Position

1. Centrality

a. people speak of a node's centrality as their position in the network.
b. alternatively, phrase a hypothesis in terms of position (an actor;s position in the network determines their power, or how quickly they adopt, or influence, or risk) and then measure it in terms of centrality
c. but position is more than centrality. centrality is an attribute of a node's position.

2. Position

a. a network is a structure: a graph. the structure contains information about a set of actors. what information? only one bit of information: who is connected to who. that is all that is given.
b. how do we locate an actor in this structure? "where" is actor 'a'? We cannot point to a geographic region, because graphs do not use space in the usual sense. how a graph is drawn is arbitrary as long as the right nodes are connected to each other.
c. a location in a network is a set of nodes to whom a node is connected to. It is literally a neighborhood. If you ask where is sally, the answer is here, she's the one connected to bill and bob and bubba.
d. if two nodes occupy the same position in the network, it means they are connected to the same people.
i. which implies they are not connected to the same people as well.

3. Structural Equivalence

a. if two nodes are connected to the same people and not connected to the same people, then they structurally identical.
i. they have the same degree, same closeness centrality, same betweenness, are members of the same cliques, etc.: identical in every way.
b. structurally substitutable. Structurally equivalent. If position in a structure affects ones risks or opportunities, then two people who occupy the same structural position will have same outcomes.
i. if there is some dependent variable whose values depend on structural attributes of a node: how far from center it is, what its degree is, etc. then structurally equivalent nodes will have the same values on that variable.
c. don't need to know what it is about a position that makes certain outcomes happen. We simply predict that structurally similar individuals will have similar outcomes.
d. structurally equivalent doctors will adopt/not adopt new medicine. structurally equivalent organizations will develop same form. mechanisms:
i. tend to mimic each other
ii. responding to same social or corporate environments. if organizations are shaped by their environments, and they have the same environments, then they will have the same shapes
e.

4. Lorrain and White

a. An equivalence relation is a binary relation R (a set of ordered pairs of nodes) such that (i) aRa (reflexivity), (ii) aRb iff bRa (symmetry), and (iii) aRb and bRc implies aRc (transitivity).
i. A consequence of these conditions is the existence of equivalence classes
ii. Every equivalence relation induces a partition,
iii. and every partition is associated with an equivalence relation
b. 1971, Lorrain and White defined structural equivalence as formal definition of same position. Two nodes a,b are structurally equivalent iff for all c, (a,c) E implies (b,c) E
c. In other words, a b iff N(a) = N(b).
d. some graph
e. for directed graph, (a,b) are structurally equivalent iff for all c, (a,c) E implies (b,c) E, and (c,a) E implies (c,b) E. In other words, a b iff Ni(a) = Ni(b) and No(a) = No(b).
f. dirhier
g. in the adjacency matrix, a b iff have identical rows and identical columns
h. actually, there is an error in all three of these definitions. consider (a,c) in the "thinking graph" (see figure below). if (a,b) in E, then they can't be structurally equivalent unless self loops. Can try a b iff N(a) - {a,b} = N(b) - {a,b},
i. but this doesn't work if a has self loop and b does not
i. best definition is: a is structurally equivalent to b iff (a,b) is an automorphism of G.
i. An automorphism is a permutation of nodes such that (x,y) E iff ( (x), (y)) E.
j. will get into automorphisms later, in the meantime, think of it this way. two nodes are structurally equivalent if when you remove their labels and redraw the graph in a different shape, you can't tell which node was which.
i. this definition works for both a,c and b,d in thinking graph.
k. what are the sets of structurally equivalent nodes in the thinking graph?

5. Measures of Structural Equivalence

a. In real life, no two actors are structurally equivalent.
b. Burt 1976: euclidean distance
c. Correlation
d. Other measures

6. Blockmodeling

a. If we permute the order of the nodes in an adjacency matrix so that equivalent nodes are next to each other, and then draw lines through the matrix so that we create little matrix blocks, we find that
i. every matrix block is either all ones or all zeros (disregarding the diagonal)
ii. the blocks with ones are called one-blocks and the others are called zero-blocks
iii. the sets of equivalent actors are also called blocks: can be confusing. I will try to say actor-blocks and matrix-blocks.
b. This suggests an alternative way to approach structural equivalence in real networks: what if we let the blocks be imperfect? 1-blocks that are not all ones, or 0-blocks that are not all zeros?
i. we could design an algorithm that finds the optimal blocking: divides the network into classes of nodes that are more or less structurally equivalent.
c. First algorithm was Breiger et al's CONCOR.
f. Today we use combinatorial optimization algorithms, such as UCINET's tabu search