Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 37
SESSION I
Social Network Theo PerspecOves
OCR for page 37
OCR for page 37
 ~
Photograph by Ben Shahn' Batches, MS, October, 1933
Library of Congress, Prints & Photographs Division, FSAOWI Collection, [reproduction number,
LCUSF33006093M4]
Finding Social Groups: A MetaAnalysis of the Southern
Women Natal
Linton C. Freeman
University of California, InJirze
I. Introcluction
For more than 100 years, sociologists have been concerned with relatively small,
cohesive social groups (Tonnies. ~ ~ 887] 1940; Durkheim ~ ~ 893] ~ 933; Spencer ~ 89597;
Cooley, 19091. The groups that conceIn sociologists are not simply categories—like
redheads or people more than six feet tall. Instead they are social collectivities
characterized by interaction and interpersonal ties. Concern with groups of this sort has
been and remains at the very core of the field.
These early writers made no attempt to specify exactly what they meant when they
referred to groups. But in the 1930s, investigators like Roethlisberger and Dickson
(1939) and Davis, Gardner and Gardner (1941) began to collect systematic data on
' The author owes a considerable debt to Moms H. Sunshine who read an earlier draft and made extensive
suggestions al] of which improved this manuscript.
DYNAMIC SOCIAL NETWO=MODEL~G ED ISIS
39
OCR for page 37
interaction and interpersonal ties. Their aim was to use the data both to assign
individuals to groups and to determine the position of each individual—as a core or
penpheral group member. But, to assign individuals to groups and positions, they needed
to specify the sociological notions of group and position in exact terms.
Over the years a great many attempts have been made to specify these notions. In
the present paper ~ will review the results of 21 of these attempts. All 21 tried to specify
the group structure in a single data set. And 11 of the 21 also went on once ~ttP.mnt~.~1 to
specify core and peripheral positions.
~ ^r—~~
My approach to this review is a kind of metaanalYsis. Schmid. Koch and
~ ~ .
LaVange (1991) define metaanalysis as ". . . a statistical analysis of the data from some
collection of studies in order to synthesize the results." And that is precisely my aim
here. A typical metaanalysis draws on several data sets from a number of independent
studies and brings them together in order to generalize their collective implications. Here
am also trying to discover the collective implications of a number of studies. But
instead of looking at the results produced by several data sets, ~ will be looking at the
results produced by several different analytic methods.
In this metaanalysis I will compare the groups and the positions that have been
specified by investigators who examined data collected by Davis, Gardner and Gardner
(1941) EDGG] in their study of southern women. My comparison draws on a number of
techniques, including consensus analysis (Batchelder and Romney, 1986, 1988, 1989),
canonical analysis of asymmetry (Gower, 1977) and dynamic pairedcomparison scaling
(Batchelder and Bershad, 1979; Batchelder, Bershad and Simpson, 19921. ~ will address
two questions: (~) do the several specifications produce results that converge in a way
that reveals anything about the structural form of the data? And, (2) can we learn
anything about the strengths and weaknesses of the various methods for specifying groups
and positions?
2. Southern Women Data Set
In the 1930s, five ethnographers, Allison Davis, Elizabeth Stubbs Davis, BurIeigh
B. Gardner, Mary R. Gardner and J. G. St. CIair Drake. collected data on stratification in
Natchez, Mississippi (Warner, 1988, p. 931. They produced the book cited above EDGG]
that reported a comparative study of social class in black and in white society. One
element of this work involved examining the correspondence between people's social
class levels and their patterns of informal interaction. DOG was concerned with the issue
of how much the informal contacts made by individuals were established solely (or
primarily) with others at approximately their own class levels. To address this question
the authors collected data on social events and examined people's patterns of informal
contacts.
In particular, they collected systematic data on the social activities of 18 women
whom they observed over a ninemonth penod. Dunn;, that periods various subsets of
40
D YN~MIC SOCIAL NETWORK MODELING ED CAL YSIS
OCR for page 37
these women had met Art a series of 14 informal social events. The participation of
women in events was uncovered using "interviews, the records of participant observers,
guest lists, and the newspapers" (DOG, p. 1491. Homans (1950, p. 82), who presumably
had been in touch with the research team, reported that the data reflect joint activities
like, "a day's work behind the counter of a store, a meeting of a women's club, a church
supper, a card party, a supper party, a meeting, of the ParentTeacher Association, etc."
This data set has several interesting properties. It is small and manageable. It
embodies a relatively simple structural pattern, one in which, according to DOG, the
women seemed to organize themselves into two more or less distinct groups. Moreover,
they reported that the positions~ore and penpheral~f the members of these groups
could also be determined in teens of the ways in which different women had been
involved in group activities.
At the same time, the DOG data set is complicated enough that some of the
details of its patterning are less than obvious. As Homans (1950, p. 84) put it, "The
patient is frayed at the edges." And, finally, this data set comes to us in a twomode
woman by event fonn. Thus, it provides an opportunity to explore methods designed
for direct application to twomode data. But at the same time, it can easily be
transformed into two onemode matrices (woman by woman or event by event) that can
be examined using tools for onemode analysis.
Because of these properties, this DGG data set has become something of a
touchstone for comparing analytic methods in social network analysis. Davis, Gardner
and Gardner presented an intuitive interpretation of the data, based in part on their
ethnographic experience in the community. Then the DGG data set was picked up by
Homans (1950) who provided an alternative intuitive interpretation. In 1972, Phillips and
Conviser used an analytic tool, based on information theory, that provided a systematic
way to reexamine the DGG data. Since then, this data set has been analyzed again and
again. It reappears whenever any network analyst wants to explore the utility of some
new tool for analyzing data.
3. The Data Source
Figure I, showing which women attended each event, is reproduced from DGG (p.
148~. DGG examined the participation patterns of these women along with additional
infonnation generated by interviews. As ~ discussed in Section ~ above, they had two
distinct goals in their analysis: (~) they wanted to divide women up into groups within on
the basis of their coattendance at events, and (2) they wanted to determine a position in
the core or periphery for each woman. As they put it (p. 1501:
Where it is evident that a group of people participate
together in these informal activities consistently, it is obvious
DYNAMIC SOCIAL NETWo=MOD~L~G ED ^4YSIS
41
OCR for page 37
42
that a cliques had been isolated. Interviewing can then be
used to clarify the relationship. Those individuals who
participate together most often and at the most intimate
affairs are called core members; those who participate with
core members upon some occasions but never as a group
by themselves alone are called primary members; while
individuals on the fringes, who participate only infrequently,
constitute the secondary members of a clique.
Not ~ P~aU"
OCR for page 37
We are faced with a dilemma then. Are we to believe the data presented in Figure
~ or those presented in Figure 2? Homans (1950) was the first to use these data in a
secondary analysis. He reported additional details that suggest that he was probably in
touch with the original research team. And in his report he used the data as they are
displayed in Figure I.
Moreover, there is additional ancillary evidence for the correctness of the data as
presented in Figure I. It tutus out that there is a contradiction in the presentation of
Figure 2 that makes it difficult to accept the data presented there as correct. Compare, for
example, the participation patterns displayed by the two women designated with red
arrows in Figure 2: woman ~ ~ and woman ~ 6. According to Figure 2, these two women
displayed identical patterns of participation. Yet, in that figure, woman ~ ~ was classified
as a primal member of her "clique'. while woman 16 was called secondary. This
contradiction implies that the correct data are those shown in Figure i.
T~ Or
s~s=P
cow:
Ape.... i..
polyp ~ 
~
~ry. . .
Go ~ ~ ~ ~ ~
Body.
al
13
4
l IS
tS
L7
8
By
10
11
12
13
t~
IS
t.
(~7
1
~~ ~ PI
1 2 3 4 5 ~ 7 8 4 10 11 ~ 13 ~
C C ~ C  ~ C
~ ~ C  ~ ~ C C 
 C C ~ ~ ~ C C C
C  ~ C C C ~ C 
p, p ~ _ p _ _
p _ p p _ p _
~ P ~ p _
 S  S S
S  S S &
S S ~   ~
P P p _ ~ At.
P P P  P P P
C ~ C ~  ~ C
C  ~ ~ C ~ C ~
 ~ ~ C lO.CI
S S
$
S

_
[S  S! q~
_
_
. _
Figure 2. Participation of the Southern Women in Events
Most analysts have apparently reached this conclusion. Most have used the data
as shown in Figure I. A few, however, have analyzed the data as presented in Figure 9.
and this produces a problem for any attempt to compare the results of one analysis with
those of another. When the two analyses are based on the use of different data sets
comparisons are, of course, not possible.
DYNAMIC SOCIAL NETWORK MODEM AND ISIS
43
OCR for page 37
~ have assumed that the "correct" data are those shown in Figure I. For the
relatively small number of results that have been produced by analyses of the data of
Figure 2, I: have asked the original analysts to redo their analyses using the Figure ~ data
or ~ have redone them myself.
4. Finding Groups and Positions in the DOG Data
4.l Davis, Gardner and Gardner's IntuitionBased Groups (DGG41)
In their own analysis Davis Gardner and Gardner did not use any systematic
analytic procedures. They relied entirely on their general ethnographic knowledge of the
community and their intuitive grasp of the patterning in Table ~ to make sense of the data.
As Davis and Warner (1939) descnbed it, they drew on ". . . records of overt behavior
and verbalizations, which cover more than five thousand pages, statistical data on both
rural and urban societies, as well as newspaper records of social gatherings . . ."
DOG drew on all this material and used it both to assign the women to groups and
to determine individuals' positions within groups. They indicated that the eighteen
women were divided into two overlapping groups. They assigned women 1 through 9 to
one group and women 9 through IS to another. They assigned three levels in terms of
core~penphery participation in these groups. They defined women ~ through 4 and 13
through 15 as core members of their respective groups. Women 5 through 7 and ~ ~ and
12 they called primary. Women ~ and 9 on one hand and 9, 10, ~6, 17 and ~~ on the
other were secondary. Note that woman 9 was specified as a secondary member of both
groups because, they said, "in interviews" she was "claimed by both" (DOG. p. ~ 5 ~ ).
4.2 Homans' IntuitionBased Analysis (HOM50)
Like Davis, Gardner and Gardner before him, Homans ~950) interpreted these
data from an intuitive perspective. Unlike those earlier investigators, however, Homans
did not have years of ethnographic experience in Natchez to draw upon. His intuitions,
therefore, had to be generated solely by inspecting the DOG data and, presumably, by
conversations with the ethnographers.
Homans implied that he had reanalyzed the data using, a procedure introduced by
Forsyth and Katz (1946) whom he cited. Forsyth and Katz had suggested permuting the
rows and columns of a data matrix so as to display its group structure as clusters around
the principal diagonal of the matrix (upper left to lower right). Their procedure required
that both the rows and columns be rearranged until as far as possible more or less
solid blocks of nonblank cells are gathered together. Such blocks of cells, they
suggested, represent "wellknit~' groups.
l:t is doubtful that Homans actually used the Forsyth and Katz procedure. DOG
had already arranged the matrix in such a way that it displayed group structure.
Seemingly they had anticipated the Forsyth and Katz approach by six years. Homans.
44
DYNAMIC SOCIAL NETWORK MODEM ED ISIS
OCR for page 37
then, did not rearrange the data matrix at all; he simply copied the arrangement of Figure
l, exactly as it was reported by DOG.
In any case, after inspecting the arrangement shown in Figure I, Homans grouped
~ 6 of the women and distinguished two levels of core and penpheral positions. In his
report Homans (p. 84) report wrote:
we generalize these observations by saying that the
women were divided into two groups. The pattern is frayed
at the edges, but there is a pattern. The first seven women,
Evelyn through Eleanor, were clearly members of one group;
numbers ~ ~ through ~ 5, Myra through Helen, were just as
clearly members of another. Some women participated
about equally with both groups but not very much with either;
Pear! twoman S] is an example. And some participated,
though not very often, only with the second group. Pearl,
Olivia, Flora [women 8, ~ 7 and ~ 8] and their like are
marginal group members.
This statement is somewhat ambiguous. ~ does assign women ~ through ~ to one
group and ~ ~ through 15, along with 8, 17 and IS to the other. Because woman ~ (PearI)
is assigned to both, the two groups overlap. In addition Homans characterized women 8
17 and IS to "marginal positions" but it is difficult to know whet he intended by the
phrase '`and their like.'? His statement, moreover, makes no mention at all of woman 9
(Ruth) or womanI6 (Dorothy). They were simply not assigned to either group or to any
position.
4.3 Phillips and Conviser's Analysis Based on Information Theory (P&C72)
Phillips and Conviser (1972) were the first to use a systematic procedure in the
attempt to uncover the group structure in the DGG data.3 They reasoned that a collection
of individuals is a group to He extent that all of the members of the collection attend the
same social events. So, to examine the DGG data. they needed an index of the variability
of attendance. They chose the standard information theoretic measure of entropy, H
(Shannon, 1964~. H provides an index of the viability of a binary (yes/no) variable. In
this case, it was applied to all the women (and all the events) in the DGG data. Thus, H
was used to provide an index of the degree to which different collections of women
attended different sets of events (and different sets of events attracted different collections
of women).
Phillips and Conviser set about to find groups by comparing various ways of
partitioning the women into subsets. They argued that any given partitioning produced
social groups if the entropy H summed for all of the subsets was less than the entropy for
3 It should be noted that Phillips and Conviser attributed the southern women data to Homans. Nowhere in
their paper did they acknowledge DGG.
DYNAMIC SOCIAL NETWORK MODEL~G ED ISIS
45
OCR for page 37
the total set of women. In such an event, the women assigned to each subset would be
relatively homogeneous with respect to which events they attended.
They evaluated the utility of any proposed partitioning by cadculating the
information theoretic measure a. a is an index of the degree to which the overall entropy
of the total collectivity H is reduced by calculating the value Hi within each of the i
designated subsets and summing the results (Garner and McGill, ~ 956~. a is large when
all the women who are classed into each subset are similar with respect to their
attendance patterns. It is maximal only when the withinsubset patterns are all identical.
To employ this approach then, it is necessary to partition the women in all
possible ways. calculate a for each partitioning, and see which partitioning produces the
largest value.
There is, however, a major difficulty with this approach. The number of possible
partitionings grows at an exponential rate with an increase in the number of individuals
examined. The number grows so rapidly that the partitions cannot all be examined, even
with as few as ~ ~ women to be considered.
So Phillips and Conviser worked out a way to simplify the problem. Like
Homans, they cited again the procedure suggested by Forsyth and Katz (19461. That
procedure rearranges the rows and columns in the data matrix in such a way that women
who attended the same events and events that were attended by the same women are
grouped together. When this is done, only those women who are close together in the
matrix are eligible to be in the same group. That being the cases only those relatively few
partitionings that include or exclude individuals in successive positions in the data matrix
need to be considered.
As ~ indicated above, the DOG data had already been arranged in the desired
order by the original authors. So, like Homans, Phillips and Conviser did not actually
have to rearrange them. They could proceed directly to partitioning. They began by
partitioning the women into two classes (1 versus 2 through IS, ~ and 2 versus 3 through
~ 8, ~ through 3 versus 4 through ~ 8, etc.~. They reported that, of all these twogroup
partitions, the split of ~ through ~ ~ versus ~ 2 through ~ ~ yielded the largest value of a.
In checking their results, however, T discovered that their result was based on an
error in calculation. When ~ recalculated ~ discovered that the maximum value of a is
actually achieved with the ~ through 9 versus JO through ~~ split. This approach simply
partitions; it cannot distinguish core or peripheral positions, nor can it permit
overlapping.
4.4 Breiger's Matrix Algebraic Analysis (BGR74)
Breiger (1974) used matrix algebra to show that the original twomode, woman by
event DGG data matrix could be used to generate a pair of matrices that are,
46
DYNAMIC SOCIAL NETWORK MODELING kD ISIS
OCR for page 37
mathematically, dual. First, multiplying the original matrix by its transpose produces a
woman by woman matrix in which each cell indicates the number of events coattended
by both the row and the column women. Second, multiplying the transpose by the
original matrix yields an eventbyevent matrix where each cell is the number of women
who attended both the row event and the column event. The womanbywoman matrix is
shown in Figure 3.4 And its dual, the eventbyevent matrix. is shown in Figure 4.
1 1 1 1 ~ 1 1 ~ 1
1 2 3 4 ~ ~ 7 8 9 0 ~ 2 3 4 ~ 6 7 ~
E L T ~ ~ F E P R V M K S N H D O F
1 EVELYN 8 ~ 7 ~ 3 4 3 3 3 2 2 2 2 2 1 2 1 1
2 LAURA 6 7 6 6 3 4 4 2 3 2 1 1 2 2 2 ~ O O
3 THERESA' 7 6 8 6 4 4 4 3 4 3 2 2 3 3 2 2 1 1
4 BRENDA 6 6 ~ 7 ~ 4 4 2 3 2 1 1 2 2 2 1 0
C~LOi It 3 3 4 4 4 2 2 0 2 1 0 0 1 1 1 ~ ~ 0
6 FRANCES 4 4 4 4 2 4 3 2 2 1 1 1 1 1 1 1 0 0
7 ELEANOR 3 4 4 4 2 3 4 2 3 2 1 1 2 2 2 1 0 0
8 PEARL 3 2 3 2 0 2 2 3 2 2 2 2 2 2 1 2 1 1
9 RUTH 3 3 4 3 2 2 3 2 4 3 2 2 3 2 2 2 1 1
10 VERNE 2 2 3 2 1 1 2 2 3 4 3 3 4 3 3 2 1 1
11 MYRA 2 1 2 1 0 1 1 2 2 3 4 4 4 3 3 2 1 1
12 KATHERINE 2 1 2 1 0 1 1 2 2 3 ~ 6 6 ~ 3 2 1 1
13 SYLVIA 2 2 3 2 1 1 2 2 3 4 4 6 7 6 4 2 1 1
14 NORA 2 2 3 2 1 1 2 2 2 3 3 ~ 6 ~ 4 1 2 2
15 HELEN 1 2 2 2 1 1 2 1 2 3 3 3 4 4 ~ 1 1 1
16 DOROTHY 2 1 2 1 0 1 ~ 2 2 2 2 2 2 1 1 2 1 1
17 OLIVIA 1 0 1 0 0 ~ ~ 1 1 1 ~ 1 1 2 1 1 2 2
18 FLORA 1 ~ 1 ~ O ~ 0 1 1 1 1 1 1 2 ~ 1 2 2
_
Figure 3. The OneMode, Woman by Woman, Matrix Produced by Matrix
Multiplication
~ 2 3 4 5 S 7 ~ 9 10 11 12 13 14
E1 E2 E3 E4 E5 ES E7 E8 E9 E10 E11 E12 E13 E14
1 E1 3 2 3 2 3 3 2 3 1 ~ ~ ~ 0 0
2 E2 ~ 3 3 2 3 3 2 3 2 ~ 0 0 0 0
3 E3 3 3 6 4 6 ~ 4 ~ 2 ~ 0 ~ 0 0
4 E4 2 2 ~ 4 4 3 3 3 2 0 0 Q O O
5 ES 3 3 6 4 8 6 ~ 7 3 0 a G ~ 0
E6 3 3 ~ 3 6 ~ ~ 7 4 1 1 1 1 1
7 E7 2 2 4 3 6 ~ 10 8 ~ 3 2 4 2 2
8 E8 3 3 ~ 3 7 7 8 14 9 4 1 ~ 2 2
~ E9 1 2 2 ~ 3 4 ~ 9 12 4 3 ~ 3 3
10 E10 ~ ~ 0 0 C 1 3 4 4 ~ 2 ~ 3 3
11 E11 ~ 0 0 ~ 0 1 2 1 3 2 4 2 1 1
12 E12 0 0 0 ~ 0 1 4 ~ 5 ~ 2 6 3 3
13 E13 ~ 0 0 0 0 1 2 2 3 3 1 3 3 3
14 E 14 0 ~ 0 ~ 0 1 2 2 3 3 1 3 3 3
Figure 4. The OneMode, Event by Event, Matrix Produced by Matrix
Multiplication
Breiger renamed DOG's "Myra." He listed her as "Myrnaq in his tables and diagrams. Breiger s
designation has been picked up in a number of later works includin;, the data set released as part of the
UCINET progran1 (Borgatti, Everett and Freeman. 1992).
DYNAMIC SOCIAL NETWORK HOD~:~ING ED ISIS
47
OCR for page 37
woman 6, but the pairedcomparison analysis places woman 5 closer to the core than
woman 6.
To evaluate the effectiveness of each of the ~ ~ orders provided by the analytic
procedures, ~ used gamma. Gamma provides an orderbased measure of agreement. T
compared each of the orders suggested by the procedures with the idealized orders
provided by canonical analysis and p~redcompar~son analysis.
Because the two idealized orders are so similar. their gammas with the orders
~ ~ ~ . ~ ~ . . .
p~uuuc<;u by one analytic procedures were, of course, nearly Identical. The results for
both the canonical and the pairedcomp~son standards are shown in Figure 14.
For both modelbased standards. Homans' order produced the highest gamma.
One must be careful, however, in looking at these values because different gamma
calculations may be built on vastly different numbers of observations. In this case, the
value of I.0 associated with Homans' work was based on only 17 comparisons in the
order of the women. In contrast the values associated with the two analyses by Newman
were based on 58 and 59 comparisons respectively. Because Homans' report contained
relatively less information about who was in the core and who was peripheral it
generated fewer predictions about positions. The predictions it did make happened to
agree with the positional information produced by both cntena. But the Newman
analyses both produced large numbers of predictions, and they were still mostly in
agreement with those produced by the cntena.
Beyond Newman. the orders produced hv n~vi.c GY~rrlner ~nr1 f'~~rAnPr
~ —J ~ ~7 _ _A ~ A, ~ ,~ ~ ~ of_ A A_
themselves, by Doreian, by Freeman and White in their first analysis, and by Skvoretz
and Faust are consistently in agreement with the cntena. Their gammas are all above .9
and they are all based on at least 43 comparisons. At the opposite extreme, both
Bonacich analyses and the Borgatti and Everett bicTique analysis do not agree very well
with the cntena.
ID Code Analysis (;amma with Number of 4:;amma wilt, Number of
Cannonical Comps PairedCompanson Comps
1 DGG41 cams, Gardner and Gardner, Eggnog. 0.962 52 0.923 52
2 HOMED Homans, Intuidon 1.000 17 1.000 17
6 BCH78 Bonacich, Boolean Algebra ~.111 9 0.~00 10
7 DOR79 Doreian, PJgebra~c Topology 0.329 28 0.929 28
8 BCH91 Bonacich, Correspondence Analysis 0.313 67 0.324 68
12 FWl;93 Freeman and White, Full Lath ce 0.953 43 0.~53 43
13 FVV293 Freeman and White, Su~Latice 0.867 45 O.870 46
14 BEl97 Borgati and Everett, BiClique O.385 39 0.350 40
17 S&F99 Skvore~and Faust,p~ 0.932 59 0.933 60
18 ROB00 Roberts, Correspondence Analysis 0.844 64 0.846 65
20 N~1 Newman, Weighted C~P4tendance 0.~6 58 O.932 59
Figure 14. Gammas Showing the Degree to which 11 Analvses A~ree`1 with the Twn
Standards In Assigning individuals to Core and Peripheral Positions
DY7 JAMIC SOCIAL NETWORK MODELING AND ANALYSIS
67
OCR for page 37
So again, we have been able to uncover something close to a consensus this time
with respect to core and peripheral positions. And we have again been able to find out
something about the extent to which each of the analytic procedures approaches that
consensus.
6. Summary and Discussion
6.l Assignment to Groups
Each of the ~ ~ analyses reported here assigned the DGG women to groups.
Consensus analysis determined the agreement among the assignments. It turned out that
there was a strong core of agreement among most of the analytic devices. The agreement
was substantial enough to allow the model to be used specify a partition of the women
into groups~ne that captured the consensus of all the analyses. At the same time, the
consensus analysis was also able to provide ratings of the "competence' of each of the
analytic procedures.
The consensual assignment of women to groups and the "competence" ratings
of the analytic methods were reported above. The "competence" scores were reflected in
the first axis of an singular value decomposition of the matches generated by the methods
in assigning pairs of women to the same or to different groups. In that earlier
examination ~ reported only the first axis. But here. it is instructive to examine the
second and third axes. They are shown in Figure 15.
Fues3
BCH78 ~ ~ BeR74
~ DOR78
BE197 ~~
r
FRE8Q
Baled
nOM5O S&F— Romo BE3er
Bat ~ ·P""
SINEWS FR193
~1
FWI93
Figure 15. Axes 2 and 3 Produced by the Singular Value Decomposition of the
Matches in the Assignments of Women
68
DYNAMIC SOCIAL NETWORK HODE~G ED ISIS
OCR for page 37
The arrangement of points representing the analytic methods in Figure 15 tells a
good deal about both the partitioning of women to groups and the competence of the
methods. The consensus put women ~ through A in the first group and women 10
through ~ ~ in the second. In Figure 15, the basis for determining why this was the "best"
partition becomes apparent. That partition was specified exactly by six of the analytic
procedures: P&C72 (The corrected version of Phillips and Conviser's inflation
theoretic algorithms, BCH91 (Bonacich's correspondence analysis), FRI93 (Freeman~s
first genetic a:[gonthm solution), BE297 (Borgatti and Everett's taboo search), BE397
(Borgatti and Everett's genetic aIgorithm) and ROB00 (Roberts' correspondence analysis
of normalized data). These analyses are all placed at a single point at the upper right of
the figure.
Other analyses that produced results that were quite close to that ideal pattern
are clustered closely around that point. For example, BBA75 and NEW01 produced the
same pattern with only one exception. They both assi;,ned woman ~ to the second group.
~ ~ ~ ~ ~ ~ A: ~ ~ 1~ ~ C) _ _ ~ ~ ~ _ .1 1 — —
=——~
r ~ ~~ ~ asslg~lt;u corn women ~ ana a to me second group. DGG41 put woman 9 in both
groups. And FW193 put woman 16 in both. Finally, S&F99 deviated only by failing to
include woman 16 in either group. Thus, in addition to the six "perfect" partitionings, six
additional procedures came very close to the ideal and are clustered in the region
surrounding these "perfect" solutions. This clustering is the key. It shows a clear
consensus around the I9, 10~8 division. This consensus is really remarkable in view of
the immense differences among the analytic procedures used.
Figure 16 relabels all the points such that their departure from the "perfect"
partitioning is displayed. Note that the I9, 10IS partition is labeled "PERFECT."
Note also that departures from that ideal are generally placed farther from the PERFECT
point as the degree of their departure grows. They are, moreover, segregated in terms of
the kinds of departure they embody. All the points that fall on the left of the vertical axis
involve methods that failed to assign two or more of the women to groups. Overall, those
points are arranged in such a way that those falling further to the left are those that are
missing more women. Immediately to the right of the vertical, are the methods that
located women in the "wrong" group. And their height indicates the number of women
classified in "error." All the way to the right are the methods that assigned women to
both groups. And, to the degree that they assigned more women in that way, they are
farther to the right. Finally, it should be noted that there are two analyses that assigned
women to multiple groups on the left. But it is clear from their placement that this
analysis was more responsive to their inability to place women in groups than it was to
their dual assignments.
DYNAMIC SOCKS NETWO=MODEL~G ED TRYSTS
69
OCR for page 37
2 MISSING AND 3 MISSING AND 1 MISSING FRFE T
20VERLAP 1 OV~RW P C
6 MISSING 4 MISSING  \  / 1 WRONG SIDE /
O \ 0 2 MISSING
O ~ WRONG SIDE /
O O O
MISSING ~ l
I ~ OWR~P
2 WRONG SIDE
3 MISSING ~
6 WRONG SIDE
Figure 16. Axes 2 and 3 of the Matches Labeled by Structural Form
OveratI, then, it is clear that there was a consensus about assigning women to
groups. Six methods agreed, and most of the others departed relatively little from that
agreedupon pattern.
6.2 Positions in Groups
In assigning positions to individuals, ~ used two, quite different, scaling
techniques. One was based on a dominance model. For each pair of women A and B. the
mode! placed A closer to the core than B. if and only if more procedures placed A closer
to the core than B. The other was probabilitybased. It placed A closer to the core than B
with some probability based on the proportion of procedures that placed A closer to the
core than B.
Despite their differences, the results of these two methods turned out to be
almost identical. They were similar enough that either could be taken as providing
something very close to an optimum assignment of individuals to positions. The
effectiveness of each of the analytic procedures was evaluated by their monotone
correlations with these optima. The results were very similar; the correlation between the
gammas produced by the dominance model and those produced by the probability mode!
was .983. So, even with without consensus amon, the procedures, 1 was able to find the
a3reedupon order~ore to penphe~y and to evaluate the ability of each method to
uncover that order.
70
DYNAMIC SOCIAL NETWO~MODEL~G kD ISIS
OCR for page 37
6.3 A Final Word
As a whole T believe that this metaanalysis has been productive. As far as
assigning women to groups was concerned, the results were dramatic. When it came to
assigning women to positions, the results were less dramatic, but still fairly convincing.
We end, then, with four strong results: (~) We have a consensual partitioning of women
into groups. (2) We have a consensual assignment of women into core and peripheral
positions. (3) We have a rating of the methods in teens of their competence in assigning
women to groups. And (4) we have a rating of the methods in terms of their ordinal
correlations with the standard positional assignments.
~ would like to wind up with two additional comparisons of the analytic
procedures examined here. These final comparisons will be restricted to the published
analyses of the DGG data; they will not include the unpublished analyses involving my
use of Osbourn's VERI or the analysis Newman ran at my request.
The first comparison is based on time. Figure ~ 7 shows the average competence
ratings of procedures published at various points of time. The data in Figure 17 show an
interesting secular trend. There has been a slow but consistent trend toward increasing
competence through time. Thus, the overall tendency in published reports using the
DOG data is clearly in the direction of greater competence.
In addition. it is possible to make some generalizations about the adequacy of the
various kinds of analytic procedures that have been used to find groups and positions
using the DOG data. Six procedures (BGR74, BCH7S, DOR79, E&B93, FWI93 and
FW293) all took essentially algebraic approaches. Five (P&C72, FRI93, FR293, BEl97
and BE297) used venous aigonthms to search for an optimal partition. Three analyses
(BBA75, BCH91 and ROB00) employed various versions of singular value
decomposition. Two (DGG41 and HOM50) were based simply on the authors'
intuititions. And three developed unique approaches. One (FRE92) looked at a kind of
transitivity. A second (BEl97) dealt with overlapping bictiques. And the third (S&F99)
developed a statistical model.
D YIJAMIC SOCIAL NETWORK MODELING AND ANAL YSIS
71
OCR for page 37
1
0.8
O.6
0.4
0.2
to
. . .
I:. ' .: , :: , .: ,., .: , ,~
.. . .. . .. . . .
I:' '' : I: . ' ' ( ,,,,:~`,,~"i , I'd ,': 'a ,.,,:'~',^~,'~'.','i,' ~ v~ .
',: ''j,,''''f''"'.,.,,';,;,. ',':. .': ;' , ( '. '.:' ' :' ' ' ' ' ' ~1 ~ i
. ' . ~ In,, ~~ .d F) ~ ~ . ~ . ~ q)y i`, ,~ ,~ ~ .' ~ '.~ ~ . ,.,, <, _
: :~ i. ~ ;. : .: . :.'^.,':; ,~:'.:~ . ~ :<~;:~ :., ;:.i~~;^~ I .'. at ~~:~':<
.. " ,"':"':'"''' 'a , ant: :,'~., A ''_",','[ ' . '': :~:~:~
~ W ` ~ ~~ : ; ~,W,,,~ .,,{  . . .,,,,, Al . . ~ 1 St.,; ,;
'~ ~~] ~ ~ ~~ :~ of; a,' ~ ~ ~ ~~: ~
, . ~ f ~ 2~r ' ~ ' '.,''.~'.;~~2 A; I' ' W.>' ~ ,' ,; ~'~v~' ',
i~ ~~
40s and SOs 70s
9193 9700
Figure 17. Competences of the Procedures Over Time
All in all, then, we have used seven distinct classes of procedures in analyzing the
DOG data. Figure ~ ~ shows the relative success of each class in terms of its average
competence.
Procedure N Average Score
Statistical hJlodel 1 0.957
Eigen Structure 3 0.954
C: ptimal Partition ~ 0.941
Transitivity 1 0.926
Cliques 1 0.916
Algebraic Duality 6 0.914
Intuition 2 0.887
Figure IS. Average Competences of the Various Classes of Procedures
A number of features of Figure 17 are worth noting. First, the statistical mode! of
the DOG data developed by Skvoretz and Faust was the winner. It won despite the fact
that. unlike most of the other procedures, it was not explicitly designed to uncover
groups. Group structure emerged as a sort of biproduct of a broader structural analysis.
72
DYNAMIC SOCIAL N~:TWORKMODEL~G AD ISIS
OCR for page 37
The statistical model is not' however, the undisputed champion. It is followed so
closely by the three singular value decomposition analyses, that it has to share the crown
with them. And the five partitioning programs are right up there near the top.
There seems to be a step between all those procedures and the next three. Clearly
transitivity, bicliques and the algebrabased approaches did not do as well. And, finally,
the intuitive judgments fall at the bottom. In part that position is due to the vagaries of
Homans' report, but DGG themselves did very little better. This result is particularly
interesting given the fact that Davis, Gardner and Gardner's interpretation of their own
data is often taken as privileged. The assumption has been that because they had a huge
amount of ethnographic experience in the community, DGG had an edge they somehow
knew the ' true" group structure. But, particularly in the light of the present results, there
is no compelling reason to award DGG any special exalted status visavis their ability to
assign individuals to groups. Indeed, their very intimacy with these IS women might
have led to various kinds of biased judgments.
References
Batchelder, W. H. and N. J. Bershad
1979 The statistical analysis of a Thurstonian mode! for rating chess
players. Journal of Mathematical Psychology, 19:3960.
Batchelder, W. H., N. J. Bershad and R. S. Simpson
1992 Dynamic p~redcomparison scaling. Journal of Mathematical
Psychology, 36: ~ 85212.
Batchelder, W. H. and A. K. Romney
986 The statistical analysis of a general Condorcet mode! for dichotomous
choice situations. In B. Grofman and G. Owen (Eds.) hnformation
Pooling and Group Decision Malting. Greenwich, Connecticut: JAl
Press, Inc.
Batchelder, W. H. and A. K. Romney
1988 Test theory without an answer key. Psychometr~ka 53:7192.
Batchelder, W. H. and A. K. Romney
1989 New results in test theory without an answer key. In E. Roskam (Ed.)
Advances in Mathematical Psychology Vol. H. Heidelberg New
York: Springer VerIag.
Birkhoff, G.
1940 Lattice theory. New York: American Mathematical Society.
Bonacich, P.
1978 Using boolean algebra to analyze overlapping memberships.
DYNAMIC SON NETWO=MODEf~G ED TRYSTS
73
OCR for page 37
Sociological Methodology 101!15.
Bonacich, P.
1990 Simultaneous group and individual centralities. Social Networks
13: 155168.
Borgatti, S. P. and M. G. Everett
~ 997 Network analysis of 2mode data. Social Networks ~ 9:943269.
Borgatti, S. P., M. G. Everett and L. C. Freeman
1991 UCINET IV, Version I.0. Reference Manual. Columbia, SC:
Analytic Technologies.
Breiger, R. L.
lL974 The duality of persons and groups. Social Forces 53: 181190.
Breiger, R. L., S. A. Boorrnan and P. Arable
1975 An algorithm for clustering relational data, with applications to social
network analysis and comparison to multidimensional scaling.
Journal of Mathematical Psychology 12:328383.
Cooley, C. H.
1909 Social Organization. New York: C. Scr~bner's sons.
Davis, A., B. B. Gardner and M. R. Gardner
1941 Deep South. Chicago: The University of Chica go Press.
Davis, A. and W. L. Warner
1939 A comparative study of American caste. In Race Relations and the
Race Problem. (Ed.) E. T. Thompson. Durham: Duke University
Press.
Deming, W. E. and F. F. Stephan
940 On a least squares adjustment of a sampled frequency table when the
expected mar ,inal totals are known. Annals of Mathematical
Statistics ~ ~ :427~1 4.
Doreian~ P.
1979 On the delineation of small group structure. In H. C. Hudson, ed.
Classifying Social Data, San Francisco: JosseyBass.
Durkheim, E.
t ] 893] 1933 The division of labor in society. Translated by G. Simpson.
New York: Free Press.
74
DYNAMIC SOCIAL METWORKMODEL~G AND ANALYSIS
OCR for page 37
Everett, M. G. and S. P. Borgatti
1992 An Extension of regular colounng of graphs to digraphs, networks
and hypergraphs. Social Networks 15:237254.
Forsyth, E. and L. Katz
1946 A matrix approach to the analysis of sociometr~c data. Sociometry 9:
340347.
Freeman, L. C.
1993 On the sociological concept of "group": a empirical test of two
models. Amencan Journal of Sociology 98: ~ 52 ~ 66.
Freeman' L. C.
1994 Finding groups with a simple genetic algorithm. Journal of
Mathematical Sociology ~ 7:22724 ~ .
Freeman, L. C.
1997 Uncovenng organizational hierarchies. Computational and
Mathematical Organization Theory 3:518
Freeman, L. C. and V. Duquenne
1993 A note on regular colorings of two mode data..Social Networks
15:437442.
Freeman, L. C. and D. R. White
1994 Using Galois lattices to represent network data. In Sociological
Methodology 1993. P. Marsden, ed. Pp. 127146. Cambridge, MA:
Blackwell.
Garner, W. R. and W. J. McGill
1956 The relations between information and vivace analyses.
Psychometnka 21: 219228.
GIover, F.
1989 Tabu search. Part I. ORSA Journal on Computing I: 190206.
Goodman, L. A.
1995 A single general methoc! for the analysis of crosscIassified data:
Reconciliation and synthesis of some methods of Pearson, Yule, and
Fisher, and also some methods of correspondence analysis and
association analysis. Journal of the American Statistical Association
91: 408428.
Gower, J.C.
1976 The Analysis of asymmetry and orthoganaTity. In J. Bana, F. Brodeau
DYNAMIC SOCIAL N~TWO~MODEL~G ED ISIS
75
OCR for page 37
and G. Romier (Eds.), Recent Developments in Statistics, North
Holland, Amsterdam.
Granovetter, M.
1973 The strength of weak ties. American Journal of Sociology 81
1303.
: 1287
HolIand, J. H.
969 Outline for a logical theory of adaptive systems. Journal of the ACM,
9:2973 14.
Hogans, G. C.
1950 The Human Group. New York: Harcourt, Brace and Company.
Jameson, K. A., M. C. Appleby and L. C. Freeman
1998 Finding an appropriate order for a hierarchy based on probabilistic
dominance. Animal Behaviour 57:991998.
Johnson, S. C.
1967 Hierarchical clustering schemes. Psychometrika 32:241254.
Koch, G. G. and L. M. LaVange
1991 An overview of statistical issues and methods of metaanalysis
Journal of Biopharmaceutical Statistics 1: 103120
Luce, R. D., and A. Perry
1949 A method of matrix analysis of group structure,'' Psychometr~ka,
14:951 16.
Mosteller, F.
968 Association and estimation in contingency tables. Journal of the
American Statistical Association 63: I28.
Newman, M.
2001 The structure of scientific collaboration networks. Proceedin;,s of the
National Academy of Science 98:404409.
Osbourn, G. C.
1999 Clustering, pattern recognition and augmentation of data visualization
using VER! algorithm. Invited talk, American Statistical Society
Meeting.
Phillips, D. P. and R. H. Conviser
1972 Measuring the structure and boundary properties of groups: Some
uses of information theory. Sociometry 35:235254.
76
DYNAMIC SOCIAL NT:TWORKMODELI7JG AND ANALYSIS
OCR for page 37
Richards, W. and A. Seary
1996 Introduction to eigen analysis of networks. Paper presented at the
INS NA Sunbelt Meeting, San Die;,o.
http://www. sfu.ca/~richards/Pages/wdr97.htm
Roberts, J. M. Ir.
2000 Correspondence analysis of twomode networks. Social Networks
22:6572.
Roethlisberger, F. J. and W. I. Dickson
1939 Management and the Worker: An Account of a Research Program
Conducted by the Western Electric Company, Hawthorne Works,
Chicago. Cambridge, Mass.: Harvard University Press.
Sailer, L. D., and S. J. C. GauTin
1984 Proximity, sociality and observation: the definition of
social groups. American Anthropologist, 86, 9198.
Shanon, C. E.
1963 The mathematical theory of communication. In C. E. Shannon and
W. Weaver, The Mathematical Theory of Communication. Urbana:
University of Illinois Press.
Skvoretz, J. and K. Faust
1999 Logit models for affiliation networks. Sociological Methodology
29:253280.
Spencer. H.
~ 89597 The Principles of Sociology. 3d ea., New York: D. Appleton.
Thurstone, L. L.
~ 927 A law of comparative judgment. Psychological Review 34:273286.
T.. . _
onuses, I.
[~855] 1936 Fundamental Concepts of Sociology (Gemeinschaft und
geselischaft). Translated and supplemented by C. P. Loomis. New
York: American Book Company.
Warner, M.
1987 W. Lloyd Warner: Social Anthropologist. New York: Publishing
Center for Cultural Resources.
Wasserman, S. and P. Pattison
1997 Logit models and Touristic regressions for social networks: I. An
introduction to Markov graphs and p> . Psychometrika. 60:401426.
DYNAMIC SOCIAL NETWORK MODEM ED ISIS
77