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Direct and indirect methods for structural equivalence
Social Networks 14 (1992f 63-90 North-HoliaI?d
63
Direct and indirect methods for structural equivalence *
Procedures far establishing a partition of a network in terms of structural equivalence can he divided into direct and indirect approaches. For the Former, a new criterion function is proposed that reflects directly structural equivalence concerns. ‘This criterion function can then be (locally) optimized to create a partition. For indirect approaches, measures of dissimilarity must be compatible with the definition of structural equivalence.
I. introduction The concept of structural equivalenee has substantive, ~a~h~~~~ical, and technical components. We focus on some of the mathematical ideas with a view to (i) establishing foundations for a direct approach for partitioning a network in terms of structural equivalence, and (ii) assessing the appropriateness of dissimilarity measures for structural equivalence in indirect approaches.
As noted by Doreian fI988), equivalence emerged as a ~ou~datiu~al concept for the analysis of social network representations of social * Revised version of a paper presented at the Eleventh Annual Sunbelt Social Network Conference, Tampa, Florida, Feb. 14-17, 1991. ’ Exchange visit at the University of Pittsburgh” Department of Sociology (November-December 19901. * Fulbright scholar at the University of Pittsburgh, Department of Sociology (September 1996February 1991). 0378~8733/92/$05.00 % I992 - Efsevier Science Publishers B.V. Ah rights reserved
structure. In practice, social actors linked by one (or more) social relation(s) are partitioned into disjoint classes. Let E=(X,,X, )...) X,,} be a finite set of units. The units are related by binary relations t==l . . . Y
R,cExE,
with determine A@“= (E, R,,
a network
R2,.-., R,)
In the following we restrict our discussion to a single relation described by a corresponding binary matrix R = ~YjjI,xn where rij = i
1
xiRXj
0
otherwise
R
In some applications rjj can be a ~~~~egative real number expressing the strength of the relation R between units X, and X,. The main procedural goal of block modeling is to identify, in a given network, &asters fclasses) of units that share structural characteristics defined in terms of R. The units within a cluster have the same or similar connection patterns to other units. They form a clustering
which is a partition of the set E:
Each partition determines an equivalence relation (and vice versa>. Let us denote by - the relation determined by partition %. A block model consists of structures obtained by identifying all units from the same cluster of the clustering G?“,The partition (ideally) is constructed by using structural information contained in R and actors in the same partitioned class are equivalent to each other in terms of R alone.
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Such actors share a common structural position within the network. (In general, the statement can be extended to include (R,}.) Drawing on the work of Nadel (19571, in an effort to discuss social positions and roles, Lorrain and White (1971) provided the first definition of equivalence: Actors are equivalent if they are connected to the rest of the network in idEnticat ways, a very stringent requirement. Such actors are said to be structurally equivalent. There have been several efforts to weaken the criterion of equivalence. See, for example, White and Reitz (1983) and Winship and Mandel (1983). A more extensive survey is provided by Pattison (1988). However, we deal only with structural equivalence in this paper. Treatment of weaker equivalences, specificaly regular equivalence, is provided elsewhere (Batagelj et al. 1992). There are important reasons for pursuing equivalence ideas for social networks. One is the idea that there are fundamental role structures with observed network behavior, especially interaction, providing indicators of the fundamental structure. Block modeling provides a way of distilling this structure from observed social relations. A very pragmatic rationale is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily. Block modeling facilitates this also. For either case, block modeling is an empirical partitioning (clustering) procedure. The adequacy of such a procedure can be pursued at two levels. One level is found in ideal structures where non-trivial structural equivalences exist. In such ideal structures, it is possible to determine if known equivalences in a network are located by a specific block modeling procedure. The second level is found in empirical (non-ideal) networks, where few exact equivalences are found. We deal with both by considering real networks as perturbed ideal networks, 1.2. Structural equivalence A permutation
vx,
YEE:
cp: E --, E is an automorphism of the relation R iff
(XRY=vp(X)Rfp(Y))
The units X and Y are structurally equivalent, we write X s Y, iff the permutation (transposition) P = (Xy) is an automorphism of the relation R (Borgatti and Everett 1989).
In other words: X and Y are structurally equivalent iff sl. XRY-YRX ~2. XR-Y w YRY ~3. QZ E E\{X, Y): (XRZ (-j YRZ) s4. VZ E E\{X, Y): (ZRX@ ZRY) or in the matrix form: X, =X, iff sl’. Yij= ‘)i ** . rri = Y;; 33’ ~3’. Qk f i, j: rik = rjk ~4’. Qk f i, j: rki = rki The matrix form of the definition of structural equivalence extended also to the case when rij are real numbers.
can be
1.3. Establishing block models There are two main approaches to block modelling problems based on the structural equivalence and its relaxations: l
l
indirect approach: reduction to the standard data analysis problems (c.luster analysis, multidimensional scaling) by determining a dissimilarity matrix between units which is compatible with the selected type of equivalence; direct approach: construction of a criterion function P(@?) which measures the fit of the clustering C to the network data, and solving the corresponding optimization problem. For this purpose a relocation procedure from cluster analysis can be adapted.
An indirect approach is one where the empirical partitioning of a network is done in two stages. First, the information in R (or more generally {R,J) is used to create a partition, and second, the partition is evaluated in terms of R. STRUCTURE (Burt 1976), one of the most used algorithms, is inherently indirect. A dissimilarity matrix is either constructed internally (with 6 options) or read from an input file. This matrix is then clustered. Confirmatory factor analysis can be used to assess the adequacy of each structural position identified in the analysis. The other widely used procedure, CONCOR (Breiger et al. 19751, is harder to classify as it has aspects of both approaches. It is indirect in the sense of creating a partition - with users having options for the
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number of splits - that can be evaluated separately. On the other hand, it is direct in the sense of seeking partitions to maximize within clusters (iterated) correlations (to + 1.0) and to minimize correlations between clusters (to - 1.0). However, there is no explicit criterion function in the usual sense of an optimization problem. We note that the criterion function we propose for a direct approach can be used to assess the adequacy of partitions obtained by CONCOR or STRUCTURE, or any algorithm
2. The indirect approach
via measuring
the equivalence
of pairs of
units
The first step of an indirect approach is the definition of a dissimilarity measure. This is a crucial step as not all dissimilarities are consistent with structural equivalence. Moreover, although the definition of structural equivalence is “local” it has “global” implications structurally equivalent units behave in the same way also to all other units. A position is defined in terms of all other units in a network. 2.1. Properties of units
The property t: E -+ R is structural if for every automorphism the relation R and every unit X E E
holds. Let t(U) be a structural property of the unit U. Then we have X=Y*t(X)=t(Y)
Some examples of such properties
are
t(U) = degree (number of neighbors~ of unit U or (see Batagelj 1990) t(U) = number of units at distance d from the unit U
cp of
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or (see Burt 1990) t(U) = number of triads of type x at the unit U. For other examples see Freeman (1978) and Hummon et al. (1990). The collection of structural properties t,, t,, . . . , t, is complete (for structural equivalence) iff for each pair of units X and Y (Vi, 1 Si
ti(X)=t,(Y))*X=Y
is satisfied. Remark. The triads are not complete:
the path of length 3 provides a simple counterexample. The extreme units have the same triadic spectrum, but they are not structurally equivalent. We can define the description of the unit U as
and the dissimilari~
between units U and V as
where D is some (standard) dissimilarity between real vectors. In the case when the dissimilarity D has the property
and the properties d(X,Y)=O=D([X].
t,, t,, . . . , t, are complete, it holds [Y])-[x]
=[Y]dfi: ti(x)=ti(Y)
o;yzEy Therefore, we finally have d(X,Y)=O-X=Y
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2.2. Properties of pairs of units A property q: E x E --j [wis structural if for every automorphism the relation R and every pair of units X, YE E it holds 4(X7 Y) = q((P(X),
cp of
cp(Y>)
For example, for the original relation R, q(U, V) = r(U, V). See Fiksel (1980) as another example: q(U, V) = number of common neighbors of units U and I/ and q(U, V) = length of the shortest path from U to I/. We note that if the union of the neighborhoods of U and V is identical to the intersection of their neighborhoods, then U and I/ are structurally equivalent (Fiksel 1980). For other examples see Burt (1988). Let q(U, V) be a structural property of units U and I/. Then, if X=Y we have ql. 4(x, YJ = 9CY x) q2. 4(x, x) = q(Y Y) q3. VZ E E\{X, Y}: q(X, Z) = q(Y, Z) q4. VZ E E\{X, Y): q(Z, X) = q(Z, Y) The property q is sensitive if the properties ql-q4 imply also that
x- f. An example of this type of property is UR*V q(u,v)={; otherwise where R* is the transitive and reflexive closure of R. Therefore we can describe each unit U by a vector [U] = [4(U, X,), 4(U, X2),...,
4(U XJ,
4(X,,
U),..., 4(X,7
VI
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and define
the dissimilarity
equiclalence
units U, I/E E as
between
where D is a dissimilarity between the corresponding descriptions. Some examples of such dissimilarities and a discussion of their appropriateness are given in Section 2.3.1. 2.3. Matrix dissimilarities 2.3.1. Dissimilarities The following is a list of dissimilarities between units X, and X,: Manhattan
for measuring
distance:
ill
dm(XiT
xj)=
(I4is-4jsI
+
I&-i_4,jl)
s=l
Euclidean
distance:
stl((4;s-
4,)"
Truncated d,(Xi,
Manhattan
Xj)=
2
+
(qsi
qs,):)
-
distance:
(Iqi,-qjsIfIq,,-q~j()
s=l s#i,j
Truncated
Euclidean
ds(Xi>Xj) =
distance
(Faust
2 ((qis-
d
qjs)' +
(4si
1988):
-
4rj)*)
s=l
s#i,j
Corrected
Manhattan-like
d,(p)(Xi,
Xj) =d,(Xi>
dissimilarity Xj) +P.(
(p > 0)
Iqi,-qjjI f Iqij-qjil)
the similarity
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Corrected Euclidean-like d,(p)(Xi,
Xj)=
d,(Xi,
71
dissimilarity (Burt and Minor 1983): xj)2+P*
((4,i-4,)2+(qij-4ji)2)
Corrected dissimilarity
dC(P)(xi7
xj)
=
{m
It is easy to verify that all expressions from the list define a dissimilarity, i.e., they have properties dl. d(X, Y> 2 0 d2. d(X, X) = 0 d3. d(X, Y> = d(Y, X) A dissimilarity d which has also the properties d4. d(X, Y>=O*X=Y d5. d(X, Y) + d(Y, Z) 2 d(X, Z) is called a distance. Each of the dissimilarities from the list can be assessed to see whether or not it is also a distance. From calculus we know that Proposition 2.1. Dissimilarities
d, and d, over R” are distances.
Proposition 2.2. If d over E has the property di, i = 1,. . . , 5, then 0 has also this property over E. Proposition 2.3. If d, over E, and d, over E, have the property di, i = 1,. . . , 5, then d, + d, over E, x E, has also this property. Proposition 2.4. If d, over E, and d, over E, have the property di, i = 1,. . . , 5, then /m over E, x E, has also this property. Remark. Because different units may have identical descriptions (rows and columns) the property d(Xi,
Xi) = 0 aXi
=Xj
does not hold so none of the listed dissimilarities can be distances in a
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strict sense. Nevertheless, we can use a slightly relaxed property
d(X,, Xi) = 0 -xi
-xj
Remark. In the case q = r all the dissimilarities from the lgt are invariant to the complementing of relation R + E x E \R = R, i.e., d(R)(X,
Y) = d~~~(X, Y>.
2.3.2. Compatibility In procedures based on structural equivalence we can expect the best results if we use a dissimilarity d which is compatible with structural equivalence, i.e., Xi=Xj-d(X;,
Xj> =0
Not all the dissimilarities from the list are compatible. If CJis sensitive, for dissimilarities d, and d, only d(X,,
Xj)=o=+xi=xi
holds. The converse does not hold. For the matrix
R=O
1
[1
01
the units d,(X,,
X, and X, are structurally
X2) = 2 and d,(X,,
equivalent,
If q is structural property, for dissimilarities Xi=Xj*d(X,,
but (for 4 = r)
X2> = a. d, and d, only
Xj)=O
holds. The converse does not hold. For the matrix
for 9 = r, d,(X,, X2> = d&X,, X2> = 0, but the units X, and X, are not structurally equivalent.
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73
If a structural property 9 is sensitive, the dissimilarities d,, d, and are compatible with the structural equivalence. To see this, suppose that X, =Xj. Then from the properties ql-q4 it follows ql’. Cjij- LJji= 0 42’. qii-qjj=o 43’. Vk # i, j: qik - qjk = 0 44’. ‘dk # i, j: qki - qkj = 0 Therefore all the terms which occur on the right side expressions in the definitions of dissimilarities d,, d, and d, are zero. Hence also d,(X[, XjJ = d,(Xi, Xj, = d,(Xi, Xjj, = 0. This establishes
d,
X;=Xj*d(Xj,
X,)=0
To prove also d(X,,
xi) = 0 axi
sxj
we shall prove the equivalent statement Xi$Xj=+d(Xi,
Xj)fO
Since 4 is by assumption sensitive, from Xi +Xj it follows that for at least one pair of units X,, X,,, {u, U) n Ii, j} f fl at least one of the properties ql-q4 does not hold. This means that qu, # q,,,or equivalentlyq,,,-q,, f 0. The dissimilarities d,, d, and d, have value zero exactly when all the terms are zero. Therefore in this case they are different from zero. 2.3.3. Triangle i~e~uali~ It is straightforward to establish which of the listed dissimilarities satisfy the triangle inequality
d(Xi, Xk) +d(Xk, Xj) 2 d(Xi, Xj) Suppose that d can be expressed in the form d(X,
Y) = 6(X,
Y) + d/(X, Y)
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where 6 depends only on i, j, k components of vectors describing units X and Y; and d’ depends on the remaining components. Then, by proposition 2.3, to prove that the triangle inequality holds it is sufficient to prove that it holds for 6 and d’. Because of Propositions 2.4 and 2.2 it is worthwhile to consider first the dissimilarities d, and d,.
The dissimilarity dk satisfies the triangle inequality. Therefore, if we consider only vectors which are equal on I’ = I\(i, j, k}, the triangle inequality holds for d, iff it holds for 6,. This is equivalent to A, 2 0 where
By complete enumeration of all possible cases for O/l matrices on the computer we found that there exist counterexamples. For example, for the matrix
we
get
A,= ~O-O~~l-l~+~l-~~+~O-O~-~O-~~-~~-~~ The same matrix is also a counterexample For the dissimilarity d,(p) we have A,(~)=~,(P)(X,,
Xk) +S,(P)(~,,
= -2 for d,.
Xj)-‘~(P)(X,~
“‘JJ * ( I 4;i - qkkI f I qik- qkiI) ’ I 4ij +p+?kk
-
qjj
I f
qkj
%)
I +
Iqji
-
qjk
I
I qkj - qjkI) + I qki- 4ji I + l qik- 4ij
- p * ( I qii - qjj I + I Lirii - qji
I ) - I 4ik -
qjk
I -
1 qki
-
qk j
1
1
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There exist caunterexamples matrix
i 01
01
for every
p >
75
0. For example, for the
01
we get
A,(P)
=P * (IO-OI+/O-OI)+ +p-(
ll-11+/o-OI)+
=1+1-(2+p)=
II-l/+10-11 Il-0(-t-IO-O/
-p
Therefore the triangle inequality does not hold for all p > 0. For p = 2 all the counterexamples have either the form 1 + 1 - 4 = -2 or the form 1+3-6= -2. Since fi+fi-fi=O and I@ + fi - 6 > 0 the dissimilarity a,(2) satisfies the triangle inequality for O/l matrices. For p = 1 all the counterexamples have either the form 1 + 1 - 3 = -1 or the form 1+2-4= -1. Since fi+fi-fi>O and fi + fi - v% > 0 the dissimilarity 6,(l) satisfies the triangle inequality for O/l matrices. We can show that the dissimilarity d,(p), p = 1, 2 also satisfies the triangle inequality for O/l matrices. By definition d, = ,,,q = km, The previous result tells us that & = 6, satisfies the triangle inequality. Because di = dk, the dissimilarity d:, and by Proposition 2.2 also iq satisfy the triangle inequality. From the equalities
it finally follows, by Proposition inequality.
2.4, that d,
satisfies the triangle
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V. Batagelj et al. / Methods for structural equicalence
Similar-y, by complete similarity
=
I, ’
satisfies de =
((4ii
-
9jj)*
the triangle
+
enumeration,
(4ij
-
inequality
4ji)')
+
we can establish
(Clik-
for O/l
4jk)2
matrices
(qki -
+
for
that the dis-
qkj)*
p = 1,
2. Because
$6;+ (d$
by Proposition 2.4, d, also satisfies triangle inequality for O/l matrices. For real vectors the dissimilarity a,(2) does not satisfy the triangle inequality. For a counterexample consider the matrix 5 0 4
9 7 6
6 4 6
for which A,=6,(X,,
X,)+6,(X,,
Xi) -S,(Xi,
Xj) = @-
+
d% - d-i%
= -1.51 For the dissimilarities 6,(l) and 6,(l) over real numbers the answer is unknown. A random search for a counterexample with more than l,OOO,OOOtrials failed. For a,(2) over real numbers Martin Juvan and Marko PetkovHek (1991) proved that the triangle inequality holds. For a, b E (0, 1) the following equality (~2 - b)* = I a -b
I
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% Batagelj et al. / Methods for siructural equicalence
holds. This implies from the list
some additional
relations
between
dissimilarities
de(P)=dc(d The last equality shows that in the case of O/l matrices it is sufficient to prove the triangle inequality only for one of dissimilarities d, and d C’ It is interesting that for symmetric matrices, q,,, = q,, for all u and u, the dissimilarities d, and d, (and therefore also d,, d, and d,) satisfy the triangle inequality. Considering symmetry, we obtain A,
=
2(
) qik
-
qi;
1+
I qij
( 1qii -
qkk
-
qkj
I -
I qik
-
qki
I>
and A, = 2~ .
+
2( I qik
- 4ij
I + I qkk
I’
I qij -
- ‘Ijj
qkj
1- 1qii - qjj I> I - I qik
- qkj
In both cases we can apply the inequality
I qik
- 9ij I + I 4ij -
qkj
I)
I a I + I b I 2 I a + b I giving
I 2 I (qik - Sij) + (4ij -
qkj)
I = I qik
-
qkj
I
and
) qii -
qkk
I+
I qkk
- qjj
I 2 1qii - qjj 1
Thus proving A 2 0. Dissimilarities 6, and 6, over symmetric matrices satisfy the triangle inequality. Using the Propositions 2.3 we prove that this holds also for dissimilarities d, and d,.
78
We can define several properties of pairs of units based on a relation. If a property is sensitive then corrected Manhattan d,, corrected Euclidean d, and corrected dissimilarity d, are compatible with structural equivalence, i.e., two units are structurally equivalent exactly when their dissimilarity is zero. This is an important result: we note that the corrected Euclidean-like dissimilarity used by Burt and Minor (1983) is compatible. Clustering methods usually do not require that the used dissimilarity is a distance. Nevertheless we discussed also the triangle inequality for the selected dissimilarities. If the property is symmetric, the triangle inequality holds for all of the considered dissimilarities. For binary properties it holds only for d, and d,.
3. Direct approach: optimization 3.1. Goodness-of-fit for block models
As described above, in an indirect approach, a block model is established, and then assessed. It seems that the fit of a block model for the network data has received very little attention. There are some exceptions. Arabie et al. (1978) suggest the product moment correlation as a measure of fit. Specifically, given a partition via a block model, the correlation between the permuted data matrix and the corresponding “ideal” matrix with perfect fat fit was proposed as a measure of fit. Carrington et al. (1980) provide example of different structures having the same correlation coefficient and suggest that the correlation is not useful as a measure of fit. In its stead they propose a measure that is constructed in terms expected densities within and between blocks. They argue (Carrington et al. 1980: 226) “_ an index of goodness-of-fit block models should measure the extent to which densities of O’s or l’s in the submatrices of the blocked data matrixfes) deviate from the densities of the blocks in the expanded image matrixtes) towards the worst possible density (a)“. The value of (Yis the threshold used by an analyst to determine O’s or l’s for the image matrix. The measure is a weighted average of the squares of differences between observed and expected densities (for the worst possible fit where the expected value
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is a) for each cell corresponding to the image matrix. However, as Carrington et al. (1980: 231) note “there is no obvious . . . way to use (the measure) to show that one model is unambiguously preferable to another.” Panning (1982) reverts to a correlation approach to measure the fit of a block model and explicitly redefines a block modeling effort as a regression analysis. The redefinition seems too forced but it does retain the idea of measuring the correspondence of a block model partition and an ideal partition with a fat fit. This is used below. 3.2. Criterion for structural equicalence Batagelj et al. (1992) propose the construction of criterion functions that reflect the notion of regular equivalence. The same approach can be used in the case of structural equivalence. From the definition of structural equivalence it follows that there are four possible ideal diagonal blocks B(C, C) Type 0. bij = 0 Type 1. bij = Sii Type 2. bij = 1 - ajj Type 3. bij = 1 where aij is the Kronecker delta function and i, j E C. For the nondiagonal blocks NC,, C,,), u # u only blocks of type 0 and type 3 are possible. Given a clustering $57= {C,, C,, . . . , C,}, let SS’(C,,, C,.) denote the set of all ideal blocks corresponding to block NC,, C,,>. Then the global error of clustering 55’can be expressed as
where
the obvious
choice for d is
d(R(C,, Cl.), B) =
c
XEC,,,YEC,
bxy -~,,I
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It is easy to verify that this defined to structural equivalence P( 5?) = 0 e $5’defines
structural
criterion
function
P(g)
is sensitive
equivalence
In addition, it is invariant to the transformation of complementing the relation R + E x E \I?. The selected dissimilarity between blocks is based on the assumption that the error 0 + 1 is equiprobable to the error 1 + 0. In the case that this assumption is not valid we can introduce (Y and p as weights for the two types of errors. Let B* be a solution of the optimization problem from the right side of the expression for p(C,,, C,,). Then we can redefine p, for example, as follows p(Cu,
C,,)=a
card{(X,
Y)EC~XC,,:
+ p card{(X,
rxr>bzy}
Y) E C,, x C,.: Y,~ < b,*l)
Selecting large a(a = 100) and small p (0 = 11, in optimizing P(‘Z) we primarily are seeking solutions with the least number of errors of the type 0 --) 1 and, as a secondary criterion, among them solutions with the least number of errors of the type 1 + 0. In the special case when the number of O’s equals the number of l’s in the block R(C,, C,.), we select l’s block for B* if a 2 /3; and O’s block otherwise. We note this case is not infrequent, empirically. For solving the block modelling problem we use a local optimization algorithm: Determine the initial clustering 5%‘; repeat: if in the neighborhood of the current clustering 22’ there exists a clustering 2%” such that P(%?‘> then move to clustering ‘39”‘. In the algorithm the neighborhood of a given clustering is defined by the following clustering transformations: l l
moving a unit from one cluster to another cluster; interchanging of two units from different clusters.
We repeat the local optimization search with several random clusterings and preserve the best 10 obtained solutions.
initial
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3.3. Example: Sampson data - liking 3.3. I. Description Sampson (1968) gathered sociometric data from a group of men in a monastery. These data, in a form with 18 individuals and 4 relations (affect, esteem, influence, and sanctioning), have been used as a test bed for many network tools. We follow this tradition, but we confine our attention to the affect (liking) relation. The 18 men, lay and clerical novices in the first group were Ambrose (91, Berthold (61, Bonaventure (51, Mark (71, Peter (41, and Victor (8). The second cohort was made up of Albert (161, Amand (131, Basil (3), Boniface (151, John Bosco (11, Elias (171, Gregory (21, Hugh (141, Louis (111, Romuald (lo), Simplicius (181, and Winfrid (12). During a period of conflict, triggered by changes designed to relax rules and simplify procedures in the monastery, most of the novices split into two opposing groups. One, labeled the Young Turks, was led by John Bosco and Gregory, with Albert, Boniface, Hugh, Mark, and Winfrid as followers. The other basic group, dubbed the Loyal Opposition by Sampson, was led by Peter. The followers were Ambrose, Berthold, Bonaventure and Louis. Sampson identified two other groups. One, labeled the Outcasts, had Basil, Elias, and Simplicius. The remaining monks composed an Interstitial group having no consistent ties with either of the basic parties to the conflict. 3.3.2. Results In the following analysis the positive affect (liking) relation from time four is used. The relation was dichotomized by coding first, second, and third choices as 1, and coding 0 otherwise. First, we present results obtained by the indirect approach. The corrected dissimilarity d,(p = 2) was calculated on original the relation LIKING. Figure 1 shows the dendrogram obtained by the agglomerative with the Ward criterion function (e.g., Gordon 1981: 41) based on the proposed dissimilarity matrix. The hierarchical clustering shows three and also five distinct clusters. We consider both and rerun the analysis with the relocation algorithm with the Ward criterion function (e.g., Gordon 1981: 44). The obtained clustering with 3 clusters is g: = {{1, 2, 7, 12, 14, 15, 16), (4, 5, 6, 8, 9, 10, 11, 13}, (3, 17, 18}}
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Boniface
15
Albert
16
Amand John
13 Bosco
Hugh
1 14
Gregory
2
Mark
7
Winfrid
12
Elias
17
Simplicius
18
Basil
3
Berthold
6
Romuald
10
Louis
11
Bonaventur
5
Victor
8
Peter
4
Ambrose
9
Fig. I. Sampson
and is Using Yound group, The
data-liking
/ indirect
approach,
Ward.
the same as that obtained with the agglomerative algorithm. Sampson’s labels, the first cluster (block) corresponds to the Turks, the second to the Loyal Opposition with the Interstitial and the third to the Outcasts. clustering with 5 clusters obtained with relocation method
@ = {{I, 14}, (2, 7, 12, 15, 16}, {4, 5, 9, II}, (6, 8, 10, 131, (3, 17, 18)) is not the same as the hierarchical solution which also means that clustering with five clusters is less appropriate structure for these data. Tables 1 and 2 display the LIKING relation matrix permuted into a form compatible with the three-block model (Table 1) and five-block model (Table 2) obtained with the indirect approach (relocation algorithm). In the tables the 0 --, 1 errors (it should be 0 but there is 1) are indicated by l and 1 - 0 errors (it should be 1 but there is 0) by *. From Table 1 it can be seen that there is only one block of the type 2, all others are of the type 0 or 1. Therefore it is not surprising that there are only 0 --, 1 errors. There are two blocks out of nine without
% Batagelj et al. / Methods for structural equivalence Table 1 Sampson
data - liking / clustering,
relation,
1 2 7 12 14 15 16 4 5 6 8 9 10 11 13 3 17 18
k = 5
1272456
4
5
6
8
9
1 0
1 1
1 3
3
000..00 .o..ooo o.o*oo. . ..oooo . 0 0 o...ooo o..oo*o 0 0 0 0 0 0 0000000 0 0 0 000.000 0 0 0 0000*00 00.0000 . 0 0 0.00000
0 0 0 0 0 0 0 0 . . . 0 . 0 0 0 0 0
0 0 0 0 0 0 0 . 0 . 0 . . . . 0 0 0
0 0 0 0 0 0 0 . 0 0 . 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 . 0 . 0 0 0 0
0 0 0 0 0 0 0 0 . . . 0 . 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 . . 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 . 0 0
0 . 0 0 0 0 0 0 ooo 0 0 0 0 0 0 0 011 1 1
1 John Bosco Gregory Mark Winfrid Hugh Boniface Albert peter Bonaventw Berthold Victor Ambrose Romuald LO& Amand Basil Elias Simplicius
83
1
1
1
.
0
.
0
0 0
‘0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0.00000
1 7
1 8
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 .
0 1
1 0
errors and a further three with only one error. The clustering with 5 clusters (Table 2) shows both types of errors. Ten out of 25 blocks have no errors. By the direct approach (local optimization) we obtained as the best solution with 3 clusters for Q = /3 = 1 the clustering @,, = (11, 2, 7, 12}, {4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16) (3, 17, 18)) and with 5 clusters the clustering g;,
= ((1, 13, 14, 15, 16) (2, 7, 12) {4,5, 9) (6, 8, 1% 11) (3, 17, 18})
The corresponding terings are presented
rearanged relation matrices for these two clusin Tables 3 and 4.
84
V. Batagelj et al. / Methods
Table 2 Sampson
data - liking / clustering,
relation,
1 1 142725645916803378 John Bosco Hugh Gregory Mark Winfrid B&face Albert Peter Bonaventur Ambrose Louis Berthold Victor Romuald Amand Basil Eli.%? Simplidus
10100*0000000000*00 14 10 0 o..o 2*0011**00000000000 700101*100000000000 12 .OllO 15 0 01110 16 0 Oll*lO 4000000001*1*000000 5 0 0 0 0 90000*00*10*0~00000 11 0.0 0 60000000111*0000000 800000001*1i~~000000 10 0 0 0 0 13 0 0 0.0 3*000000 17 0 0.0 18 0 0.0
1
for structural equivalence
k =5
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
*
* *
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
*l*
0
0.0
0
0
0
0
0
0 0
0 0
* *
0 0
0 0
0 0
. 0
0 0
0 0.
0
0 0
0 0
0 0
1 1 1 *1* 0000000*011 0 0 0 0 0 0
0 0
0 0
0 0
0 0
0101 0110
We repeated the direct approach also for (Y= 100 and p = 1. The best obtained clustering with three clusters is the following one g;,
= {{l, 2, 7, 12, 5, 16}, {4, 5, 6, 8, 9, 11}, (3, 10, 13, 14, 17, 18}}
and for five clusters g:,
= ((1, 15}, (2, 7, 12, 14, 161, {4, 5, 678, 9, ll),
{lo),
{3, 13, 17, 18j) The rearranged relation matrices for these clusterings are sented in Tables 5 and 6. The same matrix was also analyzed STRUCTURE and CONCOR. By STRUCTURE, for positions (Euclidean distance between raw relation patterns) and positions (between raw relation patterns - mean difference) we obtained following clustering with 3 clusters:
preby = 0 = 1 the
‘Z;, = {{l, 2, 7, 12, 14, 15, 16}, {4, 5, 6, 8, 9, 10, 11}, (3, 13, 17, 181)
8.5
1
1
1
1
2
7
2
4
5
6
8
9
0
1
3
1 4
1 5
1 6
0 1 *
* 0 I
* 1 0
1 1 1
0
0
0
0
0
000.00
0
0
0
0
0
0
0
0
0 0
0
a
0
0
a
0 0 0
0 0
0
. 0 0
0
0
1
John Bosccl ---iGregory 2 7 Mark Winfrid 12 4 Peter Bonaventur 5 Berthofd 6 8 Victor 9 Ambrose Romtid 10 Louis It Amand 3.3 14 Hugh Bonif&x 1.5 Albert 16 Basil 3 Eli& 17 Simplicius 18
I
I
1
0
D
0
0
0
0
.
.
0
000000000*0
. 0
0
0
0
0
.
0
0
0
0
.
0
0
0
0 0 0
0
and clustering with 5 clusters G$, = {{l, 2, 7, 12, 141, (1% ‘Ifi), (4, 5, 6, 8, 9, 111, (101, {3, 13, 17, 18)) By STRUCTURE, for positions = 2 (between patterns - covariance) we obtained the clusterings
deviation
and clustering with 5 clusters @:z = ((1, 14, 1% (2, 7, 12, 16}, (4, 5, 6, 8, 9, ll}, (lo}, {3,13, 17,18}}
relation
V Batagelj et al. / Methods for structural equicalence
86 Table 4 Sampson
data
liking / LocOpt
1
1
k = 5, a = 1, /3= 1
1
1
1
1
1
1
1 0
134562724596801378 JohnBosco Amend Hugh BClnif.%e Albert Gregory Mark Winfrid peter Bonaventur
o o . o o * * 1 1 13 0 0 0 0 0 *l* 14 .o 0.0 * *lO 15 0 0 0 0 01110 16 0 0 O.Oll* 2*00000110000000000 70000*1010000000000 12 .o 0 0 0110 40000000001**00*000 o o o o 0 0 5 0 o
By CONCOR
we obtained
0 0 0.0 0 0 0 0
0
0 1
the following
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
. 0 0 0 0
0 0. 0 0 0
0 0 o
0
0
0
0
0
0
0
0
0
0
1
0
0
0
.
0
0
0
clustering
with 3 clusters
~‘2 = ((1, 2, 7, 12, 14, 15, 16}, {4, 5, 6, 8, 9, 10, II, I3}, (3, 17, 18)) and clustering
with 5 clusters
%,J = {{l, 14}, (2, 7, 12, 15, 16}, {4, 9, II, 13}, (5, 6, 8, IO), (3, 17, I@} The direct approach was used also to obtain the best clustering with 1 to 8 clusters. In Table 7 the values of the criterion function for structural equivalence P(E’ (the global error of the clustering E’> for all mentioned clusterings and for selected values of cy and p are presented. In the case of (Y= p = 1 the criterion function actually counts number of errors in the blocks. The clustering with 3 clusters with the best criterion function is that obtained by the direct approach (P = 44). The clustering obtained with the indirect approach and by
V. Batagelj et al. / Methods for structural equivalence
Table 5 Sampson
data - liking / LocOpt
Bonaventur Berthold Victor Ambrose LO& Basil Rmnuald Amand Hugh El& Simplicius
Table 6 Sampson
5 000000 6 000000 8 0 0 000.00 9 000000 -11 . 0 3 000000 10 13 00.000 .oo*.o 14 0.0000 17 18 11 0 .
0
0
0
data - liking / LocOpt
k = 3, (Y= 100, p = 1
0
0
0
1 0 0 * 15 0 0 1 21*011**0000000 7**101*10000000 12 1 * 1 14 1 1 * 16jl* 111 4110 010 5 6 8 9
0
0
0
lO**ll llO*l* 1 * 0 * 1 * 1 0 0 0 ..oo.o 0 . 000000 0 0 0 1 0 0
1 * 0
0 1 1 0
1 0 * 0
* * 0 0
0
0
0
0
0 0
0 0
0 0
0 0
l
1 303478
1
1
1
1 II
0 ~00000 000000 0 0 0 0 0 0 000000 0 0 000000 0 0 0 0 000.00 00.0.. 0 0 00000. 000000 .oooo.
0
.
0
0
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
0
0 0
0 0
0 0
0 0
.
0
0
0
3
1 3
1 7
1 8
.
0
0
0
0 0 0 0
.ooo.o
k = 5, a = 100, fJ = 1
1 15272464568910 JohnBosco Boniface Gregory Mark Winfrid Hugh Albert Peter Bonaventur Berthold Victor Ambrose
87
*_ 1
1 * 1 0
1 1 1
0 1 * 0
1 1 *
* 0 * 0
1 * *
1 0 0
* 0 * 0 010 010
0 0
0 0 0 1
0 0
0 0 0 1
0 0
0 0 0 *
0 0
0 0 0 *
0 0
1 0 0
0 0 0 0 010 110
0
0
0
0
0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
88
V. Batageij
Table 7 The values of the criterion
et al. / Methods
function
for .structural
equivalence
for the best clusterings
n
cr=1,p=1
(Y= 100, p = 1
1 2 3 4 5 6 7 8
56 50 44 40 35 31 28 25
5600 40 14 24 28 1242 941 743 446 434
@ ??I ?A @
50 52 52 50
5000 4804 4804 5000
@ G7s51 Y:, 9;
38 46 44 38
1919 1135 1127 25 13
CONCOR are the same with 6 errors more. The worse is the STRUCTURE solution (P = 52). The same is also true for the obtained clusterings with 5 clusters. In the case of (Y= 100 and p = 1 the criterion function counts the number of 0 + 1 errors in the first two digits and on the last two the number of 1 + 0 errors. The sum of both errors givs the total number of errors. The clusterings obtained by the direct approach with (Y= 100 and p = 1 have usually less 0 -+ 1 errors and more 1 + 0 ones. The error sum is usually greater than in the case of (Y= fi = 1.
4. Conclusion In order to provide a firmer foundation for partitioning networks in terms of structural equivalence, two distinct considerations are important. First, there is a choice between direct and indirect approaches, and second, for indirect approaches, there is a choice with regard to a dissimilarity measure. Concerning the choice of a similarity measure, it is necessary to use a measure that is compatible with the concept of structural equiva-
t? Butagelj ef al. / Methods forstnrclural equir:alence
89
lence. It is shown that corrected Euclidean distance on relational data or the shortest paths matrix are both appropriate to measure structural equivalence. However, some of the popular measures of dissimilarity, notably Manhattan distance, Euclidean distance, truncated Manhattan distance, and truncated Euclidean distance, are not compatibie with structural equivalence. The direct approach uses a specified g~dness-of-fan measure as a criterion function to obtain partitions based on structural equivalence. We used a local optinli~atio~ procedure to minimize the criterion function. As a result of this analysis, we obtain partitions that differ from those of CONCOR and STRUCTURE. There is an additional benefit stemming from the criterion function we propose. It can be used as a measure of fit for any clustering obtained elsewhere, When used in this fashion, we show that the CONCOR and STRUCTURE partitions are not optimal. We suspect that for many of the published structural equivalence partitions in the literature, the values of the criterion function, when calculated, will indicate that the partitions are not optimal.
Arabie, Phipps, Scott A. Boorman and Paul R. Levitt 1978 “Constructing blockmodels: Haw and why”. Journal of Mathematical Psychology 17: 21-63.
Batagelj, 1990 Batageij, 1992 Borgatti, 1989
Vladimir “Dissimilarities between structured objects - Fragments”. Submitted. Vladimir, Patrick Doreian and AnuSka Feriigoj “An o~t~rn~~at~ona~approach to reguiaf equivalence”. So&i .Weiwu& 14: 121-135. Stephen P. and Martin G. Everett “The class of ali regular equivalences: Algebraic structure and computation”. Social Networks If: 65-88.
Breiger, Ronald L., Scott A. Boorman, Phipps Arabic 1975 “An algorithm for clustering rciational data with appfications to sociai network analysis and comparison to multidimensional scaling”. Journal of Mathematieul Psychology 12: 328-383.
Burt, Ronald S. 1976 “Positions in networks”. Social Forces 55: 93-122. 1988 “Some properties of structural equivalence, measures derived from sociametric choice data”. Social Networks 10: l-28. 1990 “Detecting role equivalence”. SO&I Ne~work.~12: 83-97. Burt, Ronald S. and Michael _T.Minor 1983 Appfied~g~~~~~ ~ndysis. BeverlyHills: Sage.
90
K Batagelj et al. / Methods for structurul equivalence
Carrington, P.J., G.H. Heil and SD. Berkowitz 1980 “A goodness-of-fit index for blockmodels”. Social Networks 2: 219-234. Doreian, Patrick 1988 “Equivalence in a social network”. Journal of Mathematical Sociology 13: 243-282. Faust, Katherine 1988 “Comparison of methods for positional analysis: Structural and general equivalences”. Social Networks 10: 313-341. Fiksel, J. 1980 “Dynamic evolution in societal networks”. Journal of Mathematical Sociology 7: 27-46. Freeman, Linton C. 1978 “Centrality in social networks: Conceptual clarification”. Social Networks I: 215-239. Gordon, A.D. 1981 Classification: Methods for the Exploratory Analysis of Multit~ariate Data. London: Chapman and Hall. Hummon, Norman P., Patrick Doreian and Linton C. Freeman 1990 “Analyzing the structure of the centrality-productivity literature created between 1948 and 1979.” Knowledge: Creation, Diffusion, Ufilizafion I I: 459-480. Lorrain, Franqois and Harrison C. White 1971 “Structural equivalence of individuals in social networks”. Journul of Mafhematical Sociology 1: 49980. Nadel, S.F. 1957 The Theory of Social Structure. London: Cohen and West. Panning, William H. 1982 “Fitting blockmodels to data”. Social Networks 4: 81-101. Pattison, Phillips E. 1988 “Network models; Some comments on papers in this special issue”. Social Networks 10: 3833411. PetkovSek, Marko, E-mail message, 18 January 1991. Sampson, Samuel F. 1968 A Novitiate in a Period of Change: An Experimental and Case Study of Social Relationships. Ph.D. Doctoral Dissertation, Cornell University. White, Douglas R. and Karl P. Reitz 1983 “Graph and semigroup homomorphisms on networks of relations”. Sociul Networks 5: 193-234. Winship, Christopher and Michael Mandel 1983 “Roles and positions: A critique and extension of the blockmodeling approach”, in: Samuel Leinhardt ted.) Sociological Methodology 1983- 1984. San Francisco: JosseyBass.
63
Direct and indirect methods for structural equivalence *
Procedures far establishing a partition of a network in terms of structural equivalence can he divided into direct and indirect approaches. For the Former, a new criterion function is proposed that reflects directly structural equivalence concerns. ‘This criterion function can then be (locally) optimized to create a partition. For indirect approaches, measures of dissimilarity must be compatible with the definition of structural equivalence.
I. introduction The concept of structural equivalenee has substantive, ~a~h~~~~ical, and technical components. We focus on some of the mathematical ideas with a view to (i) establishing foundations for a direct approach for partitioning a network in terms of structural equivalence, and (ii) assessing the appropriateness of dissimilarity measures for structural equivalence in indirect approaches.
As noted by Doreian fI988), equivalence emerged as a ~ou~datiu~al concept for the analysis of social network representations of social * Revised version of a paper presented at the Eleventh Annual Sunbelt Social Network Conference, Tampa, Florida, Feb. 14-17, 1991. ’ Exchange visit at the University of Pittsburgh” Department of Sociology (November-December 19901. * Fulbright scholar at the University of Pittsburgh, Department of Sociology (September 1996February 1991). 0378~8733/92/$05.00 % I992 - Efsevier Science Publishers B.V. Ah rights reserved
structure. In practice, social actors linked by one (or more) social relation(s) are partitioned into disjoint classes. Let E=(X,,X, )...) X,,} be a finite set of units. The units are related by binary relations t==l . . . Y
R,cExE,
with determine A@“= (E, R,,
a network
R2,.-., R,)
In the following we restrict our discussion to a single relation described by a corresponding binary matrix R = ~YjjI,xn where rij = i
1
xiRXj
0
otherwise
R
In some applications rjj can be a ~~~~egative real number expressing the strength of the relation R between units X, and X,. The main procedural goal of block modeling is to identify, in a given network, &asters fclasses) of units that share structural characteristics defined in terms of R. The units within a cluster have the same or similar connection patterns to other units. They form a clustering
which is a partition of the set E:
Each partition determines an equivalence relation (and vice versa>. Let us denote by - the relation determined by partition %. A block model consists of structures obtained by identifying all units from the same cluster of the clustering G?“,The partition (ideally) is constructed by using structural information contained in R and actors in the same partitioned class are equivalent to each other in terms of R alone.
V. Batage!j et al. / ,~eth~~~sfor structural equirwlence
65
Such actors share a common structural position within the network. (In general, the statement can be extended to include (R,}.) Drawing on the work of Nadel (19571, in an effort to discuss social positions and roles, Lorrain and White (1971) provided the first definition of equivalence: Actors are equivalent if they are connected to the rest of the network in idEnticat ways, a very stringent requirement. Such actors are said to be structurally equivalent. There have been several efforts to weaken the criterion of equivalence. See, for example, White and Reitz (1983) and Winship and Mandel (1983). A more extensive survey is provided by Pattison (1988). However, we deal only with structural equivalence in this paper. Treatment of weaker equivalences, specificaly regular equivalence, is provided elsewhere (Batagelj et al. 1992). There are important reasons for pursuing equivalence ideas for social networks. One is the idea that there are fundamental role structures with observed network behavior, especially interaction, providing indicators of the fundamental structure. Block modeling provides a way of distilling this structure from observed social relations. A very pragmatic rationale is to reduce a large, potentially incoherent network to a smaller comprehensible structure that can be interpreted more readily. Block modeling facilitates this also. For either case, block modeling is an empirical partitioning (clustering) procedure. The adequacy of such a procedure can be pursued at two levels. One level is found in ideal structures where non-trivial structural equivalences exist. In such ideal structures, it is possible to determine if known equivalences in a network are located by a specific block modeling procedure. The second level is found in empirical (non-ideal) networks, where few exact equivalences are found. We deal with both by considering real networks as perturbed ideal networks, 1.2. Structural equivalence A permutation
vx,
YEE:
cp: E --, E is an automorphism of the relation R iff
(XRY=vp(X)Rfp(Y))
The units X and Y are structurally equivalent, we write X s Y, iff the permutation (transposition) P = (Xy) is an automorphism of the relation R (Borgatti and Everett 1989).
In other words: X and Y are structurally equivalent iff sl. XRY-YRX ~2. XR-Y w YRY ~3. QZ E E\{X, Y): (XRZ (-j YRZ) s4. VZ E E\{X, Y): (ZRX@ ZRY) or in the matrix form: X, =X, iff sl’. Yij= ‘)i ** . rri = Y;; 33’ ~3’. Qk f i, j: rik = rjk ~4’. Qk f i, j: rki = rki The matrix form of the definition of structural equivalence extended also to the case when rij are real numbers.
can be
1.3. Establishing block models There are two main approaches to block modelling problems based on the structural equivalence and its relaxations: l
l
indirect approach: reduction to the standard data analysis problems (c.luster analysis, multidimensional scaling) by determining a dissimilarity matrix between units which is compatible with the selected type of equivalence; direct approach: construction of a criterion function P(@?) which measures the fit of the clustering C to the network data, and solving the corresponding optimization problem. For this purpose a relocation procedure from cluster analysis can be adapted.
An indirect approach is one where the empirical partitioning of a network is done in two stages. First, the information in R (or more generally {R,J) is used to create a partition, and second, the partition is evaluated in terms of R. STRUCTURE (Burt 1976), one of the most used algorithms, is inherently indirect. A dissimilarity matrix is either constructed internally (with 6 options) or read from an input file. This matrix is then clustered. Confirmatory factor analysis can be used to assess the adequacy of each structural position identified in the analysis. The other widely used procedure, CONCOR (Breiger et al. 19751, is harder to classify as it has aspects of both approaches. It is indirect in the sense of creating a partition - with users having options for the
67
V. Batagelj et al. / Methods for structural equkalence
number of splits - that can be evaluated separately. On the other hand, it is direct in the sense of seeking partitions to maximize within clusters (iterated) correlations (to + 1.0) and to minimize correlations between clusters (to - 1.0). However, there is no explicit criterion function in the usual sense of an optimization problem. We note that the criterion function we propose for a direct approach can be used to assess the adequacy of partitions obtained by CONCOR or STRUCTURE, or any algorithm
2. The indirect approach
via measuring
the equivalence
of pairs of
units
The first step of an indirect approach is the definition of a dissimilarity measure. This is a crucial step as not all dissimilarities are consistent with structural equivalence. Moreover, although the definition of structural equivalence is “local” it has “global” implications structurally equivalent units behave in the same way also to all other units. A position is defined in terms of all other units in a network. 2.1. Properties of units
The property t: E -+ R is structural if for every automorphism the relation R and every unit X E E
holds. Let t(U) be a structural property of the unit U. Then we have X=Y*t(X)=t(Y)
Some examples of such properties
are
t(U) = degree (number of neighbors~ of unit U or (see Batagelj 1990) t(U) = number of units at distance d from the unit U
cp of
V. Batagelj
68
et al. / Methods for structural equivalence
or (see Burt 1990) t(U) = number of triads of type x at the unit U. For other examples see Freeman (1978) and Hummon et al. (1990). The collection of structural properties t,, t,, . . . , t, is complete (for structural equivalence) iff for each pair of units X and Y (Vi, 1 Si
ti(X)=t,(Y))*X=Y
is satisfied. Remark. The triads are not complete:
the path of length 3 provides a simple counterexample. The extreme units have the same triadic spectrum, but they are not structurally equivalent. We can define the description of the unit U as
and the dissimilari~
between units U and V as
where D is some (standard) dissimilarity between real vectors. In the case when the dissimilarity D has the property
and the properties d(X,Y)=O=D([X].
t,, t,, . . . , t, are complete, it holds [Y])-[x]
=[Y]dfi: ti(x)=ti(Y)
o;yzEy Therefore, we finally have d(X,Y)=O-X=Y
69
V. Batagetj et al. / Methods for structural equivalence
2.2. Properties of pairs of units A property q: E x E --j [wis structural if for every automorphism the relation R and every pair of units X, YE E it holds 4(X7 Y) = q((P(X),
cp of
cp(Y>)
For example, for the original relation R, q(U, V) = r(U, V). See Fiksel (1980) as another example: q(U, V) = number of common neighbors of units U and I/ and q(U, V) = length of the shortest path from U to I/. We note that if the union of the neighborhoods of U and V is identical to the intersection of their neighborhoods, then U and I/ are structurally equivalent (Fiksel 1980). For other examples see Burt (1988). Let q(U, V) be a structural property of units U and I/. Then, if X=Y we have ql. 4(x, YJ = 9CY x) q2. 4(x, x) = q(Y Y) q3. VZ E E\{X, Y}: q(X, Z) = q(Y, Z) q4. VZ E E\{X, Y): q(Z, X) = q(Z, Y) The property q is sensitive if the properties ql-q4 imply also that
x- f. An example of this type of property is UR*V q(u,v)={; otherwise where R* is the transitive and reflexive closure of R. Therefore we can describe each unit U by a vector [U] = [4(U, X,), 4(U, X2),...,
4(U XJ,
4(X,,
U),..., 4(X,7
VI
70
V. Batagelj et al. / Methods for structural
and define
the dissimilarity
equiclalence
units U, I/E E as
between
where D is a dissimilarity between the corresponding descriptions. Some examples of such dissimilarities and a discussion of their appropriateness are given in Section 2.3.1. 2.3. Matrix dissimilarities 2.3.1. Dissimilarities The following is a list of dissimilarities between units X, and X,: Manhattan
for measuring
distance:
ill
dm(XiT
xj)=
(I4is-4jsI
+
I&-i_4,jl)
s=l
Euclidean
distance:
stl((4;s-
4,)"
Truncated d,(Xi,
Manhattan
Xj)=
2
+
(qsi
qs,):)
-
distance:
(Iqi,-qjsIfIq,,-q~j()
s=l s#i,j
Truncated
Euclidean
ds(Xi>Xj) =
distance
(Faust
2 ((qis-
d
qjs)' +
(4si
1988):
-
4rj)*)
s=l
s#i,j
Corrected
Manhattan-like
d,(p)(Xi,
Xj) =d,(Xi>
dissimilarity Xj) +P.(
(p > 0)
Iqi,-qjjI f Iqij-qjil)
the similarity
V. Batagelj et al. / Methods for structural equivalence
Corrected Euclidean-like d,(p)(Xi,
Xj)=
d,(Xi,
71
dissimilarity (Burt and Minor 1983): xj)2+P*
((4,i-4,)2+(qij-4ji)2)
Corrected dissimilarity
dC(P)(xi7
xj)
=
{m
It is easy to verify that all expressions from the list define a dissimilarity, i.e., they have properties dl. d(X, Y> 2 0 d2. d(X, X) = 0 d3. d(X, Y> = d(Y, X) A dissimilarity d which has also the properties d4. d(X, Y>=O*X=Y d5. d(X, Y) + d(Y, Z) 2 d(X, Z) is called a distance. Each of the dissimilarities from the list can be assessed to see whether or not it is also a distance. From calculus we know that Proposition 2.1. Dissimilarities
d, and d, over R” are distances.
Proposition 2.2. If d over E has the property di, i = 1,. . . , 5, then 0 has also this property over E. Proposition 2.3. If d, over E, and d, over E, have the property di, i = 1,. . . , 5, then d, + d, over E, x E, has also this property. Proposition 2.4. If d, over E, and d, over E, have the property di, i = 1,. . . , 5, then /m over E, x E, has also this property. Remark. Because different units may have identical descriptions (rows and columns) the property d(Xi,
Xi) = 0 aXi
=Xj
does not hold so none of the listed dissimilarities can be distances in a
V. Batagelj et al. / methods
72
forstructural equiralence
strict sense. Nevertheless, we can use a slightly relaxed property
d(X,, Xi) = 0 -xi
-xj
Remark. In the case q = r all the dissimilarities from the lgt are invariant to the complementing of relation R + E x E \R = R, i.e., d(R)(X,
Y) = d~~~(X, Y>.
2.3.2. Compatibility In procedures based on structural equivalence we can expect the best results if we use a dissimilarity d which is compatible with structural equivalence, i.e., Xi=Xj-d(X;,
Xj> =0
Not all the dissimilarities from the list are compatible. If CJis sensitive, for dissimilarities d, and d, only d(X,,
Xj)=o=+xi=xi
holds. The converse does not hold. For the matrix
R=O
1
[1
01
the units d,(X,,
X, and X, are structurally
X2) = 2 and d,(X,,
equivalent,
If q is structural property, for dissimilarities Xi=Xj*d(X,,
but (for 4 = r)
X2> = a. d, and d, only
Xj)=O
holds. The converse does not hold. For the matrix
for 9 = r, d,(X,, X2> = d&X,, X2> = 0, but the units X, and X, are not structurally equivalent.
V.Batageij etal./ Methodsfor structuralequir:alence
73
If a structural property 9 is sensitive, the dissimilarities d,, d, and are compatible with the structural equivalence. To see this, suppose that X, =Xj. Then from the properties ql-q4 it follows ql’. Cjij- LJji= 0 42’. qii-qjj=o 43’. Vk # i, j: qik - qjk = 0 44’. ‘dk # i, j: qki - qkj = 0 Therefore all the terms which occur on the right side expressions in the definitions of dissimilarities d,, d, and d, are zero. Hence also d,(X[, XjJ = d,(Xi, Xj, = d,(Xi, Xjj, = 0. This establishes
d,
X;=Xj*d(Xj,
X,)=0
To prove also d(X,,
xi) = 0 axi
sxj
we shall prove the equivalent statement Xi$Xj=+d(Xi,
Xj)fO
Since 4 is by assumption sensitive, from Xi +Xj it follows that for at least one pair of units X,, X,,, {u, U) n Ii, j} f fl at least one of the properties ql-q4 does not hold. This means that qu, # q,,,or equivalentlyq,,,-q,, f 0. The dissimilarities d,, d, and d, have value zero exactly when all the terms are zero. Therefore in this case they are different from zero. 2.3.3. Triangle i~e~uali~ It is straightforward to establish which of the listed dissimilarities satisfy the triangle inequality
d(Xi, Xk) +d(Xk, Xj) 2 d(Xi, Xj) Suppose that d can be expressed in the form d(X,
Y) = 6(X,
Y) + d/(X, Y)
71
tJ. Batagelj
et al. / Methods
far stntcturai
equitalence
where 6 depends only on i, j, k components of vectors describing units X and Y; and d’ depends on the remaining components. Then, by proposition 2.3, to prove that the triangle inequality holds it is sufficient to prove that it holds for 6 and d’. Because of Propositions 2.4 and 2.2 it is worthwhile to consider first the dissimilarities d, and d,.
The dissimilarity dk satisfies the triangle inequality. Therefore, if we consider only vectors which are equal on I’ = I\(i, j, k}, the triangle inequality holds for d, iff it holds for 6,. This is equivalent to A, 2 0 where
By complete enumeration of all possible cases for O/l matrices on the computer we found that there exist counterexamples. For example, for the matrix
we
get
A,= ~O-O~~l-l~+~l-~~+~O-O~-~O-~~-~~-~~ The same matrix is also a counterexample For the dissimilarity d,(p) we have A,(~)=~,(P)(X,,
Xk) +S,(P)(~,,
= -2 for d,.
Xj)-‘~(P)(X,~
“‘JJ * ( I 4;i - qkkI f I qik- qkiI) ’ I 4ij +p+?kk
-
qjj
I f
qkj
%)
I +
Iqji
-
qjk
I
I qkj - qjkI) + I qki- 4ji I + l qik- 4ij
- p * ( I qii - qjj I + I Lirii - qji
I ) - I 4ik -
qjk
I -
1 qki
-
qk j
1
1
V. Batagelj et al. / Methods for structural eqkalence
There exist caunterexamples matrix
i 01
01
for every
p >
75
0. For example, for the
01
we get
A,(P)
=P * (IO-OI+/O-OI)+ +p-(
ll-11+/o-OI)+
=1+1-(2+p)=
II-l/+10-11 Il-0(-t-IO-O/
-p
Therefore the triangle inequality does not hold for all p > 0. For p = 2 all the counterexamples have either the form 1 + 1 - 4 = -2 or the form 1+3-6= -2. Since fi+fi-fi=O and I@ + fi - 6 > 0 the dissimilarity a,(2) satisfies the triangle inequality for O/l matrices. For p = 1 all the counterexamples have either the form 1 + 1 - 3 = -1 or the form 1+2-4= -1. Since fi+fi-fi>O and fi + fi - v% > 0 the dissimilarity 6,(l) satisfies the triangle inequality for O/l matrices. We can show that the dissimilarity d,(p), p = 1, 2 also satisfies the triangle inequality for O/l matrices. By definition d, = ,,,q = km, The previous result tells us that & = 6, satisfies the triangle inequality. Because di = dk, the dissimilarity d:, and by Proposition 2.2 also iq satisfy the triangle inequality. From the equalities
it finally follows, by Proposition inequality.
2.4, that d,
satisfies the triangle
76
V. Batagelj et al. / Methods for structural equicalence
Similar-y, by complete similarity
=
I, ’
satisfies de =
((4ii
-
9jj)*
the triangle
+
enumeration,
(4ij
-
inequality
4ji)')
+
we can establish
(Clik-
for O/l
4jk)2
matrices
(qki -
+
for
that the dis-
qkj)*
p = 1,
2. Because
$6;+ (d$
by Proposition 2.4, d, also satisfies triangle inequality for O/l matrices. For real vectors the dissimilarity a,(2) does not satisfy the triangle inequality. For a counterexample consider the matrix 5 0 4
9 7 6
6 4 6
for which A,=6,(X,,
X,)+6,(X,,
Xi) -S,(Xi,
Xj) = @-
+
d% - d-i%
= -1.51 For the dissimilarities 6,(l) and 6,(l) over real numbers the answer is unknown. A random search for a counterexample with more than l,OOO,OOOtrials failed. For a,(2) over real numbers Martin Juvan and Marko PetkovHek (1991) proved that the triangle inequality holds. For a, b E (0, 1) the following equality (~2 - b)* = I a -b
I
77
% Batagelj et al. / Methods for siructural equicalence
holds. This implies from the list
some additional
relations
between
dissimilarities
de(P)=dc(d The last equality shows that in the case of O/l matrices it is sufficient to prove the triangle inequality only for one of dissimilarities d, and d C’ It is interesting that for symmetric matrices, q,,, = q,, for all u and u, the dissimilarities d, and d, (and therefore also d,, d, and d,) satisfy the triangle inequality. Considering symmetry, we obtain A,
=
2(
) qik
-
qi;
1+
I qij
( 1qii -
qkk
-
qkj
I -
I qik
-
qki
I>
and A, = 2~ .
+
2( I qik
- 4ij
I + I qkk
I’
I qij -
- ‘Ijj
qkj
1- 1qii - qjj I> I - I qik
- qkj
In both cases we can apply the inequality
I qik
- 9ij I + I 4ij -
qkj
I)
I a I + I b I 2 I a + b I giving
I 2 I (qik - Sij) + (4ij -
qkj)
I = I qik
-
qkj
I
and
) qii -
qkk
I+
I qkk
- qjj
I 2 1qii - qjj 1
Thus proving A 2 0. Dissimilarities 6, and 6, over symmetric matrices satisfy the triangle inequality. Using the Propositions 2.3 we prove that this holds also for dissimilarities d, and d,.
78
We can define several properties of pairs of units based on a relation. If a property is sensitive then corrected Manhattan d,, corrected Euclidean d, and corrected dissimilarity d, are compatible with structural equivalence, i.e., two units are structurally equivalent exactly when their dissimilarity is zero. This is an important result: we note that the corrected Euclidean-like dissimilarity used by Burt and Minor (1983) is compatible. Clustering methods usually do not require that the used dissimilarity is a distance. Nevertheless we discussed also the triangle inequality for the selected dissimilarities. If the property is symmetric, the triangle inequality holds for all of the considered dissimilarities. For binary properties it holds only for d, and d,.
3. Direct approach: optimization 3.1. Goodness-of-fit for block models
As described above, in an indirect approach, a block model is established, and then assessed. It seems that the fit of a block model for the network data has received very little attention. There are some exceptions. Arabie et al. (1978) suggest the product moment correlation as a measure of fit. Specifically, given a partition via a block model, the correlation between the permuted data matrix and the corresponding “ideal” matrix with perfect fat fit was proposed as a measure of fit. Carrington et al. (1980) provide example of different structures having the same correlation coefficient and suggest that the correlation is not useful as a measure of fit. In its stead they propose a measure that is constructed in terms expected densities within and between blocks. They argue (Carrington et al. 1980: 226) “_ an index of goodness-of-fit block models should measure the extent to which densities of O’s or l’s in the submatrices of the blocked data matrixfes) deviate from the densities of the blocks in the expanded image matrixtes) towards the worst possible density (a)“. The value of (Yis the threshold used by an analyst to determine O’s or l’s for the image matrix. The measure is a weighted average of the squares of differences between observed and expected densities (for the worst possible fit where the expected value
I/: Batagelj et al. / Methods
forstructural
rquic~alence
79
is a) for each cell corresponding to the image matrix. However, as Carrington et al. (1980: 231) note “there is no obvious . . . way to use (the measure) to show that one model is unambiguously preferable to another.” Panning (1982) reverts to a correlation approach to measure the fit of a block model and explicitly redefines a block modeling effort as a regression analysis. The redefinition seems too forced but it does retain the idea of measuring the correspondence of a block model partition and an ideal partition with a fat fit. This is used below. 3.2. Criterion for structural equicalence Batagelj et al. (1992) propose the construction of criterion functions that reflect the notion of regular equivalence. The same approach can be used in the case of structural equivalence. From the definition of structural equivalence it follows that there are four possible ideal diagonal blocks B(C, C) Type 0. bij = 0 Type 1. bij = Sii Type 2. bij = 1 - ajj Type 3. bij = 1 where aij is the Kronecker delta function and i, j E C. For the nondiagonal blocks NC,, C,,), u # u only blocks of type 0 and type 3 are possible. Given a clustering $57= {C,, C,, . . . , C,}, let SS’(C,,, C,.) denote the set of all ideal blocks corresponding to block NC,, C,,>. Then the global error of clustering 55’can be expressed as
where
the obvious
choice for d is
d(R(C,, Cl.), B) =
c
XEC,,,YEC,
bxy -~,,I
80
K Bata&
et al. / Methods for structural equiralence
It is easy to verify that this defined to structural equivalence P( 5?) = 0 e $5’defines
structural
criterion
function
P(g)
is sensitive
equivalence
In addition, it is invariant to the transformation of complementing the relation R + E x E \I?. The selected dissimilarity between blocks is based on the assumption that the error 0 + 1 is equiprobable to the error 1 + 0. In the case that this assumption is not valid we can introduce (Y and p as weights for the two types of errors. Let B* be a solution of the optimization problem from the right side of the expression for p(C,,, C,,). Then we can redefine p, for example, as follows p(Cu,
C,,)=a
card{(X,
Y)EC~XC,,:
+ p card{(X,
rxr>bzy}
Y) E C,, x C,.: Y,~ < b,*l)
Selecting large a(a = 100) and small p (0 = 11, in optimizing P(‘Z) we primarily are seeking solutions with the least number of errors of the type 0 --) 1 and, as a secondary criterion, among them solutions with the least number of errors of the type 1 + 0. In the special case when the number of O’s equals the number of l’s in the block R(C,, C,.), we select l’s block for B* if a 2 /3; and O’s block otherwise. We note this case is not infrequent, empirically. For solving the block modelling problem we use a local optimization algorithm: Determine the initial clustering 5%‘; repeat: if in the neighborhood of the current clustering 22’ there exists a clustering 2%” such that P(%?‘>
moving a unit from one cluster to another cluster; interchanging of two units from different clusters.
We repeat the local optimization search with several random clusterings and preserve the best 10 obtained solutions.
initial
V. Batagelj
et al. / Methods
for structural
equioalence
81
3.3. Example: Sampson data - liking 3.3. I. Description Sampson (1968) gathered sociometric data from a group of men in a monastery. These data, in a form with 18 individuals and 4 relations (affect, esteem, influence, and sanctioning), have been used as a test bed for many network tools. We follow this tradition, but we confine our attention to the affect (liking) relation. The 18 men, lay and clerical novices in the first group were Ambrose (91, Berthold (61, Bonaventure (51, Mark (71, Peter (41, and Victor (8). The second cohort was made up of Albert (161, Amand (131, Basil (3), Boniface (151, John Bosco (11, Elias (171, Gregory (21, Hugh (141, Louis (111, Romuald (lo), Simplicius (181, and Winfrid (12). During a period of conflict, triggered by changes designed to relax rules and simplify procedures in the monastery, most of the novices split into two opposing groups. One, labeled the Young Turks, was led by John Bosco and Gregory, with Albert, Boniface, Hugh, Mark, and Winfrid as followers. The other basic group, dubbed the Loyal Opposition by Sampson, was led by Peter. The followers were Ambrose, Berthold, Bonaventure and Louis. Sampson identified two other groups. One, labeled the Outcasts, had Basil, Elias, and Simplicius. The remaining monks composed an Interstitial group having no consistent ties with either of the basic parties to the conflict. 3.3.2. Results In the following analysis the positive affect (liking) relation from time four is used. The relation was dichotomized by coding first, second, and third choices as 1, and coding 0 otherwise. First, we present results obtained by the indirect approach. The corrected dissimilarity d,(p = 2) was calculated on original the relation LIKING. Figure 1 shows the dendrogram obtained by the agglomerative with the Ward criterion function (e.g., Gordon 1981: 41) based on the proposed dissimilarity matrix. The hierarchical clustering shows three and also five distinct clusters. We consider both and rerun the analysis with the relocation algorithm with the Ward criterion function (e.g., Gordon 1981: 44). The obtained clustering with 3 clusters is g: = {{1, 2, 7, 12, 14, 15, 16), (4, 5, 6, 8, 9, 10, 11, 13}, (3, 17, 18}}
82
K Batagelj et al. / Methods for structural equi~dence
Boniface
15
Albert
16
Amand John
13 Bosco
Hugh
1 14
Gregory
2
Mark
7
Winfrid
12
Elias
17
Simplicius
18
Basil
3
Berthold
6
Romuald
10
Louis
11
Bonaventur
5
Victor
8
Peter
4
Ambrose
9
Fig. I. Sampson
and is Using Yound group, The
data-liking
/ indirect
approach,
Ward.
the same as that obtained with the agglomerative algorithm. Sampson’s labels, the first cluster (block) corresponds to the Turks, the second to the Loyal Opposition with the Interstitial and the third to the Outcasts. clustering with 5 clusters obtained with relocation method
@ = {{I, 14}, (2, 7, 12, 15, 16}, {4, 5, 9, II}, (6, 8, 10, 131, (3, 17, 18)) is not the same as the hierarchical solution which also means that clustering with five clusters is less appropriate structure for these data. Tables 1 and 2 display the LIKING relation matrix permuted into a form compatible with the three-block model (Table 1) and five-block model (Table 2) obtained with the indirect approach (relocation algorithm). In the tables the 0 --, 1 errors (it should be 0 but there is 1) are indicated by l and 1 - 0 errors (it should be 1 but there is 0) by *. From Table 1 it can be seen that there is only one block of the type 2, all others are of the type 0 or 1. Therefore it is not surprising that there are only 0 --, 1 errors. There are two blocks out of nine without
% Batagelj et al. / Methods for structural equivalence Table 1 Sampson
data - liking / clustering,
relation,
1 2 7 12 14 15 16 4 5 6 8 9 10 11 13 3 17 18
k = 5
1272456
4
5
6
8
9
1 0
1 1
1 3
3
000..00 .o..ooo o.o*oo. . ..oooo . 0 0 o...ooo o..oo*o 0 0 0 0 0 0 0000000 0 0 0 000.000 0 0 0 0000*00 00.0000 . 0 0 0.00000
0 0 0 0 0 0 0 0 . . . 0 . 0 0 0 0 0
0 0 0 0 0 0 0 . 0 . 0 . . . . 0 0 0
0 0 0 0 0 0 0 . 0 0 . 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 . 0 . 0 0 0 0
0 0 0 0 0 0 0 0 . . . 0 . 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 . . 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 . 0 0
0 . 0 0 0 0 0 0 ooo 0 0 0 0 0 0 0 011 1 1
1 John Bosco Gregory Mark Winfrid Hugh Boniface Albert peter Bonaventw Berthold Victor Ambrose Romuald LO& Amand Basil Elias Simplicius
83
1
1
1
.
0
.
0
0 0
‘0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0.00000
1 7
1 8
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 .
0 1
1 0
errors and a further three with only one error. The clustering with 5 clusters (Table 2) shows both types of errors. Ten out of 25 blocks have no errors. By the direct approach (local optimization) we obtained as the best solution with 3 clusters for Q = /3 = 1 the clustering @,, = (11, 2, 7, 12}, {4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16) (3, 17, 18)) and with 5 clusters the clustering g;,
= ((1, 13, 14, 15, 16) (2, 7, 12) {4,5, 9) (6, 8, 1% 11) (3, 17, 18})
The corresponding terings are presented
rearanged relation matrices for these two clusin Tables 3 and 4.
84
V. Batagelj et al. / Methods
Table 2 Sampson
data - liking / clustering,
relation,
1 1 142725645916803378 John Bosco Hugh Gregory Mark Winfrid B&face Albert Peter Bonaventur Ambrose Louis Berthold Victor Romuald Amand Basil Eli.%? Simplidus
10100*0000000000*00 14 10 0 o..o 2*0011**00000000000 700101*100000000000 12 .OllO 15 0 01110 16 0 Oll*lO 4000000001*1*000000 5 0 0 0 0 90000*00*10*0~00000 11 0.0 0 60000000111*0000000 800000001*1i~~000000 10 0 0 0 0 13 0 0 0.0 3*000000 17 0 0.0 18 0 0.0
1
for structural equivalence
k =5
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
*
* *
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
*l*
0
0.0
0
0
0
0
0
0 0
0 0
* *
0 0
0 0
0 0
. 0
0 0
0 0.
0
0 0
0 0
0 0
1 1 1 *1* 0000000*011 0 0 0 0 0 0
0 0
0 0
0 0
0 0
0101 0110
We repeated the direct approach also for (Y= 100 and p = 1. The best obtained clustering with three clusters is the following one g;,
= {{l, 2, 7, 12, 5, 16}, {4, 5, 6, 8, 9, 11}, (3, 10, 13, 14, 17, 18}}
and for five clusters g:,
= ((1, 15}, (2, 7, 12, 14, 161, {4, 5, 678, 9, ll),
{lo),
{3, 13, 17, 18j) The rearranged relation matrices for these clusterings are sented in Tables 5 and 6. The same matrix was also analyzed STRUCTURE and CONCOR. By STRUCTURE, for positions (Euclidean distance between raw relation patterns) and positions (between raw relation patterns - mean difference) we obtained following clustering with 3 clusters:
preby = 0 = 1 the
‘Z;, = {{l, 2, 7, 12, 14, 15, 16}, {4, 5, 6, 8, 9, 10, 11}, (3, 13, 17, 181)
8.5
1
1
1
1
2
7
2
4
5
6
8
9
0
1
3
1 4
1 5
1 6
0 1 *
* 0 I
* 1 0
1 1 1
0
0
0
0
0
000.00
0
0
0
0
0
0
0
0
0 0
0
a
0
0
a
0 0 0
0 0
0
. 0 0
0
0
1
John Bosccl ---iGregory 2 7 Mark Winfrid 12 4 Peter Bonaventur 5 Berthofd 6 8 Victor 9 Ambrose Romtid 10 Louis It Amand 3.3 14 Hugh Bonif&x 1.5 Albert 16 Basil 3 Eli& 17 Simplicius 18
I
I
1
0
D
0
0
0
0
.
.
0
000000000*0
. 0
0
0
0
0
.
0
0
0
0
.
0
0
0
0 0 0
0
and clustering with 5 clusters G$, = {{l, 2, 7, 12, 141, (1% ‘Ifi), (4, 5, 6, 8, 9, 111, (101, {3, 13, 17, 18)) By STRUCTURE, for positions = 2 (between patterns - covariance) we obtained the clusterings
deviation
and clustering with 5 clusters @:z = ((1, 14, 1% (2, 7, 12, 16}, (4, 5, 6, 8, 9, ll}, (lo}, {3,13, 17,18}}
relation
V Batagelj et al. / Methods for structural equicalence
86 Table 4 Sampson
data
liking / LocOpt
1
1
k = 5, a = 1, /3= 1
1
1
1
1
1
1
1 0
134562724596801378 JohnBosco Amend Hugh BClnif.%e Albert Gregory Mark Winfrid peter Bonaventur
o o . o o * * 1 1 13 0 0 0 0 0 *l* 14 .o 0.0 * *lO 15 0 0 0 0 01110 16 0 0 O.Oll* 2*00000110000000000 70000*1010000000000 12 .o 0 0 0110 40000000001**00*000 o o o o 0 0 5 0 o
By CONCOR
we obtained
0 0 0.0 0 0 0 0
0
0 1
the following
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
. 0 0 0 0
0 0. 0 0 0
0 0 o
0
0
0
0
0
0
0
0
0
0
1
0
0
0
.
0
0
0
clustering
with 3 clusters
~‘2 = ((1, 2, 7, 12, 14, 15, 16}, {4, 5, 6, 8, 9, 10, II, I3}, (3, 17, 18)) and clustering
with 5 clusters
%,J = {{l, 14}, (2, 7, 12, 15, 16}, {4, 9, II, 13}, (5, 6, 8, IO), (3, 17, I@} The direct approach was used also to obtain the best clustering with 1 to 8 clusters. In Table 7 the values of the criterion function for structural equivalence P(E’ (the global error of the clustering E’> for all mentioned clusterings and for selected values of cy and p are presented. In the case of (Y= p = 1 the criterion function actually counts number of errors in the blocks. The clustering with 3 clusters with the best criterion function is that obtained by the direct approach (P = 44). The clustering obtained with the indirect approach and by
V. Batagelj et al. / Methods for structural equivalence
Table 5 Sampson
data - liking / LocOpt
Bonaventur Berthold Victor Ambrose LO& Basil Rmnuald Amand Hugh El& Simplicius
Table 6 Sampson
5 000000 6 000000 8 0 0 000.00 9 000000 -11 . 0 3 000000 10 13 00.000 .oo*.o 14 0.0000 17 18 11 0 .
0
0
0
data - liking / LocOpt
k = 3, (Y= 100, p = 1
0
0
0
1 0 0 * 15 0 0 1 21*011**0000000 7**101*10000000 12 1 * 1 14 1 1 * 16jl* 111 4110 010 5 6 8 9
0
0
0
lO**ll llO*l* 1 * 0 * 1 * 1 0 0 0 ..oo.o 0 . 000000 0 0 0 1 0 0
1 * 0
0 1 1 0
1 0 * 0
* * 0 0
0
0
0
0
0 0
0 0
0 0
0 0
l
1 303478
1
1
1
1 II
0 ~00000 000000 0 0 0 0 0 0 000000 0 0 000000 0 0 0 0 000.00 00.0.. 0 0 00000. 000000 .oooo.
0
.
0
0
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
0
0 0
0 0
0 0
0 0
.
0
0
0
3
1 3
1 7
1 8
.
0
0
0
0 0 0 0
.ooo.o
k = 5, a = 100, fJ = 1
1 15272464568910 JohnBosco Boniface Gregory Mark Winfrid Hugh Albert Peter Bonaventur Berthold Victor Ambrose
87
*_ 1
1 * 1 0
1 1 1
0 1 * 0
1 1 *
* 0 * 0
1 * *
1 0 0
* 0 * 0 010 010
0 0
0 0 0 1
0 0
0 0 0 1
0 0
0 0 0 *
0 0
0 0 0 *
0 0
1 0 0
0 0 0 0 010 110
0
0
0
0
0 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
88
V. Batageij
Table 7 The values of the criterion
et al. / Methods
function
for .structural
equivalence
for the best clusterings
n
cr=1,p=1
(Y= 100, p = 1
1 2 3 4 5 6 7 8
56 50 44 40 35 31 28 25
5600 40 14 24 28 1242 941 743 446 434
@ ??I ?A @
50 52 52 50
5000 4804 4804 5000
@ G7s51 Y:, 9;
38 46 44 38
1919 1135 1127 25 13
CONCOR are the same with 6 errors more. The worse is the STRUCTURE solution (P = 52). The same is also true for the obtained clusterings with 5 clusters. In the case of (Y= 100 and p = 1 the criterion function counts the number of 0 + 1 errors in the first two digits and on the last two the number of 1 + 0 errors. The sum of both errors givs the total number of errors. The clusterings obtained by the direct approach with (Y= 100 and p = 1 have usually less 0 -+ 1 errors and more 1 + 0 ones. The error sum is usually greater than in the case of (Y= fi = 1.
4. Conclusion In order to provide a firmer foundation for partitioning networks in terms of structural equivalence, two distinct considerations are important. First, there is a choice between direct and indirect approaches, and second, for indirect approaches, there is a choice with regard to a dissimilarity measure. Concerning the choice of a similarity measure, it is necessary to use a measure that is compatible with the concept of structural equiva-
t? Butagelj ef al. / Methods forstnrclural equir:alence
89
lence. It is shown that corrected Euclidean distance on relational data or the shortest paths matrix are both appropriate to measure structural equivalence. However, some of the popular measures of dissimilarity, notably Manhattan distance, Euclidean distance, truncated Manhattan distance, and truncated Euclidean distance, are not compatibie with structural equivalence. The direct approach uses a specified g~dness-of-fan measure as a criterion function to obtain partitions based on structural equivalence. We used a local optinli~atio~ procedure to minimize the criterion function. As a result of this analysis, we obtain partitions that differ from those of CONCOR and STRUCTURE. There is an additional benefit stemming from the criterion function we propose. It can be used as a measure of fit for any clustering obtained elsewhere, When used in this fashion, we show that the CONCOR and STRUCTURE partitions are not optimal. We suspect that for many of the published structural equivalence partitions in the literature, the values of the criterion function, when calculated, will indicate that the partitions are not optimal.
Arabie, Phipps, Scott A. Boorman and Paul R. Levitt 1978 “Constructing blockmodels: Haw and why”. Journal of Mathematical Psychology 17: 21-63.
Batagelj, 1990 Batageij, 1992 Borgatti, 1989
Vladimir “Dissimilarities between structured objects - Fragments”. Submitted. Vladimir, Patrick Doreian and AnuSka Feriigoj “An o~t~rn~~at~ona~approach to reguiaf equivalence”. So&i .Weiwu& 14: 121-135. Stephen P. and Martin G. Everett “The class of ali regular equivalences: Algebraic structure and computation”. Social Networks If: 65-88.
Breiger, Ronald L., Scott A. Boorman, Phipps Arabic 1975 “An algorithm for clustering rciational data with appfications to sociai network analysis and comparison to multidimensional scaling”. Journal of Mathematieul Psychology 12: 328-383.
Burt, Ronald S. 1976 “Positions in networks”. Social Forces 55: 93-122. 1988 “Some properties of structural equivalence, measures derived from sociametric choice data”. Social Networks 10: l-28. 1990 “Detecting role equivalence”. SO&I Ne~work.~12: 83-97. Burt, Ronald S. and Michael _T.Minor 1983 Appfied~g~~~~~ ~ndysis. BeverlyHills: Sage.
90
K Batagelj et al. / Methods for structurul equivalence
Carrington, P.J., G.H. Heil and SD. Berkowitz 1980 “A goodness-of-fit index for blockmodels”. Social Networks 2: 219-234. Doreian, Patrick 1988 “Equivalence in a social network”. Journal of Mathematical Sociology 13: 243-282. Faust, Katherine 1988 “Comparison of methods for positional analysis: Structural and general equivalences”. Social Networks 10: 313-341. Fiksel, J. 1980 “Dynamic evolution in societal networks”. Journal of Mathematical Sociology 7: 27-46. Freeman, Linton C. 1978 “Centrality in social networks: Conceptual clarification”. Social Networks I: 215-239. Gordon, A.D. 1981 Classification: Methods for the Exploratory Analysis of Multit~ariate Data. London: Chapman and Hall. Hummon, Norman P., Patrick Doreian and Linton C. Freeman 1990 “Analyzing the structure of the centrality-productivity literature created between 1948 and 1979.” Knowledge: Creation, Diffusion, Ufilizafion I I: 459-480. Lorrain, Franqois and Harrison C. White 1971 “Structural equivalence of individuals in social networks”. Journul of Mafhematical Sociology 1: 49980. Nadel, S.F. 1957 The Theory of Social Structure. London: Cohen and West. Panning, William H. 1982 “Fitting blockmodels to data”. Social Networks 4: 81-101. Pattison, Phillips E. 1988 “Network models; Some comments on papers in this special issue”. Social Networks 10: 3833411. PetkovSek, Marko, E-mail message, 18 January 1991. Sampson, Samuel F. 1968 A Novitiate in a Period of Change: An Experimental and Case Study of Social Relationships. Ph.D. Doctoral Dissertation, Cornell University. White, Douglas R. and Karl P. Reitz 1983 “Graph and semigroup homomorphisms on networks of relations”. Sociul Networks 5: 193-234. Winship, Christopher and Michael Mandel 1983 “Roles and positions: A critique and extension of the blockmodeling approach”, in: Samuel Leinhardt ted.) Sociological Methodology 1983- 1984. San Francisco: JosseyBass.