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Mathematics LibreTexts

12.1: Introduction to the Idea of Equivalence


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We have been examining some of the ways that structural analysts look at network data. We began by looking for patterns in the overall structure (e.g. connectedness, density, etc.) and the embeddedness of each actor (e.g. geodesic distances, centrality). Next, we introduced a second major way of going about examining network data by looking for "sub-structure", or groupings of actors that are closer to one another than they are to other groupings. For example, we looked at the meaning of "cliques", "blocks", and "bridges" as ways of thinking about and describing how the actors in a network may be divided into sub-groups on the basis of their patterns of relations with one another.

All of this, while sometimes a bit technical, is pretty easy to grasp conceptually. The central node of a "star" network is "closer" to all other members than any other member - a simple (if very important) idea that we can grasp. A clique as a "maximal complete sub-graph" sounds tough, but, again, is easy to grasp. It is simply the biggest collection of folks who all have connections with everyone else in the group. Again, the idea is not difficult to grasp, because it is really quite concrete: we can see and feel cliques.

Now we are going to turn our attention to somewhat more abstract ways of making sense of the patterns of relations among social actors: the analysis of "equivalence classes". Being able to define, theorize about, and analyze data in terms of equivalence is important because we want to be able to make generalizations about social behavior and social structure. That is, we want to be able to state principles that hold for all groups, all organizations, all societies, etc. To do this, we must think about actors not as individual unique persons (which they are), but as examples of categories - sets of actors who are, in some defined way, "equivalent". As an empirical task, we need to be able to group together actors who are the most similar, and to describe what makes them similar, and to describe what makes them different, as a category, from members of other categories.

Sociological thinking uses abstract categories routinely. "Working class", "middle class", "upper class" are one such set of categories that describe social positions. "Men" and "women" are really labels for categories of persons who are more similar within category than between category - at least for the purposes of understanding and predicting some aspects of their social behavior. When categories like these are used as parts of sociological theories, they are being used to describe the "social roles" or "social positions" typical of members of the category.

Many of the category systems used by sociologists are based on "attributes" of individual actors that are in common across actors. If I state that "European-American males, ages 45-64 are likely to have relatively high incomes" I am talking about a group of people who are demographically similar - they share certain attributes (maleness, European ancestry, biological age, and income). Structural analysis is not particularly concerned with systems of categories (i.e. variables) that are based on descriptions of similarity of individual attributes (some radical structural analysis would even argue that such categories are not really "sociological" at all). Structural analysis seek to define categories and variables in terms of similarities of the patterns of relations among actors, rather than attributes of actors. That is, the definition of a category, or a "social role" or "social position" depends upon its relationship to another category. Social roles and positions, structural analysts argue, are inherently "relational". That's pretty abstract in itself. Some examples can make the point.

What is the social role "husband"? One useful way to think about it is as a set of patterned interactions with a member or members of some other social categories: "wife" and "child" (and probably others). Each one of these categories (i.e. husband, wife, child) can only be defined by regularities in the patterns of relationships with members of other categories (there are a number of types of relations here - monetary, emotional, ritual, sexual, etc.). That is, family and kinship roles are inherently relational. The network analyst translates this idea by saying that there are "equivalence classes" of husband, wife, child, etc.

What is a "worker"? We could mean a person who does labor (an attribute, actually one shared by all humans). A more sociologically interesting definition was given by Marx as a person who sells control of their labor power to a capitalist. Note that the meaning of "worker" depends upon a capitalist - and vice versa. It is the relation (in this case, as Marx would say, a relation of exploitation) between occupants of the two roles that defines the meaning of the roles.

The point is: to the structural analyst, the building blocks of social structure are "social roles" or "social positions". These social roles or positions are defined by regularities in the patterns of relations among actors, not attributes of the actors themselves. We identify and study social roles and positions by studying relations among actors, not by studying attributes of individual actors. Even things that appear to be "attributes of individuals" such as race, religion, and age can be thought of as short-hand labels for patterns of relations. For example, "white" as a social category is really a short-hand way of referring to persons who typically have a common form of relationships with members of another category - "non-whites". Things that might at first appear to be attributes of individuals are really just ways of saying that an individual falls in a category that has certain patterns of characteristic relationships with members of other categories.