structure determined by such interactions. The ties through which any given social unit connects represent the convergence of the various social contacts of that unit. This theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is often ignored although this may not be the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form these network configurations, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics.

History

In the late 1890s, both Émile Durkheim and Ferdinand Tönnies foreshadowed the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society"). Durkheim gave a non-individualistic explanation of social facts, arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors. Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely knit networks rather than groups.

Moreno's sociogram of a 2nd grade class

 

Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently. In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski, Alfred Radcliffe-Brown, and Claude Lévi-Strauss. A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes, J. Clyde Mitchell and Elizabeth Bott Spillius, often are credited with performing some of the first fieldwork from which network analyses were performed, investigating community networks in southern Africa, India and the United Kingdom. Concomitantly, British anthropologist S. F. Nadel codified a theory of social structure that was influential in later network analysis. In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure. Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.

By the 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Also independently active in the Harvard Social Relations department at the time were Charles Tilly, who focused on networks in political and community sociology and social movements, and Stanley Milgram, who developed the "six degrees of separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who elaborated and championed the analysis of social networks.

Beginning in the late 1990s, social network analysis experienced work by sociologists, political scientists, and physicists such as Duncan J. Watts, Albert-László Barabási, Peter Bearman, Nicholas A. Christakis, James H. Fowler, and others, developing and applying new models and methods to emerging data available about online social networks, as well as "digital traces" regarding face-to-face networks.

Levels of analysis

Self-organization of a network, based on Nagler, Levina, & Timme, (2011)

 

Centrality

 

In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system. These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world is not feasible and is likely to contain so much information as to be uninformative. Practical limitations of computing power, ethics and participant recruitment and payment also limit the scope of a social network analysis. The nuances of a local system may be lost in a large network analysis, hence the quality of information may be more important than its scale for understanding network properties. Thus, social networks are analyzed at the scale relevant to the researcher's theoretical question. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level, and macro-level.

Micro level

At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.

Dyadic level: A dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship (e.g. multiplexity, strength), social equality, and tendencies toward reciprocity/mutuality. 

Triadic level: Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity/mutuality. In the balance theory of Fritz Heider the triad is the key to social dynamics. The discord in a rivalrous love triangle is an example of an unbalanced triad, likely to change to a balanced triad by a change in one of the relations. The dynamics of social friendships in society has been modeled by balancing triads. The study is carried forward with the theory of signed graphs. 

Actor level: The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor" or "ego". Egonetwork analysis focuses on network characteristics such as size, relationship strength, density, centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, are most commonly used in the fields of psychology or social psychology, ethnographic kinship analysis or other genealogical studies of relationships between individuals. 

Subset level: Subset levels of network research problems begin at the micro-level, but may cross over into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group actions or behavior.

Meso level

In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks.

Social network diagram, meso-level

 

Organizations: Formal organizations are social groups that distribute tasks for a collective goal. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships. Intra-organizational networks themselves often contain multiple levels of analysis, especially in larger organizations with multiple branches, franchises or semi-autonomous departments. In these cases, research is often conducted at a workgroup level and organization level, focusing on the interplay between the two structures. Experiments with networked groups online have documented ways to optimize group-level coordination through diverse interventions, including the addition of autonomous agents to the groups.

Randomly distributed networks: Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior.

Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" (shaded) in the scale-free diagram (on the right).

 

Scale-free networks: A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. The Barabási model of network evolution shown above is an example of a scale-free network.

Macro level

Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population. 

Diagram: section of a large-scale social network

 

Large-scale networks: Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences.

Complex networks: Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.

Theoretical links

Imported theories

Various theoretical frameworks have been imported for the use of social network analysis. The most prominent of these are Graph theory, Balance theory, Social comparison theory, and more recently, the Social identity approach.

Indigenous theories

Few complete theories have been produced from social network analysis. Two that have are Structural Role Theory and Heterophily Theory.

The basis of Heterophily Theory was the finding in one study that more numerous weak ties can be important in seeking information and innovation, as cliques have a tendency to have more homogeneous opinions as well as share many common traits. This homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. However, being similar, each member of the clique would also know more or less what the other members knew. To find new information or insights, members of the clique will have to look beyond the clique to its other friends and acquaintances. This is what Granovetter called "the strength of weak ties".

Structural holes

In the context of networks, social capital exists where people have an advantage because of their location in a network. Contacts in a network provide information, opportunities and perspectives that can be beneficial to the central player in the network. Most social structures tend to be characterized by dense clusters of strong connections. Information within these clusters tends to be rather homogeneous and redundant. Non-redundant information is most often obtained through contacts in different clusters. When two separate clusters possess non-redundant information, there is said to be a structural hole between them. Thus, a network that bridges structural holes will provide network benefits that are in some degree additive, rather than overlapping. An ideal network structure has a vine and cluster structure, providing access to many different clusters and structural holes.

Networks rich in structural holes are a form of social capital in that they offer information benefits. The main player in a network that bridges structural holes is able to access information from diverse sources and clusters. For example, in business networks, this is beneficial to an individual's career because he is more likely to hear of job openings and opportunities if his network spans a wide range of contacts in different industries/sectors. This concept is similar to Mark Granovetter's theory of weak ties, which rests on the basis that having a broad range of contacts is most effective for job attainment.

Research clusters

Communication

Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communication concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.

Community

In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.

Complex networks

Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis. Mechanisms such as Dual-phase evolution explain how temporal changes in connectivity contribute to the formation of structure in social networks.

Criminal networks

In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.

Diffusion of innovations

Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another. This line of research seeks to explain why some become "early adopters" of ideas and innovations, and links social network structure with facilitating or impeding the spread of an innovation.

Demography

In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.

Economic sociology

The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists' contributions to analyzing the impact of social structure and networks on the economy.

Health care

Analysis of social networks is increasingly incorporated into health care analytics, not only in epidemiological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.

Human ecology

Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.

Language and linguistics

Studies of language and linguistics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.

Literary networks

In the study of literary systems, network analysis has been applied by Anheier, Gerhards and Romo, De Nooy, and Senekal, to study various aspects of how literature functions. The basic premise is that polysystem theory, which has been around since the writings of Even-Zohar, can be integrated with network theory and the relationships between different actors in the literary network, e.g. writers, critics, publishers, literary histories, etc., can be mapped using visualization from SNA.

Organizational studies

Research studies of formal or informal organization relationships, organizational communication, economics, economic sociology, and other resource transfers. Social networks have also been used to examine how organizations interact with each other, characterizing the many informal connections that link executives together, as well as associations and connections between individual employees at different organizations. Intra-organizational networks have been found to affect organizational commitment, organizational identification, interpersonal citizenship behaviour.

Social capital

Social capital is a form of economic and cultural capital in which social networks are central, transactions are marked by reciprocity, trust, and cooperation, and market agents produce goods and services not mainly for themselves, but for a common good. 

Social capital is a sociological concept about the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of unfamiliarity with the local language). A positive relationship exists between social capital and the intensity of social network use. In a dynamic framework, higher activity in a network feeds into higher social capital which itself encourages more activity.

Network position and benefits

In many organizations, members tend to focus their activities inside their own groups, which stifles creativity and restricts opportunities. A player whose network bridges structural holes has an advantage in detecting and developing rewarding opportunities. Such a player can mobilize social capital by acting as a "broker" of information between two clusters that otherwise would not have been in contact, thus providing access to new ideas, opinions and opportunities. British philosopher and political economist John Stuart Mill, writes, "it is hardly possible to overrate the value ... of placing human beings in contact with persons dissimilar to themselves.... Such communication [is] one of the primary sources of progress." Thus, a player with a network rich in structural holes can add value to an organization through new ideas and opportunities. This in turn, helps an individual's career development and advancement. 

A social capital broker also reaps control benefits of being the facilitator of information flow between contacts. In the case of consulting firm Eden McCallum, the founders were able to advance their careers by bridging their connections with former big three consulting firm consultants and mid-size industry firms. By bridging structural holes and mobilizing social capital, players can advance their careers by executing new opportunities between contacts.

There has been research that both substantiates and refutes the benefits of information brokerage. A study of high tech Chinese firms by Zhixing Xiao found that the control benefits of structural holes are "dissonant to the dominant firm-wide spirit of cooperation and the information benefits cannot materialize due to the communal sharing values" of such organizations. However, this study only analyzed Chinese firms, which tend to have strong communal sharing values. Information and control benefits of structural holes are still valuable in firms that are not quite as inclusive and cooperative on the firm-wide level. In 2004, Ronald Burt studied 673 managers who ran the supply chain for one of America's largest electronics companies. He found that managers who often discussed issues with other groups were better paid, received more positive job evaluations and were more likely to be promoted. Thus, bridging structural holes can be beneficial to an organization, and in turn, to an individual's career.

Social media

Computer networks combined with social networking software produces a new medium for social interaction. A relationship over a computerized social networking service can be characterized by context, direction, and strength. The content of a relation refers to the resource that is exchanged. In a computer mediated communication context, social pairs exchange different kinds of information, including sending a data file or a computer program as well as providing emotional support or arranging a meeting. With the rise of electronic commerce, information exchanged may also correspond to exchanges of money, goods or services in the "real" world. Social network analysis methods have become essential to examining these types of computer mediated communication.

In addition, the sheer size and the volatile nature of social media has given rise to new network metrics. A key concern with networks extracted from social media is the lack of robustness of network metrics given missing data.