June 13, 2006
Structural Equivalence
Social scientists are often interested not only in actor cohesion but also in the positional
equivalence of actors, in the sense of having identical or very similar connections to other
network actors. Structurally equivalent actors are in a competitive, rather than a cohesive,
relation. For example, two cabbage growers who market their produce to the same set of
retailers are structurally equivalent and in stiff competition to sell their vegetables. Structurally
equivalent actors are completely substitutable for one another. If one farmer were to withdraw
from the food network, it could easily be replaced by a structurally equivalent farmer, leaving the
original network structure unchanged. Perfect substitutability in a social network often generates
fierce competition to obtain favorable responses from other network participants (as is well
known to grade schoolers competing for their teacher’s attention). Network scholars who use
structural equivalence methods mostly are generally interested in understanding competitive
relations rather than group cohesion (Burt 1992).
Similar to clique identification, the definition of structural equivalence is very rigorous.
In a directed binary graph, two nodes are structurally equivalent in a specific relation if they
have identical patterns of ties sent to and received from all the other nodes in the network. More
precisely, nodes i and j are structurally equivalent if, for all nodes k in the network (but not
including i or j), node i sends a tie to node k, if and only if j also sends a tie to k, and node i
receives a tie from k if and only if j also receives a tie from k (Wasserman and Faust 1994:356).
For multiple relations, this condition must hold exactly in each of the R relations for the two
nodes to be structurally equivalent. The presence or absence of direct connections between
nodes i and j is irrelevant to determining whether they are structurallyl equivalent. Rather, their
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structural equivalence is determined only by their patterns of relations with the g - 2 other
network nodes.
Nondirected binary graphs make no distinction between senders and receivers of
relations. Thus, extending the definition of digraph structural equivalence, in a nondirected
graph actors i and j are structurally equivalent if, for all other actors k, i has a tie with k if and
only if j has a tie with k. Structural equivalence can also be applied to valued graphs, in which
ranking scales rather than binary values measure the ties between nodes. Strictly speaking, in
valued graphs, two nodes are structurally equivalent only when both have excatly identical
values of their ties with all other network nodes.
The preceding definitions of structural equivalence are too rigid to be practical for
empirical network analyses. Real network data rarely contain pairs meeting such stringent
standards. Rather, many nodes may be only approximately structurally equivalent, in the sense
that their connections with the other nodes are highly similar but not exactly identical. To
capture such approximations, researchers use variable measures of nodal similarity rather than
applying a strict all-or-nothing structural equivalence measure. The more similar two nodes are
in their respective connections with all the other nodes, the greater is their structural equivalence.
Structural Equivalence Measures
Measures of the structural equivalence of pairs of network actors are based the similarity
of their relations with other network actors. Two actors are more structurally equivalent to the
extent that their patterns of present and absent ties to and from the other actors are very similar.
Assuming a binary digraph, two structurally equivalent actors will have almost identical entries
in the corresponding rows and columns of the corresponding sociomatrix. Operationalizing this
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criterion, Burt (1978) proposed a Euclidean distance measure to approximate the structural
equivalence of actors i and j:
d ij 
g
[( x
k 1
ik
 x jk ) 2  ( xki  xkj ) 2 ] (i  j  k )
(4.14)
where d ij is the Euclidean distance between actors i and j, and the x’s are the
values (either 1 or 0 for binary relations) in the sociomatrix (where the first subscript denotes the
row and the second subscript the column). Because d ij is the positive square root of the sum of
two squared difference terms, d ij  0 . If two actors have identical ties to all others, they have
perfect structural equivalence and d ij = 0. But, the larger the d ij , the less the structural
equivalence of actors i and j.
To illustrate how to compute the structural equivalence using Euclidean distance, Figure
4.8 and Table 4.2 depicting the same five-node network structure in graph and matrix forms,
respectively. Figure 4.8 shows that actors 1 and 2 are strictly structurally equivalent as they both
have direct connection to actors 3 and 4 and no ties to actor 5. In contrast, actors 4 and 5 are not
structurally equivalent because, despiteboth sending a tie actor 3, actor 4 receives ties from
actors 1 and 2, whereas actor 5 receives no ties. Using the paired values from Table 4.2, the
Euclidean distance between actors 1 and 2 is:
d12  [( x13  x23 )2  ( x31  x32 )2 ]  [( x14  x24 )2  ( x41  x42 )2 ]  [( x15  x25 )2  ( x51  x52 )2 ]
(4.15)
d12  [(1  1)2  (0  0)2 ]  [(1  1)2  (0  0)2 ]  [(0  0)2  (0  0)2 ]
d12 = 0, indicating that actors 1 and 2 are exactly structurally equivalent. Can you show that the
Euclidean distance between actors 4 and 5 = 2 ?
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Because nondirected binary graphs make no distinction between senders and receivers of
relations, computing Euclidean distance is simpler:
d ij 
g
 (x
k 1
ik
 x jk ) 2
(i  j  k )
(4.16)
When multiple relations present in the network, the distance computation involves summing
squared differences across all R relations:
R
g
[( x
d ij 
r 1 k 1
ikr
 x jkr ) 2  ( xkir  xkjr ) 2 ]
(i  j  k ) (4.17)
A second important measure of structural equivalence is Pearson’s correlation coefficient
(used in the CONCOR algorithm discussed in the blockmodeling section below).
[MEGO (my eyes glaze over): Too many complex, indecipherable formulas for an intro-level
textbook. Edition #1, page 72 shows a simpler computation formula for r.]
g
rij 
(X
k 1
g
 R i )( X jk  R j )   ( X ki  C i )( X kj  C j )
ik
k 1


  ( X ik  R i ) 2   ( X ki  C i ) 2 
k 1
 k 1

g
g
1/ 2
g
 g

  ( X jk  R j ) 2   ( X kj  C j ) 2 
k 1
 k 1

1/ 2
(i  j  k ) (4.18)
where
Ri 
1 g
 X ik
g k 1
and
Ci 
1 g
 X ki
g k 1
ik
R i and C i in the equation are the average values of the entry value for the row i and column i
respectively. If the two actors i and j are structurally equivalent, the correlation between their
respective rows and columns in the matrix will be 1. According to the formula, one can compute
that R1 = R 2 = 2/5 and C 1 = C 2 = 0, and r12 equals 1 in Figure 4.8.
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June 13, 2006
The computation of correlation coefficients for pairs of nodes in symmetric network is
simpler than it is in asymmetric network, because there is no distinction between X ik and X ki ,
and between R i and C i . The formula that computes correlation coefficients in symmetric
network is as the following:
g
rij 
(X
k 1
ik
 R i )( X jk  R j )
g
 g

  ( X ik  R i ) 2  ( X jk  R j ) 2 
k 1
 k 1

(i  j  k )
1/ 2
(4.19)
Using this formula, we developed a JAVA program to compute the correlation
coefficients between pairs of nodes in a symmetric/undirected social network shown in Figure
4_5. Table 4.3 displays the results that the coefficient between B and E is 1, whereas it is -1
between C and F.
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