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Structural Equivalence: Meaning and Definition, Computation and Application*

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 ‘Struc- turally 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 methodolo- gical 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 cate- gorization 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 un- observable 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” is used in a loose sense here, contrasted with “clique structure” for example. Heil and White (1976) provide an example of a very formal definition. The spirit is the same.

76 Lee Douglas Sailer

deal of information can be obtained much information can we tease out of tion asked here.

Definitions of structural equivalence

from individual attributes, but how relationships alone? That is the ques-

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 over- come 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 Categorical

I

-\ approach '-._

-.fcf combine -9,)

combine individuals

=%&.. od+.,_

l I

individuals -.

-. '.

CONCOR -+ Model B

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 meaning- fulness 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 correla- tions 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 sub- matrices 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.


Figure 3. Examples of asymmetry and in transitivity.

R R A 2. .3

I B

R

I t4

(a) (b)

An important point missed by BBA is iterative solution, the most important data,

__-- _----- ___

Rcl+:‘l

‘. t? ------?’ B --__

--_ _c-- ---____---

that at every iteration of any 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 max [min(Bi,, Rki, Rmi)] B;;’ = k=lm=l

5 Rki k=l


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. Consider Figure 4. We assume that every point is SR to itself, so let

B_!! = 1, ifi=j 0, otherwise

To compute B :2 we see that R 1 1 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

~- = 0.5 1+0+1+0

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 8 1

Table 1. Intermediate results from computation of R in Figure 4.

(a) First iteration

2 3 4

1.0 0.5 0.5 0.0

I

0.33 1.0 0.33 0.33

1.0 1.0 1.0 0.0

0.0 1.0 0.0 1.0

(b) Second iteration

gqqy&

(c) Third iteration

1 2 3 4

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 ~1s and 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 B- matrix 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 ~1 points, a stable solution for B (one that is guaranteed to repfi- cate itself on subsequent iterations) depends directly on the ties in the relations V~U the solution to a set of U* simultaneous linear equations in 11’ un- knowns. 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, the criterion is

n,ax . If&+1 d. I < 0.1 [I IJ - II


Figure 5. Toy example from L W: (a) the relations, (b) L W’s model, (c) B model.

P -

1

A 5

A 3 4

(a)

0 %

295 P 0 3,4

p-1 -

1

A 5 2

A 3 4

legend

i--+j = iRj

i---aj = iSj

o= blocked together

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 indi- cates 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), 1 2 3 4 5

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 ex- tracted 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 “mor- phism?) 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.


Figure 7. L WModel 1.

P N

Caley table

note: N connects to everything

Figure 8. L W Model 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 com- binatorially, 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 Boor- man 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 5 6 7

1 1 0.74 0.90 0.87 0.86 0.66 0.86

2 0.96 1 0.93 0.91 0.87 0.91 0.91

3 0.92 0.55 1 0.85 0.88 0.50 0.84

4 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.


Figure 9. 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 “cor- relate” their age, mode of dress, income, race, sex, etc. If we use these data as our initial B1, instead of I, we might accomplish two noteworthy gains. 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.


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, semi- lattices, 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 approxi- mation 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 ela- borate 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 com- putable. 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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