Mathematical sociology is a specialty within the field of sociology that uses mathematical models to build and better understand social theories. Some sociologists use these models to mathematically depict social networks, such as an individual's network of friends and colleagues.
A mathematical model can illustrate, for example, the degree of strength or weakness of a person's relationships with family members, friends or business colleagues. Sociologists then examine relevant empirical data, such as questionnaires that ask about human relationships with family, friends and acquaintances, to determine how well the mathematical model fits the empirical evidence. If the theory articulated by the mathematical model does not fit the relevant data, it is disregarded. Valid models are those that appear consistent with whatever data a given sociologist analyzes.
Many people mistakenly assume that the relationship between sociology and mathematics is limited to techniques for analyzing data to test hypotheses and answer specific research questions. It is true that sociologists, like their counterparts in other social sciences such as political science and economics, use statistical techniques and models to analyze data. This is different from mathematical sociology, which uses formal mathematical models to explain social arrangements and processes. These models can be functional equations, which suggest an outcome as a function of other variables.
For example, a mathematical model can be used to measure a person's degree of religious faith as a function of multiple variables, such as family influence, education, income, gender, ethnicity and other variables. Because it is a mathematical model, each of these variables must be expressed in some quantifiable way. Faith, for example, could be measured as a numerical scale or in a kind of proxy such as how often someone attends worship services.
Other examples of mathematical models include differential equations used in calculus, and statistical models, such as linear regression equations.
The American Sociological Association, the leading national academic association of sociologists, recognizes mathematical sociology as a specialty within the field. There are peer-reviewed academic journals devoted to mathematical sociology.
Mathematical sociology is frequently used in what is known as social network analysis, which strives to examine patterns of interaction among the members of a society. Examples include the interactions among citizens of a community, members of a religious group, employees of a business, or even members of a family. Social network analysis is based on the idea that patterns of interaction reveal a great deal about how an individual lives.
Theories of social networks are organized and expressed in mathematical terms, such as functional equations or statistical relationships. The appropriate mathematical model to analyze a given social network depends greatly on what the researcher or analyst wishes to learn.
Mathematical sociology has been used to study differences in trust within a society, the spread of business and societal fads, how powerful individuals and entities use their positions to advance their interests, and patterns of participation in collective activities, such as organizing to support a political candidate or to advance civil rights.