Mathematical sociology
Encyclopedia
Mathematical sociology is the usage of 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 often able to elicit global patterns of social structure.
, and subsequently in the late 1940s, Anatol Rapoport
and others, developed a relational
and probabilistic approach to the characterization of large social network
s in which the nodes are persons and the links are acquaintanceship. During 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. This effort led to Harary's Structure Theorem (1953), which 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).
In 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 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.
Sociologist, James S. 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. In addition, 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 provided impetus for the emergence of a good deal of effort to link rational choice thinking to more traditional sociological concerns involving social structures.
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).
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 (Berger 2000).
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.
, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
to construct social theories. Mathematical sociology
Sociology
Sociology is the study of society. It is a social science—a term with which it is sometimes synonymous—which uses various methods of empirical investigation and critical analysis to develop a body of knowledge about human social activity...
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
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
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 often able to elicit global patterns of social structure.
History
Starting in the early 1940s, Nicolas RashevskyNicolas Rashevsky
Nicolas Rashevsky was a Ukrainian-American theoretical biologist who pioneered mathematical biology, and is also considered the father of mathematical biophysics and theoretical biology.-Academic career:...
, and subsequently in the late 1940s, Anatol Rapoport
Anatol Rapoport
Anatol Rapoport was a Russian-born American Jewish mathematical psychologist. He contributed to general systems theory, mathematical biology and to the mathematical modeling of social interaction and stochastic models of contagion.-Biography:...
and others, developed a relational
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
and probabilistic approach to the characterization of large social network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
s in which the nodes are persons and the links are acquaintanceship. During 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
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, 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. This effort led to Harary's Structure Theorem (1953), which 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).
In 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 Simon
Herbert Simon
Herbert Alexander Simon was an American political scientist, economist, sociologist, and psychologist, and professor—most notably at Carnegie Mellon University—whose research ranged across the fields of cognitive psychology, cognitive science, computer science, public administration, economics,...
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.
Further developments
The model constructed by Simon raises 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.Sociologist, James S. 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. In addition, 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 provided impetus for the emergence of a good deal of effort to link rational choice thinking to more traditional sociological concerns involving social structures.
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
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
. 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).
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 (Berger 2000).
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 amongst the fastest growing areas of sociology in the 21st century. The other major development in the field is the rise of Computational sociologyComputational sociology
Computational sociology is a branch of sociology that uses computationally intensive methods to analyze and model social phenomena. Using computer simulations, artificial intelligence, complex statistical methods, and new analytic approaches like social network analysis, 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.
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). 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. Articles in Social Networks, a journal devoted to social structural analysis, very often employ mathematical models and related structural data analyses. In addition, and this is important as an indicator of 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.Further reading
- Berger, Joseph. 2000. "Theory and Formalization: Some Reflections on Experience." Sociological Theory 18(3):482-489.
- Berger, Joseph, Bernard P. Cohen, J. Laurie Snell, and Morris Zelditch, Jr. 1962. Types of Formalization in Small Group Research. Houghton-Mifflin.
- Coleman, James S. 1964. An Introduction to Mathematical Sociology. Free Press.
- _____. 1990. Foundations of Social Theory. Harvard University Press.
- Edling, Christofer R. 2002. "Mathematics in Sociology," Annual Review of Sociology.
- Fararo, Thomas J. 1973. Mathematical Sociology. Wiley. Reprinted by Krieger, 1978.
- _____. 1984. Editor. Mathematical Ideas and Sociological Theory. Gordon and Breach.
- Helbing, Dirk. 1995. Quantitative Sociodynamics. Kluwer Academics.
- Lave, Charles and James March. 1975. An Introduction to Models in the Social Sciences. Harper and Row.
- Nicolas Rashevsky.: 1965, The Representation of Organisms in Terms of Predicates, Bulletin of Mathematical Biophysics 27: 477-491.
- Nicolas Rashevsky.: 1969, Outline of a Unified Approach to Physics, Biology and Sociology., Bulletin of Mathematical Biophysics 31: 159-198.
- Rosen, Robert. 1972. Tribute to Nicolas Rashevsky 1899-1972. Progress in Theoretical Biology 2.
- Leik, Robert K. and Barbara F. Meeker. 1975. Mathematical Sociology. Prentice-Hall.
- Simon, Herbert A. 1952. "A Formal Theory of Interaction in Social Groups." American Sociological Review 17:202-212.
- Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge University Press.
- White, Harrison C. 1963. An Anatomy of Kinship. Prentice-Hall.
See also
- PositivismPositivismPositivism is a a view of scientific methods and a philosophical approach, theory, or system based on the view that, in the social as well as natural sciences, sensory experiences and their logical and mathematical treatment are together the exclusive source of all worthwhile information....
- StatisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
- Computational sociologyComputational sociologyComputational sociology is a branch of sociology that uses computationally intensive methods to analyze and model social phenomena. Using computer simulations, artificial intelligence, complex statistical methods, and new analytic approaches like social network analysis, computational sociology...
- Peter BlauPeter BlauPeter Michael Blau was an American sociologist and theorist. Born in Vienna, Austria, he immigrated to the United States in 1939. He received his PhD at Columbia University in 1952, and was an instructor at Wayne State University in Detroit, Michigan from 1949–1951, before moving on to teach...
- Harrison WhiteHarrison WhiteHarrison Colyar White is the emeritus Giddings Professor of Sociology at Columbia University. White is an influential scholar in the domain of social networks. He is credited with the development of a number of mathematical models of social structure including vacancy chains and blockmodels...
- Nicolas RashevskyNicolas RashevskyNicolas Rashevsky was a Ukrainian-American theoretical biologist who pioneered mathematical biology, and is also considered the father of mathematical biophysics and theoretical biology.-Academic career:...
- Society for Mathematical BiologySociety for Mathematical BiologyThe Society for Mathematical Biology is an international association co-founded in 1972 in USA by Drs.George Karreman, Herbert Daniel Landahl and by Anthony Bartholomay for the furtherance of joint scientific activities between Mathematics and Biology research communities,...
- Interpersonal tiesInterpersonal tiesIn mathematical sociology, interpersonal ties are defined as information-carrying connections between people. Interpersonal ties, generally, come in three varieties: strong, weak, or absent...
- Duncan Watts
- James Samuel Coleman
- James D. MontgomeryJames D. MontgomeryJames D. Montgomery is professor of sociology and economics at the University of Wisconsin–Madison. He received his Ph.D. in economics from Massachusetts Institute of Technology. He has applied game-theoretic models and non-monotonic logic to present formal analysis and description of social...
- Thomas FararoThomas FararoThomas J. Fararo is Distinguished Service Professor Emeritus at the University of Pittsburgh. After earning a Ph.D. in sociology at Syracuse University in 1963, he received a three year postdoctoral fellowship for studies in pure and applied mathematics at Stanford University...
- Social networkSocial networkA social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
- Network ScienceNetwork scienceNetwork science is a new and emerging scientific discipline that examines the interconnections among diverse physical or engineered networks, information networks, biological networks, cognitive and semantic networks, and social networks. This field of science seeks to discover common principles,...
- Barabasi-Albert Model
External links
- Mathematical Sociology Section Home Page
- Mathematical Sociology - an indepth explanation by Phil Bonacich [DEAD link!]
- The Society for Mathematical Biology
- Bulletin of Mathematical Biophysics
- European Society for Mathematical and Theoretical Biology (ESMTB)