EQUIVALENCE

1)      Introduction

a)      Part of grand anthropological tradition in network theory

c)      A person’s role is not an inherent attribute but is fundamentally a relation. I am a teacher because of the relationships and activities I have with students, and students are students because of relations with teachers -- it is not something inherent in the individuals

i)        Over time, societies have named these roles and come to see them more as individualistically

d)      Society as a collection of positions or roles with characteristic relations to other positions and roles.

e)      Given data on social relations among a set of persons, should be possible to infer the social roles that people are playing

i)        Even if the roles do not have names.

2)      Structural equivalence

a)      Lorrain and White. Two nodes are structurally equivalent if they are connected to (just) the same third parties on the (just) the same relations

b)      Set of all structurally equivalent nodes in a network forms a structural equivalence class.

c)      If we order the rows and columns of an adjacency matrix so that members of of same SE class are next to each other, the matrix will form a blockmodel

i)        Each block will be all ones (“1-blocks” or all zeros  (“0-blocks”) (not counting the diagonal).

ii)       Now we can see how the classes relate to each other

d)      Even better, we can produce an image graph in which the nodes are SE classes and the ties are the relations between classes.

i)        Some call this the reduced blockmodel

ii)       Provides a concise model of the network in which no information is lost

e)      Can interpret these blockmodels

i)        Clique systems

ii)       Core/periphery systems

iii)     Prestige systems

3)      Why do we care about structural equivalence?

a)      Identifying sets of actors who are playing similar structural roles – i.e., are similarly connected to others

i)        Note that the actors within a set need not be connected to each other: they are together because they are connected to the same OTHERS

b)      According to Burt, structurally equivalent actors see each other as relevant reference points for social comparison. As a result, they imitate each other.

i)        Micro-basis for some elements of the kind of institutional theory that focuses on why organizations resemble each other

c)      SE actors also have identical environments. Therefore, one can apply a kind of evolutionary ecology argument to argue that SE actors will have similar behaviors because they are reacting to the same environments.

d)      Finally, SE actors are highly proximate, and so material can diffusion from one to the other (or from third parties to each of the SE actors).

4)      Working with structural equivalence in practice

a)      In real networks, actors are seldom perfectly equivalent

b)      Two basic approaches: profile similarities and imperfect blocks

c)      Profile similarity approach just measures the extent of equivalence for each pair of actors

d)      Imperfect block approach tries to place actors into discrete and mutually exclusive classes, so as to maximize the perfection of the 1-blocks and 0-blocks

5)      Problems with structural equivalence

a)      Theoretically different from cohesive subsets just as classes are different from sociological groups. But the actual definition confounds cohesion with similarity. Can be seen as a version of 2-cliques.

i)        So the methods tend to find groups rather than true classes

b)      Misspredicts certain kinds of class homogeneities, as in experimental exchange networks