The Duality of Persons and Groups
Author(s): Ronald L. Breiger
Source: Social Forces , Dec., 1974, Vol. 53, No. 2, Special Issue (Dec., 1974), pp. 181-190
Published by: Oxford University Press
Stable URL: https://www.jstor.org/stable/2576011
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The Duality of Persons and Groups*
RONALD L. BREIGER, Harvard University
ABSTRACT
A metaphor of classical social theory concerning the "intersection" of persons within groups and of
groups within the individual is translated into a set of techniques to aid in empirical analysis of the
interpenetration of networks of interpersonal ties and networks of intergroup ties. These techniques are
useful in'the study of director interlocks, clique structures, organizations within community and national
power structures, and other collectivities which share members. The "membership network analysis"
suggested in this paper is compared to and contrasted with sociometric approaches and is applied to
the study by Davis et al. (1941) of the social participation of eighteen women.
Consider a metaphor which has often appeared
in sociological literature but has remained
largely unexploited in empirical work. Individuals
come together (or, metaphorically, "intersect"
one another) within groups, which are
collectivities based on the shared interests, personal
affinities, or ascribed status of members
who participate regularly in collective activities.
At the same time, the particular patterning of
an individual's affiliations (or the "intersection"
of groups within the person) defines his points
of reference and (at least partially) determines
his individuality.1
The following discussion consists of a translation
of this metaphor into a set of techniques
which aid in the empirical analysis of the interpenetration
of networks of persons and networks
of the groups that they comprise. My
usage of the term "group" is restrictive in that
I consider only those groups for which membership
lists are available-through published
sources, reconstruction from field observation
or interviews, or by any other means. Such
groups include corporation boards of directors
(J. Levine, 1972), organizations within a community
or national power structure (Lieberson,
1971; Perrucci and Pilisuk, 1970), cliques or
organizations in a high school (Bonacich, 1972;
Coleman, 1961), and political factions.
Donald Levine (1959:19-22) writes that
"the concept of dualism" is a key principle
"underlying Simmel's social thought." Levine
explicates Simmel's dualism as "the assumption
. . . that the subsistence of any aspect of human
life depends on the coexistence of diametrically
opposed elements." My own usage
of the comparable term "duality" is specified
with respect to Equations 3 and 4 below.2
THE BASIC CONCEPTION
Consider a set of individuals and a set of
groups such that the value of a tie between any
two individuals is defined as the number of
groups of which they both are members. The
value of a tie between any two groups is de*
For their criticism and encouragement, I am
indebted to Harrison White, Gregory Heil,
Francois Lorrain, and Scott Boorman. For seminars
which first introduced me to Simmel's
thought, I am indebted to Kurt H. Wolff. Thanks
are due Professor White for support through NSF
Grant GS-2689.
1 Simmel (1955) entitled one of his essays "The
Intersection of Social Circles," but Reinhard Bendix
changed the title in translation because "a
literal translation of this phrase . . . is almost
meaningless . . . Simmel often plays with geometric
analogies; it has seemed advisable to me to
minimize this play with words ." (Simmel,
1955:125). For an assertion that Simmel's original
title is not at all inappropriate, see Walter's essay
(1959). For a more complete explication of the
"dualism" inherent in Simmel's thought, see the
essays by D. Levine, Lipman, and Tenbruck in
Wolff (1959). A similar metaphor was put forward
in America by Charles H. Cooley (1902:148),
who wrote that "A man may be regarded as the
point of intersection of an indefinite number of
circles representing social groups, having as many
arcs passing through him as there are groups."
Much later, Sorokin (1947:345) observed that
"the individual has as many social egos as there
are different social groups and strata with which
he is connected." On the "much neglected" development
of the concept of "social circle" since
Simmel's writings, see Kadushin (1966).
2The "directional duality principle" enunciated
by Harary et al. (1965) is to be distinguished from
my conception. The former principle consists in
reversing the directionality of lines in a graph; in
the method of this paper, the lines in one graph
are transformed into the points of its dual graph,
and vice versa.
[ 181 ]
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182 / SOCIAL FORCES / vol. 53:2, dec. 1974
fined conversely as the number of persons who
belong to both. A fictitious example is provided
in Figure l.A-1 and 1.A-2, where individuals
are named by capital letters and their groups
are named by integers. In concrete applications
we might take U.S. Congressmen as the individuals
and their committees as the groups, or
schoolchildren as the individuals and their
cliques as the groups, and so forth.
We may construct a matrix of interpersonal
ties (denoted P) and a separate matrix of intergroup
ties (G) in the usual way (Figure 1.B):
let the (i, j) th entry of P indicate the number
of groups to which both person i and person j
belong, and let the (i, j) th entry of G indicate
the number of persons who are members both
of group i and group j. Each matrix is square;
its row- and column-headings are identical
strings of the names of all persons (in the P
matrix) or all groups (in the G matrix) under
study. These matrices are mutually noncomparable
in the following ways: they represent
different levels of structure (persons and
groups); they are not of the same dimension;
and they differ in their cell-by-cell entries.
Although these differences between the interpersonal
network and the intergroup network
are quite evident, the P and G matrices
nonetheless stand in intimate relation to one
another. Following Simmel (1955: 125-8, 147),
think of each tie between two groups as a set
of persons who form the "intersection" of the
groups' memberships. In the dual case, think
of each membership tie between two persons
as the set of groups in the "intersection" of
their individual affiliations.
Define a binary adjacency matrix A (Figure
1.C) whose (i, j)th entry is "1" if person i is
A-1. Interpersonal network A-2. Intergroup network
A B C D E F 1 2 3 4 5
A 1 0 0 1 0 0 3 0 1 0 2
B 0 01 0 00 1,I10 1000
D 110 1 012 30 1 02 1
E 10 00 101 4j0 1 20 1
F 0 0 02 10 5101 1 10
B-1. Matrix representation (P) of Figure 1. A-1 B-2. Matrix representation (g) of Figure 1. A-2
1 2 3 4 5
A 0 0 0 Q 1
Bl 1 0 0 0 0
C1 1 1 0 0 0
D l 1 1 1 1
E IO O 1 0 0
F 00 1 1 0
C. The binary adjacency matrix (A) of person-to-group
affiliations
Figure 1. Fictitious Data
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Dualityof Persons &Groups / 183
affiliated with group j; "O" otherwise. Where
there are p persons and g groups under consideration,
A has dimension p X g, while the
P and G matrices have dimension p X p and
g X g respectively.
Notice that if we intersect any rows i and j
of the A matrix (that is, lay one row atop the
other, according the value "1" only to those
entries which are "1" in the same column of
each row) and count the number of ones in
the intersection, we discover the (i, j)th entry
of the P matrix of Figure l.B-1 (and dually
for the intersection of pairs of columns of A
with respect to the G matrix of Figure 1.B-2).
This result is purely definitional. As will be
seen below, it will be useful to formulate the
definition in matrix notation.
a
Pii = EAikA3k (1)
k=1
and similarly for ties between groups:
p
Gi; = E AkiAk (2)
k=l
The matrix A T of group-to-person ties is
equivalent to A except that its rows are interchanged
or "transposed" with its columns; that
is, AT is of dimension g x p and AT'j =Aji for
any i and j. Hence we may rewrite the above
equations using the person-to-group "translation"
matrix A and its transpose to obtain the
fundamental equalities:
P= A(AT) (3)
G= (AT)A (4)
where the multiplication is ordinary (inner
product) matrix multiplication. Thus: two distinct
matrices, one of person-to-person relations
(P) and one of group-to-group relations (G),
are uniquely defined by and derivable from a
single "translation" matrix (A) of person-togroup
affiliations.3
COMPARISON OF MEMBERSHIP NETS AND
SOCIOMETRIC NETS
There are crucial sociological and mathematical
differences between the approach of this paper
and that of conventional sociometry.4 An
elaboration of both types of differences will
help to clarify the nature and potential utility
of each approach.
Erving Goffman (1971:188), in his discussion
of "tie signs," writes that "the individual
is linked to society through two principal social
bonds: to collectivities through membership
and to other individuals through social relationships.
He in turn helps make a network of
society by linking through himself the social
units linked to him."
I disagree with Goffman in that I see no
reason why individuals cannot be linked to
other individuals by bonds of common membership
(as in interlocking directorates) or to
collectivities through social relationships (as in
"love" of one's country or "fear" of a bureaucracy).
Moreover, Goffman's focus on the individual
as his unit of analysis is a one-sided
departure from Simmel's insight into duality.
This demurral notwithstanding, I fundamentally
agree that there are two types of social ties:
membership and social-relations. Following
Goffman's terminology, I will refer to my approach
as "membership network analysis" in
contrast to the conventional "social-relations
network analysis" typified by sociometry. A
similar vocabulary is hinted by the anthropologist
S. F. Nadel (1957:91, 95) in his discussion
of "relational roles" and "membership
roles":
3 Notice that the products in Equations 3 and 4
differ from the P and G matrices of Figure I in
that the former have non-zero main diagonal entries.
(The main diagonal of a square matrix
consists of cells [1, 1], [2, 2], and so on to [p, p]
or [g, g]). Implications of this difference are discussed
in the following section.
4 For a review of sociometric and related
methods, see Glanzer and Glaser (1959).
. . .[B]elonging to a subgroup, being involved in
its regular activities and rules of behaviour, has
all the characteristics of rcse performance. Which
means that the names describing persons in terms
of the subgroups they belong to are true role
names. And this means, further, that these membership
roles, whether explicitly named or not, correspond
to r elational roles, since the very nature
of groups depends on the relationships between the
people comprising them. . . . The two networks
[membership and rielational], in other words, can
exist side by side and interpenetrate.
All sociometric approaches specify that the
points or "nodes" of a graph are actors (persons
or-much more rarely-collectivities) and
that the lines or "ties" of the graphs are social
relationships (affect, avoidance, "helping," influence,
etc.). Actors and relationships are conceived
as irreducible phenomena. When the relationships
are those of membership, however,
this conception is radically at odds with Simmel's
image (Wolff, 1959:350) in which "the
fact of sociation puts the individual into the
dual position . . . that he is both a link in the
organism of sociation and an autonomous organic
whole." With respect to the membership
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184 / SOCIAL FORCES / vol. 53:2, dec. 1974
network, on the other hand, persons who are
actors in one picture (the P matrix) are with
equal legitimacy viewed as connections in the
dual picture (the G matrix), and conversely
for groups. Formally, we have two classes (one
for all people and one for all groups under
consideration) of finite sets (each person is
associated with the set of groups to which he
belongs, and conversely for groups) and an
axiom that the intersection of any two sets belonging
to either class is contained in the power
set of the other class.5
A second axiom of the membership network
is symmetry: if person a is connected to person
b by virtue of a shared membership, then
b is connected to a as well. If two groups
share at least one member, they are mutually
related. This implies reflexivity: a person who
belongs to any group relates to himself by that
fact, and similarly for any group with members.
The main diagonal of a sociomatrix consists
solely of zeroes if irreflexivity has been imposed
(as is usual). This represents a crucial
contrast with the membership network. As Harrison
White (1971:31) has stated, "whether to
assign self-choices ('loops') in a generator
graph . . . is a fundamental theoretical issue,
not a technicality of computation, as it has
often been regarded." One advantage of the
membership net is the intuitively clear meaning
of reflexivity: the number of ties between a
person and himself is the number of groups to
which he belongs (and conversely in the dual
matrix: the number of ties between a group
and itself is the number of members), whereas
to state in a sociometric analysis that a person
''esteems" or "avoids" himself (say) three times
has no meaning that has been developed. Moreover,
the sum of the main-diagonal entries in P
always equals the corresponding sum in G, as
the affiliations not only create the differences
between the two networks but unify them as
well. (More formally: the sum of any row in
the "translation" matrix A gives the number of
groups to which a particular individual belongs;
hence, by Equation 3, the vector of row-marginals
of A is equivalent to the main diagonal
of P; similarly, the vector of column-marginals
of A is equivalent to the main diagonal of G
by Equation 4; moreover, the sum of rowmarginals
must equal the sum of column-
marginals.)
As a further theoretical implication of reflexivity,
consider the group-to-group matrix G.
A lower bound on the total number of persons
who belong to all groups (i.e., a lower bound
on the dimension of the P matrix) is given by
the largest-valued cell on the main diagonal of
G. An upper bound is given by the sum of
main-diagonal cells in G. (That is: if all the
persons belonging to all groups are found to
belong to any single group, then the lower
bound is the actual number of persons; at the
opposite extreme, if no groups overlap, the
upper bound is the actual number of persons.)
And conversely in consideration of the dual
(P) matrix.
AN APPLICATION OF DUAL ANALYSIS:
SOCIAL PARTICIPATION IN 'OLD CITY'
In empirical work we might define some
minimal level of connectivity among (say)
groups, excluding any group connected to at
least one other by at least k links, and then examine
the dual person-to-person matrix resulting
from this selection. The goal is to look
for patterned relations among persons; the
strategy is to perform operations on the (groupto-group)
matrix dual to our interest. The value
of k is set according to the "graininess" or
connectivity ratio (defined below) desired in
the resulting matrix.
Consider the study by Davis et al. (1941) of
the social participation of eighteen women in
"Old City." The method employed in their investigation
is discussed in somewhat greater
detail by Homans (1950). The researchers
compiled a table with eighteen rows-one for
each woman-and fourteen columns, one for
each "event" (such as a club meeting, a church
supper, a card party, and so on), held during
the course of a year, for which it could be
determined that various of the women were
present. The goal of the study was to determine
the clique structure among the women.
At the start of the analysis the rows were
arranged arbitrarily and the columns chronologically,
as in the A matrix of Figure 2a which
I have adapted from Homans' presentation and
in which the (i, j)th entry represents the presence
or absence of woman i at event j. The
reader will observe that the A matrix fits precisely
my conception of a "translation matrix."
The researchers were aware that they could
derive the woman-to-woman relations from A,
but they chose not to do so. A glance at the P
matrix of Figure 2b will, I believe, indicate
why. The researchers were attempting to dis-
5"Power set" denotes the set of all possible
sets of the given elements; e.g., the power set of a
set containing three objects consists of eight sets,
including the empty and universal sets.
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Duality of Persons & Groups / 185
cover the clique cleavages among the women;
however, connectivity in the P matrix is 91
percent.6 Since everyone was connected to
virtually everyone else, identification of subgroupings
became problematic. As Homans
(1950:82-3) describes it:
The chart in its rough form will not reveal very
much. (If you do not believe this, try making
such a chart for yourself.) For one thing, the
columns are probably arranged in the chronological
order of events, and the women are probably
in no particular order at all. But then we
begin to reshuffle lines and columns. As far as
columns are concerned, we put in the center the
columns representing the events . . . at which a
large number of the women were present, and we
put toward the edges the columns representing the
events . . . at which only a few of the women
were present. As far as lines are concerned, we
put toward the top or bottom the lines representing
those women that participated most often together
in social events. A great deal of reshuffling may
have to be done before any pattern appears.
There can be no doubt that the researchers
were operating with an implicit conception of
duality, although they were uninterested in
event-to-event relations. A more explicit conception
might have led them to a much less
time-consuming approach (particularly as no
computer was available) as follows. Begin with
the unpermuted A matrix of Figure 2a, even
though it is in "rough form." By Equation 4,
create the matrix-call it G-of membership
overlaps among events (see Figure 3a). Impose
the assumption that only those events which
have zero overlap with at least one other event
are likely to separate the women into socially
meaningful subgroupings. Therefore, by inspection
of the G matrix (which is dual to the
matrix of our interest), note each column
which contains no "zero" entry (i.e., columns
4, 6, 8, and 12) and eliminate the corresponding
column in the A matrix, creating the modified
translation matrix A2 of woman-to-event
relations. By Equation 3, create P2, the new
matrix of woman-to-woman relations (Figure
3b), which may be thought of as the "skeleton
structure" of the original P matrix. Inspection
of P2 will show that connectivity has been
significantly reduced to 30 percent-but is this
reduction meaningful? The answer is affirmative
with one minor qualification: although the
two cliques (of sizes seven and five, respectively)
that Homans describes7 are contained
person for person in the graph of P2 (see
Figure 4), each clique in the latter graph also
contains one additional woman (Ruth and
Verne, respectively), whom Homans (1950:
84) describes as "marginal" to both cliques.
("The pattern is frayed at the edges, but there
is a pattern.")8
DUALITY AND TRANSITIVITY
While the analysis of the previous section was
predicated on knowledge of the "translation"
matrix A, this section indicates that information
about "reachability" in either the personto-person
or the group-to-group matrix may be
derived from knowledge of its dual matrix. In
the graph of person-to-person ties, two persons
are mutually "reachable" along a path of length
n if there exists a sequence of n contiguous
ties between them (that is, if there exist n-I
intermediate persons on a connected path from
one person to the other). The number of person-to-person
ties of length n between every
two persons is given by entries of the binarized
P matrix raised to the nth power (Harary et
al., 1965). With reference to the fictitious data
of Figure l1A-1, for example, persons B and D
are connected by one 2-path (B-C-D; this is
the shortest path), but also by all (2 + 3k) paths
(k any positive integer), including two
5-paths (B-C-D-E-F-D and B-C-D-F-E-D; these
6That is, of the 153 = (1/2)(18)(17) possible
binary ties among the eighteen women, we find by
inspection of P that 139 ties actually existed. The
ratio 139/153 is 0.91.
7The clique membership reported by Homans
(1950) is as follows. Clique 1: Evelyn, Laura,
Theresa, Brenda, Charlotte, Frances, Eleanor.
Clique 2: Myrna, Katherine, Sylvia, Nora, Helen.
Women not clearly belonging to either clique:
Pearl, Ruth, Verne, Dorothy, Olivia, Flora.
8 As Homans (1950) notes, the analysis of Davis
et al. follows the logic of Forsyth and Katz, which
-as several authors (see Glanzer and Glaser,
1959 for a review) have observed-involves much
awkward and tedious manipulation. More recent
methods of clique detection (methods of Festinger
and of Luce and Perry, reviewed by Glanzer and
Glaser, 1959:326-28; see also Alba, 1973) are
applicable only to square sociomatrices: e.g., to
the P matrix of Figure 2b rather than to the rectangular
A matrix of Figure 2a. Since connectivity
in the P matrix approaches unity (91 percent),
the problem for clique detection is the reduction
of connectivity-hence the concern for operations
on the (group-to-group) matrix dual to the sociomatrix,
rather than with powers of the latter. A
new algorithm (Breiger et al., 1974) for detecting
structure in multiple relational matrices combines
this duality approach with the blocking and structural
equivalence concepts of Harrison White
(1974; White and Breiger, 1974) and has yielded
highly interpretable results on this data and on
various other social network data.
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186 / SOCIAL FORCES / vol. 53:2, dec. 1974
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Eleanor 0 1 0 1 0 0 o 1 0 0 0 1 0 0
Brenda o 1 o 1 0 o 1 1 o 1 0 1 1 0
Dorothy 0 0 0 0 0 1 0 0 0 0 0 1 0 0
Verne 0 0 0 1 1 1 0 0 0 0 0 1 0 0
Flora 1 0 0 0 0 1 0 0 0 0 0 0 0 0
Ol ivia 1 0 0 0 0 1 0 0 0 0 0 0 0 0
Laura 0 1 1 1 0 0 1 1 0 1 0 1 0 0
Evelyn 0 1 1 0 0 1 1 1 0 1 0 1 1 0
Pearl 0 0 0 0 0 1 0 1 0 0 0 1 0 0
Ruth 0 1 0 1 0 1 0 0 0 0 0 1 0 0
Sylvia 0 0 0 1 1 1 0 0 1 0 1 1 0 1
Katherine 0 0 0 0 1 1 0 0 1 0 1 1 0 1
Myrna 0 0 0 0 1 1 0 0 1 0 0 1 0 0
Theresa 0 1 1 1 0 1 1 1 0 0 0 1 1 0
Charlotte 0 1 0 1 0 0 1 0 0 0 0 0 1 0
Frances 0 1 0 0 0 0 1 1 0 0 0 1 0 0
Helen 1 0 0 1 1 0 0 0 1 0 0 1 0 0
Nora 1 0 0 1 1 1 0 1 1 0 1 0 0 1
Figure 2a. The A Matrix Indicating Presence ("1") or Absence ("0") of Each of Eighteen Women at Each of
Fourteen Social Events. Adapted from Homans (1950:83). (Row headings name the women. Column headings
name each event in chronological order.)
El Br Do Ve Fl 01 La Ev Pe Ru Sy Ka My Th Ch Fr He No
Eleanor 4 4 1 2 0 0 4 3 2 3 2 1 1 4 2 3 2 2
Brenda 4 7 1 2 0 0 6 6 2 3 2 1 1 6 4 4 2 2
Dorothy 1 1 2 2 1 1 1 2 2 2 2 2 2 2 0 1 1 1
Verne 2 2 2 4 1 1 2 2 2 3 4 3 3 3 1 1 3 3
Flora 0 0 1 1 2 2 0 1 1 1 1 1 1 1 0 0 1 2
Ol ivia 0 0 1 1 2 2 0 1 1 1 1 1 1 1 0 0 1 2
Lau ra 4 6 1 2 0 0 7 6 2 3 2 1 1 6 3 4 2 2
Evelyn 3 6 2 2 1 1 6 8 3 3 2 2 2 7 3 L4 1 2
Pearl 2 2 2 2 1 1 2 3 3 2 2 2 2 3 0 2 1 2
Ruth 3 3 2 3 1 1 3 3 2 4 3 2 2 4 2 2 2 2
Sylvia 2 2 2 4 1 1 2 2 2 3 7 6 4 3 1 1 4 6
Katherine 1 1 2 3 1 1 1 2 2 2 6 6 4 2 0 1 3 5
t4lyrna 1 1 2 3 1 1 1 2 2 2 4 4 Lf 2 0 1 3 3
Theresa 4 6 2 3 1 1 6 7 3 4 3 2 2 8 4 4 2 3
Charlotte 2 4 0 1 0 0 3 3 0 2 1 0 0 4 4 2 1 1
Frances 3 4 1 1 o 0 4 4 2 2 1 1 1 4 2 4 1 1
Helen 2 2 1 3 1 1 2 1 1 2 4 3 3 2 1 1 5 4
Nora 2 2 1 3 2 2 2 2 2 2 6 5 3 3 1 1 4 8
Figure 2b. The P Matrix of Woman-to-Woman Relations, Derived from Matrix A by Equation 3. (Each off-diagonal
entry is the number of events at which two given women were jointly present. Each main-diagonal entry is the
total number of events attended by a single woman.)
are termed degenerate paths). Similarly for the
group-to-group ties: the number of n-paths between
every two groups is contained in the
matrix Gn.
Suppose we know the G matrix but do not
know the P matrix (for example, suppose we
are given information on director-interlocks
between corporations but we have no knowledge
of director-to-director ties). Suppose
further that we have person-to-group information
for only two (or several, say p* << p) of
the p persons. We can find the 1-paths among
these (two or several) persons by A (AT) where
A has p* rows and one column for each group
in G. But it appears that we cannot find paths
of length two or more among our p* people
because we don't know who the intermediate
persons are, or how these intermediaries are
connected to others. In this case we are aided
by
Lemma 1. PI = A(Gn-')AT ; Gn = AT(pn-')A
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Duality of Persons & Groups / 187
The proof follows from associativity and substitution
of Equations 3 and 4; e.g.,
pn = (AAT)n = A(ATA)n-'AT = A(Gn-1)AT.
In this manner, we can determine the number
of paths of any length among our p* people
by examining the dual paths in G. What of
the number of groups that a person can reach
(and conversely the number of persons that a
group can reach)?
Lemma 2. PnA = (GnAT)T
Proof: PnA = (AAT)nA -AGn = ((AGn)T)T
= ((Gn)T(A)T)T = (GnAT)T.
The assertion of Lemma 2 is that if we play
out all chains of person-to-person ties as far as
we like and then observe the groups that the
last persons reach, we come out with the same
endpoints as if we had played out all group-togroup
chains to the same length and then
looked at persons reached by the last groups.
The extension of lengths of paths in any
graph has a natural limit; there exists some
minimal m such that each node reaches all
other nodes it will ever reach by paths of
length m (at most): that is, converting the
values of ties to their binary form and conceiving
matrix multiplication as Boolean (Harary
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 4 0 0 2 2 3 0 1 2 0 1 1 0 1
2 0 8 3 6 0 3 6 6 0 3 0 7 4 0
3 0 3 3 2 0 2 3 3 0 2 0 3 2 0
4 2 6 2 10 4 5 4 5 3 2 2 8 3 2
5 2 0 0 4 6 5 0 1 5 0 3 5 0 3
6 3 3 2 5 5 12 2 4 4 1 3 9 2 3
7 0 6 3 4 0 2 6 5 0 3 0 5 4 0
8 1 6 3 5 1 4 5 8 1 3 1 7 3 1
9 2 0 0 3 5 4 0 1 5 0 3 4 0 3
10 0 3 2 2 0 1 3 3 0 3 0 3 2 0
11 1 0 0 2 3 3 0 1 3 0 3 2 0 3
12 1 7 3 8 5 9 5 7 4 3 2 14 3 2
13 0 4 2 3 0 2 4 3 0 2 0 3 4 0
14 1 0 0 2 3 3 0 1 3 0 3 2 0 3
Figure 3a. The G Matrix of Event-to-Event Relations, Derived from Matrix A by Equation 4. (Each off-diagonal
entry is the number of women who participated in both of two given events. Each main-diagonal entry is the
total number of women who attended a given event. Observe that only columns 4, 6, 8, and 12 have no zero
entry.)
El Br Do Ve Fl 01 La Ev Pe Ru Sy Ka My Th Ch Fr He No
Eleanor 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0
Brenda 1 4 0 0 0 0 3 4 0 1 0 0 0 3 3 2 0 0
Dorothy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Verne 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1
Flora 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1
Olivia 0 0 0 0 1 1 0 0 0 0 0 o 0 0 0 0 1 1
Laura 1 3 0 0 0 0 4 4 0 1 0 0 0 3 2 2 0 0
Evelyn 1 4 Q Q 0 0 4 5 0 1 0 0 0 4 3 2 0 0
Pearl 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ruth 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0
Sylvia 0 0 0 1 0 0 0 0 0 0 4 4 2 0 0 0 2 4
Katherine 0 0 0 1 0 0 0 0 0 0 4 4 2 0 0 0 2 4
Myrna 0 0 0 1 0 0 0 0 0 0 2 2 2 0 0 0 2 2
Theresa 1 3 0 0 0 0 3 4 0 1 0 0 0 4 3 2 0 0
Charlotte 1 3 0 0 0 0 2 3 0 1 0 0 0 3 3 2 0 0
Frances 1 2 0 0 0 0 2 2 0 1 0 0 0 2 2 2 0 0
Helen 0 0 0 1 1 1 0 0 0 0 2 2 2 0 0 0 3 3
Nora 0 0 0 1 1 1 0 0 0 0 4 4 2 0 0 0 3 5
Figure 3b. The New Matrix P2 of Woman-to-Woman Relations, Formed by Eliminating Columns 4, 6, 8, and 12
of Matrix A and Then by Applying Equation 3. For the Graph of P2, See Figure 3.
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188 / SOCIAL FORCES / vol. 53:2, dec. 1974
Eleanor Frances
Ruth Charlotte
Evelyn Bed
Laura Theresa
Verne
Katherine v;
Sylvia
Figure 4. The Graph of Matrix P2. (For simplicity, the ties are shown in their binary form.)
et al., 1965), there is some minimal m for
which the matrix pm+1 is contained in the
union of the first m powers of P, and some
minimal n for which GI'+1 is contained in the
union of the first n powers of G. The matrices
P and G are then said to have reached transitive
closure.
Theorem. If P reaches transitive closure at the
mth power and G reaches transitive closure at
the nth power, then the absolute difjerence of
m and n is at most 1.
Here is a sketch of the proof. It follows from
Lemmas 1 and 2 that the matrix which is the
union of the first k powers of PiA (i = 1, . . .,
k) specifies (for minimal k) all groups ever
reached by each person if and only if the union
of the first k powers of G iAT specifies all per-
k
sons ever reached by each group (P1+1A C u
k i=1
PiA if and only if Gk+1AT C U GiAT). Since
the nodes on a path from a person to a group
may be conceived as an alternating sequence
of persons and groups, all persons reach all
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Duality of Persons & Groups / 189
groups they will ever reach (by paths not exceeding
length k, at most) only if they have
just reached all persons they will ever reach
(P reaches transitive closure at the kth power)
or if they are about to do so at the next remove
(P reaches transitive closure at the k+1st
power). And similarly for G (G reaches transitive
closure only at the kth or the k+ 1 st
power).
PRIMARY AFFILIATIONS AND ASYMMETRIC TIES
We have, until now, imposed symmetry on a
network of membership ties; indeed, most
writers (e.g., Bonacich, 1972; Perrucci and
Pilisuk, 1970) conceive such ties as symmetric
only. There are, however, cases (such as corporate
interlocks or coalitions among parties or
factions) in which it is more interesting to conceive
of an asymmetric tie from one person or
group to another. This creation of asymmetric
orientations out of the symmetry of group
membership was formulated by Simmel (1955:
138, 155) in terms of primary and secondary
affiliations.
[O]ne group appears as the original focus of an
individual's affiliation, from which he then turns
toward affiliation with other, quite different groups
on the basis of his special qualities, which distinguish
him from other members of his primary
group. His bond with his primary group may well
continue to exist ...
An infinite range of individualizing combinations
is made possible by the fact that the individual
belongs to a multiplicity of groups. . . . The instinctive
needs of man prompt him to act in these
mutually conflicting ways: he feels and acts with
others but also against others.
The generalization of this asymmetry occurs
in Simmel's discussion (1955:62) of competition.
"Modern competition is described as the
fight of all against all, but at the same time it
is the fight of all for all"-and thus results not
in the chaos of Hobbes (which necessitates external
control) but in an intrinsically ordered
interweaving of relations based on "the possibilities
of gaining favor and connection."
Here is a method for building asymmetry
into the basic approach of this paper. Begin
with p people, g groups, a p X g matrix F
whose (i, j)th entry is "1" if person i has group
j as his primary affiliation and is "0" otherwise,
and a p X g matrix A (as above) showing all
affiliations of each person. Partition the memberships
in A among primary and secondary
(i.e., all other) affiliations. Define the p X g
matrix S of secondary affiliations by S
A n -F.
Let us say that two people mutually influence
each other if they share a common primary
affiliation. Our substantive conceptualization of
a particular problem (for example, influence
among directors of corporations) might suggest
specifying that an asymmetric tie exists from
person i to person j ("i is influenced by j") if
a group which is is primary affiliation is a secondary
affiliation for j (the assumption here
being that directors of higher-status corporations
are more sought after to lend their prestige
to the boards of other corporations).9
Following from this conception is a matrix
P' of asymmetric ties among persons:
PI= FFT U FST = F(FT U ST) = FAT
By reasoning analogous to that of Equations
3 and 4, we find the dual matrix G' of asymmetric
ties among groups: G' = ATF. Moreover,
the above reasoning on duality and transitivity
is easily extended to the asymmetric case;
as the reader may verify, for example,
(pI)n = (FAT)n = F(ATF)n-'AT = F(G')n'-lAT
is the analogue to Lemma 1 for the asymmetric
case.
9 Mace (1971:90) quotes this observation of a
company official: "You want to communicate to
the various publics that if any company is good
enough to attract the president of a large New
York bank as a director, it just has to be a great
company."
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Testing Theoretical Hypotheses: A PRE Statistic
KENNETH J. BERRY, Colorado State University
THOMAS W. MARTIN, Colorado State University
KEITH F. OLSON, Colorado State University
ABSTRACT
Departures from statistical independence are conjoined with an assessment of predictive accuracy in a
coefficient of association for nominal-level 2 x 2 contingency tables which is both interpretable as a PRE
measure and consistent with research hypotheses of manifold forms. Measurement assumptions and
operating characteristics of the measure are delineated; definitional and computational formulae are
derived from classical probability theory; comparisons with other relevant statistics are made; and the
test of significance is shown to be the traditional chi-square test.
Social scientists have long recognized that the
assessment of association at the nominal level
presents an especially difficult problem in both
experimental and survey research. In order to
arrive at a useful and meaningful measure of
association for nominal-level variables, two
considerations must be kept in mind: (a) the
form of the empirical test must be identical
with the form of the research hypothesis (see
Costner, 1965; Duggan and Dean, 1968; Francis,
1961; Kang, 1972, 1973; Leik and Gove,
1969) and (b) the measure of association
should be "'operationally interpretable' in
terms of the proportional reduction in error of
estimation made possible by the relationship"
(Costner, 1965:342). The purpose of this
paper is to introduce K, a measure of association
for dichotomized qualitative variables
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