Structural Equivalence - Eclectic Anthropology Server

73
Social Networks, 1 (1978) 73-90
@Elsevier
Sequoia
S.A., Lausanne
- Printed
in the Netherlands
Structural Equivalence:
Meaning and Definition, Computation
Application*
and
Lee Douglas Sailer
University of cizlifornia, Irvine**
This paper presents a generalization of the concept of ‘Structural equivalence “* the key concept in algebraic approaches to the study of social
networks. Two points in a graph or set of relations will be called ‘Structurally related” if they are connected in the same ways to structural1.y
related points.
It is suggested that this new definition suitably weakens Lorrain and
White’s cutegorical approach, and is more appropriate. than CONCOR.
St~~t~~ra~ relatedness is compared to these approaches via several simple
examples.
Introduction
This paper is concerned with the algebraic approach to network analysis
(though
it isn’t always called that) exemplified
by White (19631, Boyd
(1969), Lorrain and White (197 l), Boyd et al. (197 l), Breiger et al. (1975),
Heil and White (1976), White et al. (1976), Boorman and White (1976),
D. White and Boyd (1977), and Boyd and Sailer (1978). These algebraic
ideas provide an underpinning for theories of social structure. These theories,
in turn, may be used to derive computational
definitions which accurately
reflect interesting aspects of the system under investigation. It is possible to
argue that the approach in this paper is consistent with the writings of Linton
(1936), Nadel(1957),
Merton (1957) and Goodenough
(1969).
The concept of “structural
equivalence” (sometimes called here SE) is
examined in four stages. First, the relationship
between structural
equivalence and standard
sociological concepts is discussed. Second, various
*In no way could this paper have been written without
the patience, persistance,
and perspicacity
of John Boyd and Douglas White. In addition, many of these ideas were a group creation of a seminar
at UC Irvine in 1977. This research was supported
in part by NSF grant #BNS 76-08386,
D. R. White,
H. Nutini, L. Brudner, Principal Investigators.
**Division of Social Sciencks, University of California, Irvine, Calif. 92717, U.S.A.
74
Lw Doughs Sailer
more detailed definitions
are contrasted.
Third, the details of one definition and a computational
procedure
for realising it are described. Fourth,
and finally, some applications are proposed.
Social structure,
roles and status
“Social structure
is the network of actually existing social relations”,
or at least so said Radcliffe-Brown
(1943: 190). “Social structure”
may be
taken to contrast with such systems as environment,
language, and beliefs.
Another common contrast is with function, content .and process. To understand my interpretation
of the quote above a careful distinction
must be
made. A “social relation” is not the tie between specific people, but rather a
set of ties in an entire population. The “network” is the patterning of social
relations over a set of persons, or positions, or groups, or organizations.
The
individual is important primarily as the vehicle for the extensive definition of
the relations in which we are interested. People are not social structure; the
interaction
of people may be structured,
but here I am concerned with the
interaction
of the interactions.
The interplay between patterns of kin ties
and patterns of economic ties, for example, is more interesting than (or at
least separate from) the relationship between two specific groups of people.
From this view, the specific actors are transitory.
They may change roles
through time even though the “social structure”
remains the same. (See
Nadel 1957: 16 - 17 for an early statement similar to this.)
Goodenough
(1969), following Linton, Merton, and Nadel, distinguishes
carefully between “status”, the rights and obligations of a role, and “social
identity”,
for Goodenough
the label for the special position occupied by one
of the actors in a specific relationship. ’ For Goodenough,
the methodological task is to find the rules which will translate informants’ beliefs about
the role structure (where “role” is the totality of status, social identity, and
other constructs) into behavior.
The approach here is different, but congruent. From data on the occurrence of interactions
(or of beliefs about them), can we define models for
which it will be easy to find rules relating the model to behavior? The social
scientist has two tasks before him then, to categorize
relations, and to
describe the relational interactions.
These tasks may entail the initial categorization of the actors in the network, i.e., by function.
Most occurrences in this paper of the technical (i.e., mathematical)
term
“relation” may be loosely replaced by the term “role”. For the purposes of
this paper, a role is a set of appropriate
behaviors exhibited by a pair of
actors in a particular context. A role may also be defined as a cover term or
gloss for certain individual attributes (which may be network attributes),
‘The terminology
used here is Goodenough’s.
The tarn
“role”
is used loosely
Structural equivalence
75
that is, a role is a function fulfilled by an individual. Important to the concept of role is the notion that a particular individual has certain connections
to other individuals, and that those others must themselves be in certain
positions.
Here it is assumed that roles are patterns of ties in observable and unobservable relations. The importance
of the reciprocal nature of roles is
thus sidestepped. The relations represent the actual occurrence of behavior,
and if subsets of the populations
are found to be reciprocal (e.g., doctors
and nurses), then let this be an external validation of the method. Certainly,
if “husband” and “wife” are relations in a data set, we would expect to find
that husbands have wives, but this is a different aspect of the problem. A
thorough treatment of this topic is necessary.
What is structural equivalence?
One of the easiest things to see in a network is who is connected to whom.
That is probably why so much effort has been expended in the development
of cluster and clique detection techniques. Cliques, however, are not to be
confused with roles. Here is an example of the difference. One would expect
the cliques in a kin network to be entities such as families, clans, lineages,
etc. The roles of interest in a kin network, though, are kin-types, e.g., father
and son. Families are certainly interesting structures in their own right, but
they are not roles. Rather, they are nodes in a higher order relation. Clusters
and cliques, as structures, correspond to such concepts as the family or clan.
The mathematical
structure which corresponds to the role, such as father or
boss, is the “block”. A “blockmodel”
is a set of such blocks and the relationships between
them.2 The cluster concept
is still relevant, however. A
“block” can be defined as a set of actors clustered together by virtue of their
structural equivalence.
Blocks are not defined by the amount of intra-role interaction,
as are
clusters, but by the intrinsic nature of the other blocks with which they
connect; e.g., judges interact with layers more than they do with other
judges; crooks interact with victims in a different way than they do with
other crooks. Notice that it is assumed that “status”, the rights and obligations of a role, is contained in or expressed by the patterns of ties labeled by
the name of the role. That is, it is exactly the actual exercise of rights and
obligations (and/or the expectations
of them) that we take as data, using
these data to discover the roles themselves.
Two people in the same role are substitutable.
This is what structural
equivalence
is, substitutability
with regard to relational ties. Relational
structure does not totally determine SE properties, though. Certainly, a great
*“Blockmodel”
and White (1976)
is used in a loose sense here, contrasted
with “clique structure”
for example.
provide an example of a very formal definition.
The spirit is the same.
Heil
76
Lee Douglas Sailer
deal of information
can be obtained from individual attributes,
but how
much information
can we tease out of relationships alone? That is the question asked here.
Definitions
of structural equivalence
Lorrain and White ( 197 1:63) label as “structurally
equivalent” any two
points that are related in the same ways to the same other points. That
is, for i, i, and k in a set N, and relations (sets of ordered pairs) R,, R,, . . . .
R, in N X N, a relation S is a structural equivalence
if iSj implies that, for
every k, iRk = jRk and kRi = kRj for each relation Ri.,
Of course, hardly anything or anybody is ever structurally
equivalent to
anybody else in the noisy and complex world of social relations. To overcome this obstacle. Lorrain and White apply what they call the “categorical
approach”.
They use various criteria to reduce the number of relations
derivable from the data, thus aggregating relationship data to enable nodes to
be blocked together.
Figure 1.
Map of the “‘algebraic approach” to the analysis of social structure.
combine
relations
DATA
-\
combine
individuals
'-._
-.fcf
-9,)
=%&..
od+.,_
I
CONCOR
B
Categorical
approach
-.
combine
individuals
-.
'. l
I
-+ Model
combine
relations
Furthermore,
they claim (p. 79) that this approach treats the blocking of
the nodes and the aggregation of the relations simultaneously,
as suggested
by the analogue of Category Theory. This, in fact, is not true. They identify
relations (morphisms in their terminology)
first, with no reference to blocks
at all, and only subsequently use this information
to identify nodes. Later in
this paper a technique
for blocking nodes first (and hence relations) that
coordinates
neatly with the so-called categorical approach is proposed. The
point is illustrated in Figure 1. Lorrain and White (hereafter LW) follow the
upper path from data, through the identification
of (clusters of) relations, to
a model. This paper takes the lower path. It is theoretically
desirable that this
diagram commute (at least conceptually,
if not computationally).
The ideal
approach takes the direct path from data to model.
Briefly, a blockmodel
is a reduced version of the original data, probably
with both fewer nodes (individuals)
and fewer relations. The task of the
investigator is to aggregate together SE nodes while preserving the meaningfulness of the relations. Breiger ct al. (1975) (hereafter BBA) rightly point
Structuralequivalence
71
out that such aggregations will rarely exist in practice. So they propose the
CONCOR algorithm as an attempt to make the best of an unclear situation.
There are several problems with CONCOR, however. Though in general
the ideas in BBA are quite sound, it is possible to come away from a reading
with a profound misunderstanding
of what CONCOR does. It is stated that
CONCOR is a blocking algorithm (as opposed to a clustering algorithm) and
that it may be applied to raw data (instead of first computing a proximity
measure of some kind). Let me explain why I think these two statements
are incorrect, beginning with the second.
In truth, CONCOR does seem to begin with raw network data, but what
does it do with it? First, it computes the Pearson product moment correlation for each pair of nodes based on their patterns of indegree. Below it is
suggested that ordinary correlation is a natural, if cumbersome,
measure of
SE, in the sense of LW, when the data are coded in a certain way. The point
is this: the first step taken by the CONCOR algorithm is to compute an
index of the structural equivalence of every pair of points. That CONCOR
works on raw data is an illusion, since any other index of structural equivalence could be input at this point, and CONCOR’s first step skipped.
CONCOR’s second (and each subsequent)
step is to compute correlations on the structural
equivalence matrix (from the previous step) until
the process converges to a partition of the nodes into two sets. What is
CONCOR doing? We may paraphrase LW - two points are SE if they are
related to the same points. In the same vein, we may paraphrase phase two
of the CONCOR algorithm - two points are SE if they are structurally
equivalent to the same points. This is totally different!
It is like saying that
lawyers are equivalent because they are equivalent to lawyers. The right
concept is that lawyers are equivalent because they are related to judges
and clients. (There is still a further generalization
of the concept described
below.)
Still, BBA is on the right track. Since SE, as defined by LW, is too strict
to be of any use, some weakening or generalization
must be found. LW
achieves this weakening by identifying relations. BBA generalizes the basic
definition, but inappropriately.
In passing, let me mention the definition of “lean fit” in BBA. Lean fit
is the notion that, for blocks, it is only important
to maximize zero submatrices in the relations, since nodes in the same block are not necessarily
connected
to each other, but the nodes to whom they are not connected
must be the same. This would be a very useful concept if it were utilized
by CONCOR (as it is by Heil and White (1976)). Notice, however, that the
claim in BBA (p. 333) is only that CONCOR seems to produce a result that
approximates
lean fit; there is no stronger logical connection between lean
fit and CONCOR in the paper.3
3Schwartz (1977) rejects CONCOR
on formal statistical
and computational
grounds aside from
the criticism of this paper. His criticism, using principal components
factor analysis, is thought to be
consistent with the ideas here.
78
Lee Lb&as
Sailer
An earlier version of the computational
formula presented below inspired
John Boyd to suggest a definition4
of SE that may be paraphrased
thus:
two points are SE if they are related in the same ways to points that are SE.
This is the proper generalization
of the LW definition
missed by BBA.
For example, two judges need not be connected
to the same crook in
order to be classified into the same block (as seems to be required by the LW
definition);
they need only each be connected to some crook, since crooks
are SE. This is why SE appears in the latter half of the definition as well as
in the first half.
Since “equivalent”
has such a well-established
meaning in mathematics,
we refer to Boyd’s concept as “structural relatedness” (SR).5 To understand
how SR works, refer to Figure 2 as you follow along. Assume that we are
only interested in arcs coming into nodes (following BBA) and that for the
moment we have only one relation, R. B is a structural relatedness relation
if and only if iBj implies that whenever there exists a k such that kRi, there
exists an m such that mRj and kBm. (This process is mirrored in the computational
version below.) Look at i and j. In order for i to be structurally
related to j the following must be true: for each k that chooses i there must
also be an m, to whom k is structurally related, that choosesj.
B has the desired property of weakening the original definition of SE. Its
main feature is that it does so in an intuitively
appealing way. Also, this
definition
can be generalized naturally to outdegree,
and to multiple relations as well. The apparent circularity of the de~nition may be avoided by
some tricky but equivalent rewording.
It also turns out that Boyd’s SR is not necessarily symmetric or transitive.
In Figs. 3(a) and (b) are simple examplesof how this might occur. In Fig. 3(a),
point 2 has a tie corresponding
to every tie that 3 has, so 3 is structurally
related to 2, but the converse does not hold. Likewise, the wavy lines in
Fig. 3(b) do not imply the existence of the broken lines, which would make
B transitive.
4Actually,
a whole array of possible definitions
has appeared,
some transitive,
some for both
in$egree and outdegree,
etc. In this paper, mainly one fairly general version is considered.
The iabel “B” sometimes refers to a relation, sometimes
to the algorithm
for deriving an SR relation, sometimes to a matrix of real numbers. It should be clear from the context.
Structural equivalence
79
Examples ofasymmetry and in transitivity.
Figure 3.
A
R
R
2.
I
R
__-- _-----
___
Rcl+:‘l
.3
B
I
‘.
t4
(a)
--__
t?
------?’B
--_ ---____---
_c--
(b)
An important
point missed by BBA is that at every iteration of any
iterative solution, the most important data, those most relevant to the task
of finding blocks in R, are still stored in the relation R. CONCOR throws
away R completely
after the first iteration, yet the information
that any
algorithm for finding blocks must utilize is contained in the ties in R. The
B-algorithm
below uses as much information
as possible at all times. Its
best guess at the structural relatedness of i and j is saved after each iteration
and used along with R to produce the next estimate.
Computing
B
There are many simple ways to estimate a sort of first-order SE. The
matching coefficient,
for example, and other measures described by Jardine
and Sibson (197 l), Sokal and Sneath (1963), and Hartigan (1975) could
possibly be interpreted
as measures of the degree to which two patterns of
connectivity
are the same. Pearson’s product
moment
correlation,
for
example, counts the number of identical connections
made by two points,
and then normalizes this count based on the number of choices made. It has
some disadvantages when applied to other than dichotomous
data, though
(Sailer 1972). None of these seems appropriate
for iteration, since it is not
clear what repeated application means. How should we compute a measure
like B? Here is a scheme which is intimately related to the definition presented above. Again, for indegree of a single relation alone, follow Figure 3
and this formula:
5
B;;’
=
max [min(Bi,,
k=lm=l
5
k=l
Rki
Rki, Rmi)]
80
Lee Douglas Sailer
where Rii is the “value” of the tie between i and j, typically a zero (0) or a
one (1). The min and max have been used so that, if the ties carry a value
from the interval [ 0, 11, instead of the set (0, I}, the definition is consistent
with elementary fuzzy set techniques (Zadeh 1965).
Every point is SR to itself, so we let B ’ = I, the identity matrix. We
assume that t iterations have been completed so that BFj is known for all i
and j. To compute Bii” , we find a k that is connected to i (a k such that
Rki > 0, otherwise min(Rki, Rnli) = 0 and nothing is summed), then find the
tn such that the path from k to m (via BL, ) to j (via R) is maximized, without
counting any m to j path in excess of the k to i path; we add the magnitude
of this path to the numerator.
In the denominator
we wish to find the best
m that could have had a tie to j, to standardize in the fuzzy case where k to
i was not very large, so that a poor m will suffice; the best possible m is k
itself, so we add min(Rki, Bik) = Rki to the denominator;
we repeat this for
every k related to i, and for every i and j.
Figure 4.
Calculate B jbr this relation.
This would be easier to follow in a simple example.
assume that every point is SR to itself, so let
B_!! =
Consider
Figure 4. We
1, ifi=j
0, otherwise
To compute B :2 we see that R 11 is nonzero, but there is no arc into point 2,
R MZ, such that min(B,,,
Rll, RmP) is nonzero. We sum a zero in the
numerator
and one in the denominator,
indicating that 1 is not SR to 2.
For R,, we sum zero in both the denominator
and the numerator. For RJ1,
we find that min(B,,, Rsl, R,,) = 1, so we sum 1 in the numerator and denominator;
this is evidence that 1 is SR to 2. For Rql we repeat step R21.
Thus
B:z =
0+0+1+0
~1+0+1+0
= 0.5
This process is repeated for i, j = 1, 2, 3, 4.
In Table l(a) are the results of the first iteration. Point 1 is SR to itself,
and to a lesser extent to 2 and 3 as well. Point 2 is slightly SR to everything.
Point 3 is completely SR to 1 and 2; and point 4 is SR to 2.
Notice that, for the calculation of Bf2, it was taken as evidence that 1 was
not SR to 2 that there was no point SR to 1 related to 2. Now, at the
second iteration, point 2 is (partially) such a point. B:, will thus be larger.
Structural equivalence
Table 1.
Intermediate
results from computation
81
of R in Figure 4.
(a) First iteration
2
3
4
1.0
0.5
0.5
0.0
0.33
1.0
0.33
0.33
1.0
1.0
1.0
0.0
I0.0
1.0
0.0
1.0
3
4
(b) Second iteration
gqqy&
(c) Third iteration
1
2
The results of the second iteration are in Table l(b). The final solution
(within 10W5) is in Table l(c).
Let me point out that in this example the matrix for each iteration is
monotonically
related to every other. If this were always true, there would
be no advantage to the iterative technique. Happily, this is not always the
case. Sometimes, when B:j is relatively very small, Bii’ ’ becomes very large,
especially for small t. For example, we might not have guessed from Table
l(a) that point 1 was totally SR to points 2 and 3, as it turned out to be by
Table l(c).
Once you understand
this mess, the rest is easy. For outdegree, reverse
all occurrences of ki and mj to ik and jm. For both indegree and outdegree,
compute the numerators
(ni and n,) and the denominators
(di and d,) as
above and then compute B:
B = (ni + n,)/(di
+ d,)
In the case of multiple relations we may compute all the ~2s and ds (one
pair for each relation), and then sum them in the obvious way.
Of course, there is a variety of options here. There are many ways to
combine the ~1sand ds into joint indexes of B and different ways of assessing
the strength of a path through two different relations. My feeling is that the
US and u’s are the building blocks. I see little difference between, say,
B = (ni + II,)/(Lfi + d,)
and
B = 0.5 f(ni/di) + (~,/d,)l
It would even be reasonable
to analyze indegree, outdegree,
and separate
relations separately,
and compare the answers individually. Current efforts
are directed this way.
There are two properties
of B to be discussed. One is the convergence
property of this B-algorithm. In a personal communication,
Steven Seidman
has provided a very simple proof of convergence.
Since the entries in the Bmatrix may never be larger than one (1 .O), and it can be shown that in each
iteration they may stay the same or get larger, but’never smaller, it is clear
that the process must eventually
stop. This is not a totally
satisfactory
solution, of course, since it is possible that the algorithm eventually converges to a matrix of all ones (especially in the sense that every human is
structurally
equivalent to every other). However, it seems likely that those
pairs of points which are really SR converge very early in the sequence, and
so it is possible to stop at some appropriate point chosen by ud i7oc criteria.6
There is also a demo~lstration
that interesting stable results can exist. For
a relation on ~1points, a stable solution for B (one that is guaranteed to repficate itself on subsequent iterations) depends directly on the ties in the relations V~Uthe solution to a set of U* simultaneous
linear equations in 11’ unknowns. For large II, operations research techniques find solutions to such
systems by iterative methods involving only II X n matrices much like the
algorithm for computing B.
The other is that CONCOR and B are each affected by the scale on which
the relations
are measured. There are various ways to handle this. For
example, it is possible to incorporate into any algorithm of this sort the technique of converting all values to ranks before computing. This is essentially
the technique
used to derive some nonparametric
statistical techniques.
Examples
and applications
The most important
question left to answer is “Does R work?“. 1 don’t
really know yet. There are reassuring results and others not so reassuring.
What I have are several examples, and a speculation.
As their first example, Lorrain and White analyze the two simple relations
in Figure S(a). I find it easy to think of P as “employs”
and P-’ as “is
61n the applications
below,
n,ax . If&+1
d. I < 0.1
[I
IJ
II
the criterion
is
Structural equivalence
Figure 5.
83
Toy example from L W: (a) the relations, (b) L W’s model, (c) B model.
-P
-p-1
1
1
A
5
A
5
A
3
2
A
3
4
4
(a)
0
%
P
295
legend
i--+j
=
iRj
i---aj =
iSj
o=
blocked together
03,4
employed by”. What kind of people are there in this system, and how are
they related? After considerable
calculation
and cogitation,
Lorrain and
White arrive at the result in Figure 5(b); there is a simple hierarchy.
Computation
of B for these two relations produces the matrix in Table 2
that is represented
in pictorial form by Figure 5(c). The wavy arrow indicates that 5 is structurally
related to 2, ie., 2 is substitutable
for 5. Point 5
is definitely
not substitutable
for 2 because 5 has no employees! Actually,
Bzs is substantially
greater than zero, so that with a lower cutoff exactly the
same result as Figure 5(b) may be obtained.7
Table 2.
B matrix for P and P-’ from Figure S(a),
5
1
2
3
4
7Since B produces
a matrix of real numbers, it is necessary to convert it somehow to a graphically
pleasing structure.
Typically,
a cutoff value is arbitrarily
chosen to transform
B into a relation which
can then be displayed as a hierarchical
tree, pre-order, or whatever.
In this example, B retained some additional info~~lation about P and P-’
that the categorical approach lost. This is remarkable because one would
assume that the formal algebraic approach is more “accurate” than the fuzzy
computational
approach.
Yet, R appears to be more informative
(in this
example), and in a way consistent with LW results. When less information
in
B is retained it produces the same result.
Lorrain and White mention (p. 64) that nodes in a cycle (such as a mutual
exchange system with more than two participants)
are isomorphic, but that
this is different from SE. They are correct that their concept of SE is not the
same as isomorphism,
but it should be. All the nodes in a simple cycle are
defil~itely playing the same role. They are sLlbstitutable back in the sense
that a role is played by pairs (or more) of actors. To substitute
1 for 2 in
F’igurc 6, for example, impbcs that after the natural permutation
has been
made an SE state has been reached.
Figure 6.
C~gcles,isonlor~~histrls,mod&, anlbi,uity.
data
/'-2\
Figure 6 illustrates one of the possible problems with B. Notice that there
are three images of the data that satisfy the definition of R, and that two of
them are in a real sense orthogonal.
The B-algorithm
finds a solution like
the one called A, which becomes solution C as cutoff thresholds are lowered.
What happens to solution B? The difference between A and R can be interpreted as separate dimensions of social space. In LW, greater direct control
is available to the investigator,
so that such problems do not occur. (This
may also mean that the investigator may be able to avoid undesirable results.)
Some analogous technique
for the conlputational
approach in this paper
wouId be useful.
The Sampson
monastery
data
Sampson (1969) has provided Lorrain and White, Breiger, Boorman, and
Arabie, D. White and Boyd, Heil and White, and this author with a body of
well-collected,
substantively
motivated
network
data
from
a failing
monastery.
BBA, HW, and LW each analyze the eight relations and replicate
some of the results obtained by Sampson by more traditional
methods.
D. White and Boyd (1978) extend the analysis using entailments instead of
the more strict equality, with equally promising results.
Here is an analysis of the small version of Sampson’s data found in LW to
illustrate the technique further, and more importantly,
to show that it in
fact makes the diagram in Figure 1 commute.
A generally positive relation, P, and a generally negative one, N, are extracted from Sampson’s data from near the end of his stay in the monastery.
A complete investigation
of P and N (and the LW analysis) is provided by
Boyd and Sailer ( 1978). Of relevance here are the comparable results using 8
in place of the categorical approach. The argument is not that these two
approaches
are different,
but that they are two sides of the SAME coin.
Their results must be the same, if the algebraic approach is to be internally
consistent.
LW follow this procedure:
Starting with two relations, P and N, they
generate all the possible distinct compositions
(e.g., PP, NN, NP, NN, PPP,
PPN, etc.). They find that there are “ . ..probably
at least several hundred”
compounds
(p_ 72), and so keep only those generated by strings of length
less than five or six Ps and Ns, leaving finally only seventy compound relations.
At this point they begin identifying
relations (which they call “morphism?)
on the basis of the relative size of their intersections,
e.g., P and
PPP share eight of the twelve possible ties and are thus defined to be equal.
Likewise, NN is identified with NNN. Because of these identifications,
the
seventy compound relations are reduced to only five distinct ones.
Mainly for expository
purposes, these five are reduced in the simplest
possible nontrivial way to two separate models of two relations each. That
is, when two relations are identified, several other pairs of relations must
also be combined in order to retain a consistent semigroup or category structure. In all cases but two, this reduced semigroup turns out to be the trivial
one with only one element. For the two reductions for which the semigroup
has two relations, there is no particular rationalization
aside from the fact
that they are easy to analyze.
Because of the identification
of many pairs of relations (and the subsequent creation of new relations in the reduced version which represent the
original relations), there is a partition induced on the original seven elements.
Within each block of this partition the individuals cannot be distinguished on
the basis of their ties in the two relations of the reduced set.
86
Lee Douglas Sailer
Figure 7.
L WModel 1.
P
N
Caley table
note:
Figure 8.
N connects to everything
L WModel 2.
P
N
N:--+
These two models, their partitions and relations, are shown in Figures 7
and 8-a
As discussed above, the categorical approach, despite its elegance, is combinatorially,
and hence computationally,
intractable.
Even for this small
example LW must severely limit the growth of the “category”
to strings of
length five or six9 and, more importantly,
their sociometric
strategies are
ad hoc at best. In the blockmodeling
work of White et al. (1976), and Boorman and White (1976), we see further tacit admission that this approach is
too cumbersome.
There they turn to CONCOR for relief. We now consider
the result of using B here in an analogous way.
*Notice that the labels “I”’ and “N” are definitely
not the original P and N. Rather, the set of all
relations
which has been identified
with P has induced the relation labeled “E” in Figures 7 and 8,
and then “P” was chosen as a reasonable label. (Likewise for “N”.)
9As LW points out (p. 90), it really does not make any difference
to the final result in this case,
though it could do in general.
Table 3.
B-matrix, computed for P and N
1
2
3
4
1
2
3
4
5
6
7
1
0.96
0.92
0.74
1
0.55
0.90
0.93
1
0.87
0.91
0.85
0.86
0.87
0.88
0.66
0.91
0.50
0.86
0.91
0.84
0.95
0.71
0.93
1
0.95
0.74
0.85
5
0.89
0.60
0.91
0.88
1
0.62
0.90
6
0.92
0.93
0.91
0.92
0.94
1
0.96
7
0.94
0.81
0.90
0.88
0.94
0.83
1
The first step is to compute B for the two relations P and N. This result
is shown in Table 3. Here are three points:
(1) As explained above, the B-matrix in Table 1 could be considered THE
model of the social roles in the system generated by P and N. This matrix
alone contains explicitly the various substitutabilities
of pairs of actors and
hence their positions in the relation data. B contains no info~ation
not
contained in P and N, and, if humans were only matrix perceivers, we would
need to go no further in our analysis.
(2) B is non-transitive and non-symmetric
(in Zadeh’s (1965) fuzzy sense).
This is not surprising since we know that the ties are somewhat amorphous.
When the system is more strictly structured, say in a formal organization, we
would expect B to be closer to a (fuzzy) equivalence relation.
(3) The values in B are all close to one (1.0). This is caused, again, by the
relative homogeneity
of the ties in this particular case.‘O
Since we are not able to decode the type of data structure in Table 3, we
reduce it to a less complicated
one. Two points are equivalent to the extent
that they are mutually related, which we can compute from B by replacing
each element by the smaller of itself and its reciprocal pair. Likewise, we can
then dichotomize
on some appropriate cutoff value (e.g., 0.95) and perform
a transitive closure. Call this relation“p”.
The blocks in the partition associated with the resulting equivalence relation are (1,3, 5, 7), (4), and (2,6).
This is similar to the second partition found by LW, the blocks of items
3 and 4 having been switched. Suppose we look at the relations P/o and N/o
in Figure 9.
Even though the partitions are slightly different, the semigroups are the
same! In the same way that P and N interact in the LW semigroup, so do
they here. To the extent that semigroups provide a reasonable model of
social structure in the monastery, B captures the same structure as the “categorical” approach, as expected.”
%‘here is co ncern over this tendency of the values in B to become so large. Various ways to control
this are being considered. Notice, though, that it doesn’t really matter! Don’t be misled by the
absolute size of the values; their relative size is important.
“Boorman and White (1976) provide a detailed explanation of why semigroups might provide
interesting models of social structure.
88
Lee Douglas
Figure 9.
Sailer
Quotient relations induced hv 0.
P:,
N: ---*
But look at the relative computational
complexity
associated with these
two approaches.
The LW approach is at least an order of magnitude more
expensive, with the cost-ratio increasing rapidly with the size of the problem.
This is why H. White and his colleagues must repeatedly resort to the use of
CONCOR.
Incorporating
attribute
irzfornzatiorl
All actors are substitutable
for themselves. This is axiomatic. Recall that
in the algorithm for computing B, B’ is initiallized by the identity matrix.
What other initial structures make sense?
This relational approach is academic in the sense that it arbitrarily rejects
any information
provided by individual attributes. The fact that two people
are rich might be as important
in establishing them in a structurally
equivalent position as the fact that they both have employees. It is possible to
convert attributes
to relations and to use these derived relations as you
would ordinary ones, but it is not necessary to do so. Suppose we have an
initial guess at the structural relatedness of two individuals based on the
similarity of their individual attributes. For example, for each pair we “correlate” their age, mode of dress, income, race, sex, etc. If we use these data
gains.
as our initial B1, instead of I, we might accomplish two noteworthy
First, the algorithm might converge more rapidly, having been given a head
start, so to speak. Second, by starting from a B based on otherwise unused
data, we might be able to control the kind of ambiguous solutions discussed
in the section on the Toy example in Figure 6.
This section raises another comforting thought, as well. To the extent that
the individual similarity is a useful analytical principle, it seems that we have
been using the concept of structural equivalence all along, and in a form that
is quite naturally integrated into the present one.
Structural equivalence
89
Conclusions
I think of the matrix of real numbers between zero and one produced by
B as the real model of structural relatedness. It captures a great deal of information about the role structure reflected in the relations. It is not reduced
enough to make the information
it contains available to the human eye,
however. Hierarchical
clusters, multidimensional
scales, trees, graphs, semilattices, and other structures might each be appropriate to represent the role
structure of a particular domain. But it is desirable that the data structure
used for reduction be kept separate from the concept of structural equivalence itself. That is a major part of what I have tried to do here.
It is hoped that B, or more likely some computationally
economic approximation of it, will provide a method of preprocessing a large variety of network
data sets before applying nearly any other analyses. The concept that B and
its variants attempt to capture pervades the quantitative
analysis of social
structure. The B-matrix provides an attractive starting point for more elaborate analysis (see Boyd and Sailer 1978).
Finally, it is clear that any study of actually occurring social networks is
bound to require the use of computers. We must therefore consciously direct
our theory building apparatus toward theories that are in some sense computable. Such an approach is assumed here. The very existence of structural
equivalence as a concept depends on the notion of processing large amounts
of data in a routine way.
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