An Information Flow Model for Conflict and Fission in Small Groups1

Abstract

Data from a voluntary association are used to construct a new formal model for a traditional anthropological problem, fission in small groups. The process leading to fission is viewed as an unequal flow of sentiments and information across the ties in a social network. This flow is unequal because it is uniquely constrained by the contextual range and sensitivity of each relationship in the network. The subsequent differential sharing of sentiments leads to the formation of subgroups with more internal stability than the group as a whole, and results in fission. The Ford-Fulkerson labeling algorithm allows an accurate prediction of membership in the subgroups and of the locus of the fission to be made from measurements of the potential for information flow across each edge in the network. Methods for measurement of potential information flow are discussed, and it is shown that all appropriate techniques will generate the same predictions.

Full text

An Information Flow Model for Conflict and
Fission in Small Groups
1
WAYNE W. ZACHARY
Data from a voluntary association are used to construct a new formal model for a traditional
anthropological problem, fission in small groups. The process leading to fission is viewed as
an unequal flow of sentiments and information across the ties in a social network. This flow
is unequal because it is uniquely constrained by the contextual range and sensitivity of each
relationship in the network. The subsequent differential sharing of sentiments leads to the
formation of subgroups with more internal stability than the group as a whole, and results in
fission. The Ford-Fulkerson labeling algorithm allows an accurate prediction of membership
in the subgroups and of the locus of the fission to be made from measurements of the potential
for information flow across each edge in the network. Methods for measurement of potential
information flow are discussed, and it is shown that all appropriate techniques will generate
the same predictions.
THE PROBLEM OF HOW and why fission takes place in small
bounded groups has long been a central issue in social anthropology,
even though the small groups studied have often been described under
some other rubric, such as kinship. Fission in kinship groups has been
studied from a variety of perspectives, especially those of descent theory
(e.g. Evans-Pritchard 1940; Middleton and Tait 1958; Peters 1960;
Forde 1964), and ecological adaptation (e.g. Reay 1967; Kelly 1968;
Rappaport 1969; Nelson 1971). Another type of small bounded group
frequently studied by anthropologists is the voluntary association (e.g.
Mangin 1965, 1970; Doughty 1970, Goode 1970), although seldom with
regard to fission or faction formation. In this paper I present data from
such a group, a university-based karate club, in which a factional
division led to a formal separation of the club into two organizations.
The process leading to this fission is analyzed using a new model of
fission, based on a social network approach. This model is a formal one,
taken from the family of mathematical structures known as capacitated
networks, and was developed directly from the ethnographic material
outlined in the following section. The model allows the locus of fission
within the group to be accurately predicted (greater than 97% accuracy
for the data reported here). Moreover, this result is not limited to
voluntary associations or to American culture, but rather is applicable
1 I would like to thank Henry Selby, Lucy Garretson, and David Smith who read and
commented on this and several earlier versions of this paper, as well as Russell Bernard, Peter
Killworth, Lori Corwin, and Arthur Murphy who commented on earlier versions oniy. I would also
like to thank George Benko and Robert Leskovec who helped me collect the original data. Still,
any errors and misrepresentations are my own doing. Finally, I would like to thank the National
Science Foundation which supported me as a Graduate Fellow during the analysis of the data.
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Vol. 33, 1977

CONFLICT AND FISSION IN SMALL GROUPS 453
to bounded social groups of all types in all settings. Also, the data
required can be collected by a reliable method currently familiar to
anthropologists, the use of nominal scales.
THE ETHNOGRAPHIC RATIONALE
The karate club was observed for a period of three years, from 1970
to 1972. In addition to direct observation, the history of the club prior to
the period of the study was reconstructed through informants and club
records in the university archives. During the period of observation, the
club maintained between 50 and 100 members, and its activities
included social affairs (parties, dances, banquets, etc.) as well as
regularly scheduled karate lessons. The political organization of the
club was informal, and while there was a constitution and four officers,
most decisions were made by concensus at club meetings. For its classes,
the club employed a part-time karate instructor, who will be referred to
as Mr. Hi.
2
At the beginning of the study there was an incipient conflict
between the club president, John A., and Mr. Hi over the price of
karate lessons. Mr. Hi, who wished to raise prices, claimed the authority
to set his own lesson fees, since he was the instructor. John A., who
wished to stabilize prices, claimed the authority to set the lesson fees
since he was the club's chief administrator.
As time passed the entire club became divided over this issue, and
the conflict became translated into ideological terms by most club
members. The supporters of Mr. Hi saw him as a fatherly figure who
was their spiritual and physical mentor, and who was only trying to
meet his own physical needs after seeing to theirs. The supporters of
John A. and the other officers saw Mr. Hi as a paid employee who was
trying to coerce his way into a higher salary. After a series of
increasingly sharp factional confrontations over the price of lessons, the
officers, led by John A., fired Mr. Hi for attempting to raise lesson prices
unilaterally. The supporters of Mr. Hi retaliated by resigning and
forming a new organization headed by Mr. Hi, thus completing the
fission of the club.
During the factional confrontations which preceded the fission, the
club meeting remained the setting for decision making. If, at a given
meeting, one faction held a majority, it would attempt to pass
resolutions and decisions favorable to its ideological position. The other
faction would then retaliate at a future meeting when it held the
majority, by repealing the unfavorable decisions and substituting ones
2 All names given are pseudomyms in order to protect the informants' anonymity. For
similar reasons, the exact location of the study is not given.

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JOURNAL OF ANTHROPOLOGICAL RESEARCH
favorable to itself. Thus, the outcome of any crisis was determined by
which faction was able to "stack" the meetings most successfully.
The factions were merely ideological groupings, however, and were
never organiztionally crystallized. There was an overt sentiment in the
club that there was no political division, and the factions were not
named or even recognized to exist by club members. Rather, they were
merely groups which emerged from the existing network of friendship
among club members at times of political crisis because of ideological
differences. There was no attempt by anyone to organize or direct
political strategies of the groups, and, in general, there was no barrier to
interaction between members of opposing factions. Only at times of
direct political conflict did individuals selectively interact with others
who shared the same ideological position, to the exclusion of those
holding other positions. This selective association during confrontations
is what brought the factions together only at crisis moments.
Political crisis, then, also had the effect of strengthening the
friendship bonds within these ideological groups, and weakening the
bonds between them, by the pattern of selective reinforcement. A series
of political crises, like that which preceded the fission had the effect of
"pulling" apart the network of friendship ties which held the club
together, until the group completely and formally separated.
There are several reasons for formalizing this ethnographic descrip-
tion of the fission process into a mathematical model. First, the
description is clarified. The formalization of the data necessary for the
construction of a mathematical model requires that the intuitively
based description be made precise and that all the relationships among
descriptive categories be made clear. Second, the intuitive conclusions
drawn can be formally stated and included in the model. The case
study can then be generalized on the basis of the formal properties of
these conclusions. In particular, the conclusions drawn about this group
force a rejection of a standard anthropological social network model and
require the construction of a more powerful network model—a network
flow model. This model is new to anthropology and suggests several
important new avenues of investigation in small-group studies. Third,
hypotheses about the fission process, intuited from observation of the
club, may be rigorously stated within the terms of the model and
mathematically tested. Alternatively, assumptions about the basic
conditions of the political process could be stated and then simulated
within the model to assess their validity. Only the former approach will
be used here, however.
The model is constructed by formalizing those relationships, or
those components of the system, believed to be the most significant or
the most explanatory. The feature of the karate club that appeared most

CONFLICT AND FISSION IN SMALL GROUPS 455
important in the ethnographic data was the network of friendship
relationships among club members. While only (affective) friendship is
considered here, any other dyadic relationship, such as effective
friendship, patron-client relationships, or kinship ties, could have been
used to specify the model. This flexibility gives the model a great
generality.
A formal model of the friendship network within the club, one that
contains sufficient complexity and precision to allow the testing of
propositions about the political activities and the fission process, can be
constructed. The model-building process will begin from the point of
the social network model, an intuitive, nonformal model which can
easily be transformed into a mathematical one.
SOCIAL NETWORKS
The idea of the social network model is now well-established in the
social science literature (Barnes 1968; Wolfe and Whitten 1974; Bott
1971; and Mitchell 1969), and is thus familiar to most anthropologists.
In a social network model, the social relationships among the individ-
uals in some bounded group are represented by a graph. Each
individual in the group is represented by a point on the graph, with a
line (called an edge) being drawn between any two points on the graph
if, and only if, the relationship under consideration exists between the
corresponding individuals. The analysis of patterns of social relationship
in the group is then conducted on the graph, which is merely a
shorthand representation of the ethnographic data. Barnes (1969), for
example, has programmatically suggested the investigation of such
features as the density (the ratios of actual to possible edges in the
graph) or the average length of closed circuits in the graph.
An alternative, more formal, representation of the social network is
as a square matrix of ones and zeroes. In this form, each individual is
represented by a row and column in the matrix; for example, the ith
individual being represented by the ith row and the ith column. For
each cell in the matrix, a one is assigned to it if, and only if, an edge was
drawn between the points corresponding to the row and column
designating that cell. A zero is assigned otherwise. The matrix can then
be manipulated by matrix algebraic procedures to uncover formal
relationships which are not superficially evident in the data. Examples
of techniques for using the matrix form to analyze the structure of social
relationships are presented in Flament (1963), Holland and Leinhardt
(1970), and Lorraine and White (1971).
Both the graph and the matrix representations can be constructed
for the karate club. The graph representation of the relationships in the
club (shortly before the fission) is given in Figure 1. A edge is drawn if

456 JOURNAL OF ANTHROPOLOGICAL RESEARCH
FIGURE 1
Social Network Model of Relationships in the Karate Club
This is the graphic representation of the social relationships among the 34 indi-
viduals in the karate club. A line is drawn between two points when the two
individuals being represented consistently interacted in contexts outside those of
karate classes, workouts, and club meetings. Each such line drawn is referred to as
an edge.
two individuals consistently were observed to interact outside the
normal activities of the club (karate classes and club meetings). That is,
an edge is drawn if the individuals could be said to be friends outside
the club activities. This graph is represented as a matrix in Figure 2. All
the edges in Figure 1 are nondirectional (they represent interaction in both
directions), and the graph is said to be symmetrical. It is also possible to
draw edges that are directed (representing one-way relationships); such

CONFLICT AND FISSION IN SMALL GROUPS 457
FIGURE 2
MATRIX OF RELATIONSHIPS IN THE CLUB: THE MATRIX E
Individual Number
1111111111222222222233333
1234567890123456789012345678901234
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
0
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
1
1
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
1
0
0
0
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
0
1
1
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
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1
0
0
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1
0
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0
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1
0
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0
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1
1
1
1
0
0
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1
1
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1
1
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0
This is the matrix representation of the graph shown in Figure 1. The rows and col-
umns represent individuals in the club. An entry, determined by a row/column pair,
is valued at 1 if an edge was drawn in Figure 1 between the two individuals repre-
sented by the row and column. The entry is valued at 0 otherwise. For example, the
entry in the fifth row, seventh column, is a 1, indicating there is an edge existing be-
tween individuals 5 and 7. Notice that the matrix is symmetrical--i.e., the seventh
row fifth column is also valued at 1, since it identifies the same edge. Because, by
definition, no individual can interact with himself, zeroes occupy the main diagonal
(from row/column 1 to row/column 34). Later, this matrix will be referred to by the
symbol E.

458
JOURNAL OF ANTHROPOLOGICAL RESEARCH
graphs are called asymmetrical. However, only symmetrical networks will
be considered in this paper.
Only 34 individuals are presented in Figures 1 and 2. Although club
membership was near 60 at that time, none of the 26 members not
represented interacted with other club members outside the context of
meetings and classes. Because these individuals would only be uncon-
nected points in the graph, and rows and columns of zeroes in the
matrix, they are not included. These individuals belonged to neither
faction, and did not participate in the politics of the club. Most joined
neither club at the time of the fission, preferring to quit the study of
karate altogether because of the political conflict.
Density analysis, or the various forms of matrix analysis, are not
conducted here because they, like all social network analyses, are static.
Lorraine and White (1971) have pointed out that an analysis of the
relationships between edges in the network is the only kind of analysis
possible on graph and matrix representations. Such an analysis is purely
structural (by the classic Jacobsonian definition) and must ignore all
processual aspects of the social activity represented by the network.
Since the political process leading to the fission rather than the political
structure of the group is the focus of this paper, neither the graph nor
the matrix representation is adequate. However, further information
can be included along with the matrix form to make a processual
analysis possible.
CAPICITATED NETWORKS
In constructing a processual model, the network shown in Figures 1
and 2 must be considered only as it represents the social constraints of
the system within which the political process operates. The model built
here is based on the friendship structure within the club, and the
feature of the political process which operated through these friendship
relations was the transmission of political information (both tactical,
e.g., when a meeting would occur, and ideological) during times of
political crisis. This process brought the factions into existence during
crisis periods, and was the mechanism for the positive feedback
relationship which existed between crisis and factional organization. In
this relationship, crises caused the reinforcement of only those relation-
ships within factions, thus making the factions more defined, and also
making another confrontation more likely.
From the point of view of information flow, the ties in the network
can be thought of as channels across which information may flow. The
absence of an edge precludes any direct passage of political information
between the two individuals represented by the potential edge. It is
obvious that the flow, or potential for flow, is not identical across all

CONFLICT AND FISSION IN SMALL GROUPS 459
edges in the network. Some individuals are much closer friends than
others, some see each other only in the presence of members of the other
factions. All these differing relationships are represented in the same
way in Figures 1 and 2. In order to represent the network accurately as a
net of channels for information flow, the different potentials for
information flow of the different edges in the network must be
quantified and included in the model.
By quantifying the strengths/weaknesses of the edges in the network,
a mathematical model can be specified that is formally adequate for the
investigation of information flow in the karate club, and its effect on the
fission process. The matrix representation of the social network shown in
Figures 1 and 2 will be retained as one component of this model. The
matrix form is chosen because of its greater ease of algebraic manipula-
tion. The matrix in Figure 2 will be referred to as the existence matrix,
and all entries of value one will be referred to as existing edges in the
network. A second matrix must be created which quantifies the relative
strengths/weaknesses of the existing edges in the network. This second
matrix is called the capacity matrix.
The complete mathematical model of the karate club can now be
specified as an ordered triplet (V,E,C), where V is the set of individuals
included in the network, E is an existence matrix, and C is a capacity
matrix. The elements of set V are called nodes in the network, and the
triplet (V,E,C) is called a capacitated network. The mathematical
development of capacitated networks is presented in Ford and Fulker-
son (1962), Hu (1969), Maki and Thompson (1974), and Zachary
(1977).
Each value of the matrix C can be interpreted as representing a
"capacity" or "value" of maximum possible flow for the corresponding
edge in the existence matrix E. It can be shown mathematically that
most features of flow in networks, (in this case flow of political
information) are functions of the edge capacities in the network (see
Ford and Fulkerson 1962, or Zachary 1977). In particular, the maxi-
mum flow in the network is completely determined by the values in the
matrix of edge capacities.
Unfortunately, there is no way known to quantify what has been
referred to as "political information," and by extension, there is no sure
way to specify actual capacities of information flow for the edges in the
network. However, by making certain assumptions about the social
settings in which information was communicated, this problem can be
circumvented.
Political information was communicated in contexts outside the
regular activities of the club, and club members interacted in a number
of such contexts. Since political activity was not overtly recognized by

460
JOURNAL OF ANTHROPOLOGICAL RESEARCH
club members, the transmission of political information was then
incidental to normal social interaction. It can be assumed that the
amount of information transmitted was related to the number of
contexts in which the information might have been communicated.
Mathematically, this can be stated as assuming that the amount of
information communicated is a function of the number of contexts in
which communication could take place. If, from a knowledge of the
karate club, some specific relationship between the amount of informa-
tion and the number of contexts can be determined, a procedure for
assigning values to C based on the number of contexts can be devised.
The functional relationship is at least monotone increasing. That is, the
greater the number of contexts in which a pair interacted, the more the
information that could be passed between them. The data indicate that
the relationship is roughly linear. For instance, twice as much informa-
tion could be passed over time between individuals who interacted in
four contexts as between individuals who interacted in only two
contexts.
Some ethnographic description of this proposed linearity can be
provided. A typical message unit transmitted through the network was
the news of a club meeting. The "goal" of any faction member was to
pass along such information to as many members as possible of the same
faction and to as few as possible of the opposing faction. However, as
previously stated, the communication of such items of information was
incidental to normal interaction. The assumption of linearity, then,
claims that the likelihood of this item of information being communi-
cated to any individual increases linearly with the number of contexts
in which interaction with that individual takes place.
The assumption of linearity can also be justified mathematically.
Given a monotone increasing function, it can be reasonably approxi-
mated by a single linear function if there is no severe change in the rate
of increase of the function in the interval under consideration. The data
provide no indication that such a change in rate of increase in
information flow occurs, and thus the assumption of linearity can be
accepted.
The linearity of the relationship between information flow and the
number of contexts of interaction suggests a method for generating
values for the matrix C. Since information cannot be quantified, but has
been shown to be a linear function of a variable that can be, the
quantifiable variable may be used in its place, and the analysis to follow
will not be effected, as proved in Zachary (1977, see Theorem 4).
3
A
3 In particular, Zachary (1977: Theorem 4) shows that any capacity matrix which is a linear
multiple of the unknown but "true" capacity matrix will produce the same minimum cut as the
"true" matrix when the maximum flow-minimum cut labeling procedure (used throughout the

CONFLICT AND FISSION IN SMALL GROUPS 461
specific procedure can now be outlined for generating the values of C. A
finite set of possible contexts, chosen on the basis of observation of the
group, will be used as the domain of a scale variable. Then, the
relationship between each pair of individuals in the network is
examined against this (nominal) scale. A value, equal to the total
number of contexts from the scale in which the two individuals
interacted, is then assigned to the corresponding entry in C. Eight
contexts are included in the domain of the scale applied to the edges in
the karate club network. They are:
(1) Association in and between academic classes at the university.
(2) Membership in Mr. Hi's private karate studio on the east side of
the city where Mr. Hi taught nights as a part-time instructor.
(3) Membership in Mr. Hi's private karate studio on the east side of
the city, where many of his supporters worked out on weekends.
(4) Student teaching at the east-side karate studio referred to in (2).
This is different from (2) in that student teachers interacted with each
other, but were prohibited from interacting with their sutdents.
(5) Interaction at the university rathskeller, located in the same
basement as the karate club's workout area.
(6) Interaction at a student-oriented bar located across the street from
the university campus.
(7) Attendance at open karate tournaments held through the area at
private karate studios.
(8) Attendance at intercollegiate karate tournaments held at local
universities. Since both open and intercollegiate tournaments were held on
Saturdays, attendance at both was impossible. *•
This scale was applied to the relationships between all pairs of
individuals in the karate club, using data compiled over the three years
of direct observation of interactions in the club. For each existing edge
in E (Figure 2), the pair of individuals involved interacted in at least
one of the above eight contexts. The quantified matrix of contexts is
given in Figure 3, and is the third component in the capacitated
network model (V,E,C).
NETWORK FLOWS
The flow of information in the club can now be analyzed with the
fully specified model (V,E,C), by using the family of mathematical
problems known as network flows, or Ford-Fulkerson problems, after the
remainder of the paper) is applied. A linear multiple is a linear function in which the zero in the
domain (here, information flow) is mapped into the zero in the range (here, number of contexts of
interaction). Since zero potential for information flow clearly maps into zero contexts of
interaction, the linear function used is really a linear multiple.

462
JOURNAL OF ANTHROPOLOGICAL RESEARCH
FIGURE 3
QUANTIFIED MATRIX OF RELATIVE STRENGTHS OF THE RELATIONSHIPS
IN THE KARATE CLUB: THE MATRIX C
Individual Number
1111111111222222222233333
1234567890123456789012345678901234
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
0
4
5
3
3
3
3
2
2
0
2
3
1
3
0
0
0
2
0
2
0
2
0
0
0
0
0
0
0
0
0
2
0
0
4
0
6
3
0
0
0
4
0
0
0
0
0
5
0
0
0
1
0
2
0
2
0
0
0
0
0
0
0
0
2
0
0
0
5
6
0
3
0
0
0
4
5
1
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
2
0
3
3
3
0
0
0
0
3
0
0
0
0
3
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
2
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
5
0
0
0
3
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
2
5
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
4
4
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
5
0
0
0
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
0
3
4
0
0
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
2
2
0
0
0
3
3
0
0
0
0
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
3
0
0
0
0
0
0
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
5
3
3
0
0
0
0
0
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
0
0
0
0
0
0
0
0
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
2
0
0
0
0
0
0
0
0
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
4
0
0
0
0
0
3
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
2
2
0
0
0
0
0
0
0
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
0
0
0
0
0
0
0
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
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
4
0
3
0
0
5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
3
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
2
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
2
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
3
0
0
0
0
0
0
0
0
4
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
4
0
0
0
0
0
4
2
0
2
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
3
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
7
0
0
2
0
0
0
4
4
0
0
3
0
0
0
0
0
4
0
0
0
0
0
3
3
0
0
1
0
3
0
2
5
0
0
0
0
0
3
3
4
0
5
0
0
0
0
0
0
0
0
3
2
0
0
0
3
2
4
0
0
2
1
1
0
0
4
0
0
2
4
2
2
3
4
5
0
The scale given in the text (p. 461) was used to assign values of relative strength/
weakness of the relationships in the club. This matrix gives the values assigned to
each edge specified in matrix E (Figure 2). The value represents the number of con-
texts in which interaction took place between the two individuals involved. The
row/column ordering is the same as in Figure 2.
two mathematicians who first extensively studied flows in networks
(Ford and Fulkerson 1962).
An item of political information (for example, a decision to call a
meeting) originated with one or several individuals and was spread
through the club. This can be interpreted in the model as a flow from
one node to all other nodes in the network. The effect of the basic

CONFLICT AND FISSION IN SMALL GROUPS 463
structure of E and C (that is, the paths and strengths of relationships in
the club) on the passage of this item of information through the network
can be observed by noting the paths that the item will take in a given
communication. The information (the calling of a meeting) usually
began with one of the factional leaders, Mr. Hi or John A. This
individual will be called the "source of the information," or simply, the
source. Given the course of political conflict in the club, it would be
advantageous to the source if individuals in the opposing faction did not
receive this information. Further, the source would benefit more if the
information would be deprived to individuals proportionately to their
affiliation with the other faction—the more closely a person was aligned
with the other side, the less the source would benefit from his or her
knowing about the meeting. By this reasoning, the leader of the other
faction should be the last to receive the information. He will therefore
be called the "sink of the information," or simply, the sink—the terms
"source" and "sink" being standard terms from network flow theory.
A hypothesis to be investigated is that no overt attempt may have
had to be made to avoid mentioning the news in the presence of
members of the other faction because the network tended to prevent
such meetings from occurring. The hypothesis suggests that the sink's
faction generally would not receive the information, since members of
opposing factions were less likely to associate, or did so in fewer different
contexts, than members of the same faction.
Not all individuals in the network were solidly members of one
faction or the other. Some vacillated between the two ideological
positions, and others were simply satisfied not to take sides. These
individuals are key nodes in the network, in that it was through them
that information was likely to pass from one faction to the other. The
feedback principle can be used to state that as the frequency of crisis
increases, these individuals are likely to be included in the network ties
of the members of one faction to the exclusion of those of the other
faction, regardless of whether they are politically part of that faction. It
would be expected that these individuals would join the club formed by
the faction with which they were associated when the split in the club
occurred. A second, more general, hypothesis can be framed from this
discussion: that a bottleneck in the network, representing a structural
limitation on information flow from the source to the sink, will predict
the break that occurred in the club at the time of the fission. The term
"bottleneck" refers to the first hypothesis, which claims the existence of
some structural feature in the network inhibiting information flow
between factions.
The validity of both hypotheses can be evaluated by a single
mathematical technique called the maximum flow—minimum cut labeling

464
JOURNAL OF ANTHROPOLOGICAL RESEARCH
procedure, developed by Ford and Fulkerson (1962) when they established
that the value of the maximum flow in a capacitated network is strictly
determined by the values assigned to the capacities of the edges in the
network. Intuitively stated, they proved that the maximum flow is equal
to the capacity of the smallest possible break in the network separating
the source from the sink.
A break in the network is conceptualized as a division of the set of
nodes V into two subsets, one containing the source and the other, the
sink. The edges connecting elements of one subset to elements of the
other are called the cut. The sum of the capacities of all edges in the cut
is its capacity. From Ford and Fulkerson, the maximum flow is equal to
the capacity of the minimum cut. The minimum cut, then, represents a
bottleneck in the network and possesses the properties described in the
two hypotheses.
The minimum cut is located by means of an algorithm which begins
by starting a flow moving in the network from the source to the sink.
This flow is then systematically incremented until no more flow can be
added because it is equal to the capacity of some edge on each possible
path from the source to the sink. The minimum cut is comprised of
those edges on each path from the source to the sink for which the flow
equals the capacity. If the flow is equal to the capacity of more than one
edge on the path, only the edge closest to the source is in the cut.
Mathematically, the first hypothesis states that the members of one
faction will all be on one side of the minimum cut and that the
members of the other faction will all be on the other side. The second
hypothesis states that the minimum cut will separate the club members
who joined Mr. Hi's club after the split from those who joined the
officers' club.
NETFLOW (Smillie 1969) is an APL language computer program
which carries out the maximum flow—minimum cut labeling procedure
of Ford and Fulkerson. NETFLOW was run on the model of the karate
club, (V,E,C). The program requires that the first row/column of
matrices E and C represent the source and that the last row/column
(here, row/column 34) represent the sink. Hence, row/column 1 in these
matrices designates Mr. Hi, and row/column 34 designates John A. The
results of NETFLOW support both hypotheses, and are summarized in
Table 1. In all cases, except individual number 9, individuals on the
source side of the cut belonged to Mr. Hi's faction (or no faction), and
joined the club formed by his supporters after the split. Individuals on
the sink side of the cut belonged to John A.'s faction (or no faction) and
joined the club founded by the officers. Person number 9 was a weak
supporter of John but joined Mr. Hi's club after the split. This can be
explained by noting that he was only three weeks away from a test for
black belt (master status) when the split in the club occurred. Had he

CONFLICT AND FISSION IN SMALL GROUPS
465
TABLE 1
RESULTS OF INITIAL NETFLOW RUN
INDIVIDUAL
NUMBER
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
SIDE OF CUT
Source
Source
Source
Source
Source
Source
Source
Source
Sink
Sink
Source
Source
Source
Source
Sink
Sink
Source
Source
Sink
Source
Sink
Source
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
FACTION
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Strong
John - Weak
None
Mr. Hi - Strong
Mr. Hi - Strong
Mr. Hi - Weak
Mr. Hi - Weak
John - Strong
John - Weak
None
Mr. Hi - Weak
None
Mr. Hi - Weak
John - Strong
Mr. Hi - Weak
John - Strong
John - Weak
John - Weak
John - Strong
John - Strong
John - Strong
John - Strong
John - Strong
John - Strong
John - Strong
John - Strong
John - Strong
CLUB AFTER
FISSION
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi 's
Mr. Hi's
Mr. Hi's
Officers'
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Officers'
Officers'
Mr. Hi's
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
This table summarizes the results of the first run of NETFLOW, using matrices E and
C as input. "Individual Number" identifies the individual with the corresponding
row/column in the matrices. "Side of Cut" refers to the subset of V to which the
individual was assigned by NETFLOW, either the source side or the sink side. "Fac-
tion" gives the factional affiliation of the individual, either with that of John A.,
that of Mr. Hi, or none. The strong/weak designations in this column indicate
whether the individual was a strong or a weak supporter of the faction's ideological
position. Finally, "club after fission" indicates which club was joined after the fis-
sion, either that formed by Mr. Hi, or that formed by the officers of the original
club.

466
JOURNAL OF ANTHROPOLOGICAL RESEARCH
joined the officers' club he would have had to give up his rank and
begin again in a new style of karate with a white (beginner's) belt, since
the officers had decided to change the style of karate practiced in their
new club. Having four years of study invested in the style of Mr. Hi, the
individual could not bring himself to repudiate his rank and start again.
In fact, the inclusion of individual 9 in the sink side of the cut is
supportive of both hypotheses. He was a weak political supporter of
John A., and during the crises preceding the fission, he became
increasingly associated with the members of John's faction to the
exclusion of those of Mr. Hi's. He was clearly a structural part of John's
faction and should have been assigned to the sink side of the cut by the
labeling procedure. That he did not follow the factional affiliation and
join the officers' club resulted from an overriding interest, not shared by
other club members, in remaining with Mr. Hi—his black belt. It is
important that all individuals with no factional alliance joined the club
founded by the faction to which they were assigned by the labeling
procedure. This is the faction to which they structurally, though not
necessarily ideologically, belonged.
The results of NETFLOW verify the first hypothesis by demonstra-
ting that the factions in the club could be represented by the minimal
cut in the network model. The second hypothesis is validated by
demonstrating that the membership of the clubs could be predicted by
the minimal cut in the network model. These results are not mathemati-
cally conclusive, however, unless the uniqueness of the minimal cut can
be established. There could be several possible cuts in the network
having the same minimal capacity. It must be proven that the cut
chosen by NETFLOW is the only minimal cut in the network.
Fortunately, Ford and Fulkerson (1962:ch. 2) also established a pro-
cedure by which the uniqueness of a minimum cut can be proven.
In this procedure, the entire original network is reversed, so that the
source becomes the sink, and the sink, the source. The labeling
procedure is then applied to the reversed network; if the same minimum
cut is found, then it is unique. This procedure can be accomplished by
reversing the row/column ordering of matrices E and C, (so that
row/column 1 becomes row/column 34, row/column 2 becomes row/
column 33, etc.), and then running the NETFLOW program on these
reversed matrices.
4
The results of this run are summarized in Table 2.
As can be seen from Table 2, the same cut is found in the reversed
network as in the original network. The minimum cut is then unique, in
that there is no other cut in the network with the minimum capacity.
The two hypotheses can be formally accepted on the basis of this
uniqueness. The test of the two hypotheses is summarized in Table 3.
4 Note that the matrix reversal operation used here bears no similarity or relation to either
the matrix inversion or the matrix transposition operations frequently used in matrix algebra.

CONFLICT AND FISSION IN SMALL GROUPS
TABLE 2
RESULTS OF SECOND NETFLOW RUN
467
INDIVIDUAL NUMBER
IN ORIGINAL
MATRIX C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
INDIVIDUAL NUMBER
IN REVERSED
MATRIX C*
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
SIDE OF CUT
IN ORIGINAL
MATRIX C
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Source
Sink
Source
Sink
Source
Source
Sink
Sink
Source
Source
Source
Source
Sink
Sink
Source
Source
Source
Source
Source
Source
Source
Source
SIDE OF CUT
IN ORIGINAL
MATRIX C*
Source
Source
Source
Source
Source
Source
Source
Source
Source
Source
Source
Source
Sink
Source
Sink
Source
Sink
Sink
Source
Source
Sink
Sink
Sink
Sink
Source
Source
Sink
Sink
Sink
Sink
Sink
Sink
Sink
Sink
This table summarizes the results of the second run of the program NETFLOW, this
time using the reversed matrices as input. Column 1 of the table gives the order of
the individuals as used in Table 1 and in matrices E and C. Column 2 gives the indi-
vidual numbers in the reversed matrices. Column 3 indicates the side of the cut to
which the individual was assigned by the first run of NETFLOW (from Table 1), and
column 4 lists the side of the cut to which the individual was assigned by the second
run, using the reversed matrices as input. In each case, the entry in column 3 is the
opposite the entry in column 4. Thus, the same cut is defined by both runs, and the
minimum cut is unique.
CONCLUSIONS
It has been shown how the flow of political information through the
club interacts with the political strategy of the factions to pull apart, or
bottleneck, the network at the factional boundary. This boundary
corresponds to an ideological as well as an organizational division in the

468
JOURNAL OF ANTHROPOLOGICAL RESEARCH
TABLE 3
EVALUATION OF THE HYPOTHESES
INDIVIDUAL
NUMBER IN
MATRIX C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
FACTION
MEMBERSHIP
FROM DATA
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
John
John
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
John
John
Mr. Hi
Mr. Hi
John
Mr. Hi
John
Mr. Hi
John
John
John
John
John
John
John
John
John
John
John
John
FACTION
MEMBERSHIP
AS MODELED
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
John
John
Mr. Hi
Mr. Hi
Mr. Hi
Mr. Hi
John
John
Mr. Hi
Mr. Hi
John
Mr. Hi
John
Mr. Hi
John
John
John
John
John
John
John
John
John
John
John
John
TOTALS 34 hits, 0 misses
100%
hits, 0% misses
HIT/
MISS
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
CLUB AFTER
SPLIT
FROM DATA
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Officers'
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Officers'
Officers'
Mr. Hi's
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
CLUB AFTER
SPLIT AS
MODELED
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Officers'
Officers'
Mr. Hi's
Mr. Hi's
Mr. Hi's
Mr. Hi's
Officers'
Officers'
Mr. Hi's
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Mr. Hi's
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
Officers'
33 hits, 1 miss
97%
hits, 3% misses
HIT/
MISS
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Miss
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
Hit
This table gives the results of the NETFLOW runs used to test the two hypotheses
(see pp. 462 ff.). The faction membership (column 2) and the club joined after the
fission (column 5) entries were taken from the ethnographic data. These columns
merely state what the individuals actually did. Column 3 gives the faction member-
ship as predicted by the model (based on which side of the minimum cut the individ-
ual was placed). Column 4 gives the accuracy of each of these predictions. The model
was 100% accurate in predicting faction membership. Column 6 gives the member-
ship in the two clubs formed after the fission, again as predicted by the model (based
on which side of the minimum cut the individual was placed). Column 7 gives the re-
sults of these predictions. The model was 97% accurate in predicting club member-
ship after the split. Thus, both hypotheses can be accepted.

CONFLICT AND FISSION IN SMALL GROUPS
469
club. It is possible to extend the argument used in the description of the
fission process to include the ideological division in the same feedback
framework used to describe the development of the organizational
division.
Conflict in any setting must be bounded by some sharing of rules
which allows the combatants or contestants to know what and how they
are contesting. This knowledge can be at any level, from the rules of a
game, to the shared meanings of ritual items for the gumsa and gumlao
Kachin (Leach 1954). Whatever the level, these rules must continue to
be understood by both players of the game, or the conflict ceases as the
activity becomes undefined for the players.
In the karate club, both factions shared an ideological position
toward the club in the early stages of the conflict, but followed
increasingly divergent positions. Their divergent positions made the
basis for conflict increasingly unclear to the participants. It also became
more and more difficult for members of the two factions to understand
how the opposing side could even consider themselves as part of the
same club. Just as a faction tended to know less and less about the
political activity of its opposition, it tended to understand less and less
its common ground with the opposition, until even the existence of the
club ceased to serve as a basis for unity.
It can be concluded, then, that not just political information about
club meetings was communicated in the friendship network outside the
club. In these contexts, club members communicated their perceptions
and understandings of the nature of the club in various conscious and
unconscious ways, as well. If the bottlenecking of the network prevented
the flow of political information between factions, it also prevented the
sharing of perceptions of the club held by various club members. In
particular, it tended to allow sharing of these items to a greater extent
within the factions than across the factional boundary.
A feedback relationship, then operated in two directions, one
organizational and one ideological. The strategy of the factions (whose
membership was based on ideology) acted to strengthen the cut in the
network, which itself was the organizational basis for the factions'
existence. The cut in the network acted to intensify the ideological
divisions on which club members based their factional affiliation. These
feedback relationships constituted a vicious cycle, acting as a mechan-
ism which virtually assured the fission in the club. While the content
of this mechanism was specific to the karate club, its form is a general
feature of communication patterns in small groups, and corresponds to
a feature inherent in the type of model used.
This conclusion has broad implications for the anthropological
study of small groups. The positive feedback process can be formally

470 JOURNAL OF ANTHROPOLOGICAL RESEARCH
represented as a feature of a model of the group, the minimum cut in a
capacitated network. This type of model can be constructed for any
group, and the existence and the significance of the minimum cut, or
bottleneck, can be tested. The results of this study certainly suggest that
whenever a unique minimum cut exists in a capacitated network
representation of a small group, it will act as a barrier to group unity,
and under certain conditions, will act as a selective factor for group
fission.
As suggested above, the model can be generalized to include other
types of social relationships than that used here, effective friendship.
Moreover, the method used to generate values for the edge capacities
can also easily be described generally as constructing a nominal scale,
choosing subsets of elements from the nominal scale, and assigning the
number denoting the size of the subset to the edge in question. This
method places minimal restrictions on both ethnographers and infor-
mants. Other equally simple procedures could also be devised.
The capacitated network model constructed here is mathematically
much more powerful than the usual social network model. There are
two reasons for this increased power, one mathematical and one
anthropological. Mathematically, more information is included in the
capacitated network model, in the form of the strengths/weaknesses of
the edges given in matrix C. Models containing more information are
always more powerful than those containing less, the example of the
increased power of known-distribution statistics as compared to
distribution-free statistics being perhaps the best-known to anthropolo-
gists.
Anthropologically, the approach used to build the model is proces-
sual rather than static. The network model is constructed from a view of
the social system as a processor of information—a cybernetic
machine—rather than as a static set of relationships. While the
structural features of the network are investigated, it is the structure of a
cybernetic process that is really being studied. It is from this informa-
tion processing approach that such things as feedback and information
flow can be discussed and included in the model.
Further refinement of the mathematical assumptions built into the
model is possible. The most plausible of the assumptions made is that
the relationship between the amount of information and the number of
contexts is monotone increasing. While any monotonic function which
preserves addition may be used, the present assumption of linearity can
be justified both ethnographically and as a standard computational
device. While this approximation has proven sufficient for this study,
further ethnographic methods must be developed for specifying the
relationship accurately in other cases.

CONFLICT AND FISSION IN SMALL GROUPS 471
Finally, the method used to build the network flow model differs
from that used in much of the current model-building work in network
theory (e.g. Hunter 1974; Killworth and Bernard 1975) in that it began
with an existing set of data rather than a set of purely theoretical
propositions. The model built here is a formalization of ethnographic
description of the club, and all assumptions made are ethnographically
justified. Only after it had been fully constructed was the model
generalized, and the accuracy of the predictions made by the model are
certainly dependent on this data-based aspect. Many of the mathemati-
cally more sophisticated models (Killworth and Bernard 1975, for
example) are constructed solely from theory and intuition, and it is
openly admitted that the results may not be close approximations of
social reality (Killworth and Bernard 1975: ch. l). Because of its data
orientation, the network flow model has the advantages of simplicity
and accuracy of representation.
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DEPARTMENT OF ANTHROPOLOGY
TEMPLE UNIVERSITY
PHILADELPHIA, PA. 19122