$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 15: Regular Equivalence

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Regular equivalence is the least restrictive of the three most commonly used definitions of equivalence. It is, however, probably the most important for the sociologist. This is because the concept of regular equivalence, and the methods used to identify and describe regular equivalence sets correspond quite closely to the sociological concept of a "role". The notion of social roles is a centerpiece of most sociological theorizing.

• 15.1: Defining Regular Equivalence
Formally, "Two actors are regularly equivalent if they are equally related to equivalent others." That is, regular equivalence sets are composed of actors who have similar relations to members of other regular equivalence sets. The concept does not refer to specific other actors, or to presence in similar sub-graphs; actors are regularly equivalent if they have similar ties to any members of other sets.
• 15.2: Uses of the Concept
The regular equivalence approach is important because it provides a method for identifying "roles" from the patterns of ties present in a network. Rather than relying on attributes of actors to define social roles and to understand how social roles give rise to patterns of interaction, regular equivalence analysis seeks to identify social roles by identifying regularities in the patterns of network ties - whether or not the occupants of the roles have names for their positions.
• 15.3: Finding Equivalence Sets
The formal definition of regularly equivalence says that two actors are regularly equivalent if they have similar patterns of ties to equivalent others. Consider two men. Each has children (though they have different numbers of children). Each has a wife (usually different persons). Each wife, in turn, also has children and a husband (that is, they have ties with one or more members of each of those sets). Each child has ties to one or more members of the set of "husbands" and "wives".
• 15.S: Regular Equivalence (Summary)
The regular equivalence concept is a very important one for sociologists using social network methods, because it accords well with the notion of a "social role". Two actors are regularly equivalent if they are equally related to equivalent equivalent others. Regular equivalences can be exact or approximate. Unlike the structural and automorphic equivalence definitions, there may be many valid ways of classifying actors into regular equivalence sets for a given graph.