A MATHEMATICAL MODEL FOR SINGLE
FUNCTION GROUP ORGANIZATION
THEORY WITH APPLICATIONS TO

SOCIOMETRIC INVESTIGATIONS

Thesis for the Degree of Ph. D.
MICHIGAN STATE COLLEGE
James Henry Powell
1954

 

LIBRARY

Michigan State
University

 

 

 

This is to certify that the

thesis entitled
A Mathematical Model for Single Function Group
Organization The ory wi th Applications to
Sociometric Investigations

presented bl]

Iir. James Henry Powell

has been accepted towards fulfillment
of the requirements for

Doctor of PhilOSOphy Mathematical Statistics

degree in

  

 

Major professcg/i’f‘

Date November 5, 1954

 

0—169

 

 

 

 

MSU

LIBRARIES
“

 

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

James Henry Powell
candidate for the degree of

Doctor of Philosophy

Final examination, Nbvember 5, 1954, 3:00 P.M., Physics-
Mathematics Building

Dissertation: A.Mathematical Model for Single Function

Group Organization Theory with Applications
to Sociometric Investigations

Outline of Studies

Major subject: Mathematical Statistics
Minor subjects: Algebra, Analysis, Geometry

Biographical Items
Born, flay 21, 1926, Columbus, Ohio

Undergraduate Studies, Michigan State College, 1943-4#,
cont. 1946-49

Graduate Studies, Michigan State College, 1949-52, cont.
1953—54, University of California at Berkeley, 1952-53

IExperience: Graduate Assistant, Michigan State College.
l9h9—52, Special Graduate Research.Assistant,
Michigan State College, l952—5N, Member
United States Navy, l944—h6

Member of Phi Kappa Phi, Pi Mu Epsilon, Sigma Xi

A imm-IATICAL It-IODIJL FOR SIIJGLE F UI‘ICTl'Ol-E GROUP ORGANIZATION

THEORY ~HITH APPLICATIONS TO SOCIOIETRIC Il-fV'LSTIGATIOIJS

BY

JAIES FEI‘IRY POWELL

A TEESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1954

MATH. [184' ‘

of" at;
‘u \
,.

Liv-8'5!
'1

Acmzonmnsm-Irs

The author wishes to express his sincere thanks to
Dr. Leo Katz, his major professor, for suggesting
the problem and for his continuous help in every
phase of this work.

The writer also deeply appreciates the financial
support of the Office of Naval Research.which made

it possible for him to complete this investigation.

I“, f ( ;‘- '_' .4
("‘;-_}‘k ngi

A.LA"'“LaTICAL"ODmL F R SILCLE FUICTION GROUP ORCARIZATION

THEORY FJITH APPLICATIONS TO SOCIOIETRIC IEVBSTIGATIONS

By

James Henry Powell

AN ABSTRACT

Submitted to the School of Graduate Studies of Iichigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of mathematics

Year 1954

Approved 1‘

 

l
JAIJES IEIRY POJFCLL ABSTRACT

This thesis is concerned with the one-dimensional theory of
group organization as a complex of irreflexive binary'relationships,
taking values zero and.one, between the pairs of individuals. The
basic problems in connection with this theory are

(i) the investigation of the appropriate universe or

universes of discourse,

(ii) the determination of the null distributions for

certain proposed indices of the group structure,
and

(iii) the development of simple, reasonably exact methods
for use by field investigators.

In the first part of this thesis, a decomposition of the total
sample space is given which clarifies the first kind of problem.
Also, certain bipartitional functions tabulated by David and Kendall1
and standard combinatorial.methods augmented.by a theorem developed
in this part provide the machinery for counting the number of dis-
tinct points in each of the subsets in the decomposition of the
total.sample space. This machinery makes it possible to obtain the
necessary probabilities for construction of certain null distri—
butions. These probabilities are obtained by dividing the number

of points in the disjoint subsets in the framework of the given

decomposition by the total number of points in the appropriate

 

lSee reference [7] in Bibliography.

2

J .IES HEI‘IRY IUIJ'ELL ABSTRACT

universe of discourse.

The second part of this thesis shows how the theory develOped
in the first part gives the null distributions for certain indices
employed in group organization theory. In particular, the proba-
bility distributions of (a) indices on group expansiveness, (b) the
number of isolates and (c) maximun 8:] are given in detail as illus-
trative examples of the manner in which the general theory is applied
to produce probability distributions.

In the third part, simple, apprommate distributions are sug-
gested for certain of the variables treated in the second part of
this thesis. In addition, a test criterion for one aspect of group
structure is proposed along with a simple, approximate distribution

for it.

TABLE OF CORTERTS

Page
II:I1::.CDUCTICII OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO.

0:1. General model for group organization ........
0:2. Sociometric investigations 00.000000000000000
0:5. Review of work on related problems ..........
0:4. Previous work on the present problem ........
0:5. Statement of problem and plan of attack .....

PAET I. TILE—JOEY 0.0.0.0...OOOOOOOOOOOOOOOO..OOOOOOOOOOO

(O (O ~3030Q33F4 F’

1:1. Preliminaries eeeeeeeeeeoeeeeeeeeoeeeeeeeeeeo
1:2. A decomposition of the space of all negraphs

(or n X n.matrices) oeeeoeeeeeeeeeeeeeeeeoe 10
1:3. Enumeration of n—graphs (or n x n.matrices).. 16
1:4. The number of matrices (graphs) in «0(r,g)... 17
1:5. Probability distributions ............:...... 54

1:6. Femarks ono.oeoeooecoooeeoeeeeeeeoeoeoeoeoeee 35

FILE-T II. APPLICATIOIJS .00...0.0.0.0....OOOOOOOOOOOOOO. 57

2:1. Preliminaries ooeoeeoeooeeeoooeeooeoeoeeeoeee 37
2:2. Distributions of indices on expansiveness of

a SOCial gTOUP eeeeooeeoeeoeeeeoooeeeeeeeee 57
3. Distribution of the number of isolates ...... 59
4. Distribution Of maximum 5 eeoeoeeeeeeeeeeeee 43

5. Remarks oooeeeeooeoeeoeeoooeeeeoeoeoeeeeeeooe 45

2
2
2

PART III. APPROKIHATE DISTRIBUTIONS .................. 47

5:1. Preliminaries eeeeeeooe00000000000000.0000... 47
5:2. Approximate distribution of number of

isolates eoeeeeeeeeeeeoeeeeeeeeeeeeeeeeeeee 48
5:5. Approximate distribution of maximum 3 ...... 53
3:4. Concentration Of ChOice eeeeeeeeooeoeeeoeeoee 56

SLRJMIJULY 0.00.00.00.0000......0.000000000000000000000000 60

BIBLIOGIIAPILY OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 65

o r ~
I
u 0 I I

. . a n
, - , .
a v - a
r , I ‘
- . . a

n -
r w
. o
.. .
q

C I
. -
.n :-
n I.
t 1
a c-
, :
1: ~
f i
g l
,. _

u a
V Q
g ,
p s
-\ A
O .
“ O
‘ Q
A a
¢ -
o R

TABLE

TABLE

TABLE

I.

II.

III.

* IV.

' V.

VI.

VII.

VIII.

LIST OF malts

Page

A decomposition of the space of all n—graphs..

The enumeration in the sample space decompo-
Sition 00009000000000.0000oooooooooooooooooo
The enumeration of points in the decomposition
0f (3(18) into subspaces h’(18,§‘) .........
Probability distribution for the number of
isolates in a group of eight individuals,
each making One ChOiCG 0.0000000000000000...
Cumulative probability distribution for max 5 .
in a group of eight individuals, each making

one choice oooooooooooooooooooooo0000.00.00.

Comparison of the exact and approximate distri—
butions of the number of isolates for n =- 4,

g a (1,1,2,2) oooooooooooooooooooooocoo-o...

Comparison of the exact and approximate cumu-
lative distributions of maximum 5 for n=8,

:‘<18 j

) .00.0..OOOOOOOOOOOOOOOOOOOOO...O...

Comparison of the exact and approximate cumu—
lative distributions of index of concen-

tration of choice for n = 8, g 3 (18) ......

15

26

55

42

45

52

55

59

n o y,
A . .
0 0 l
C O f

LIST OF FIGURES

Page

FIGURE 1. Two representations of organizational configu-
rations 0n 6 pOintS 000000000000000000000000.a... 10

FIGURE 2. Representations of points in u)(2,l,2,2; 5,2,1,l).. 25

FIGURE 5. Two representations of organizational configur
ration on four points 00000000000.000000000000000 4O

INTRODUC TI 0N1

0:1. General model for you}: organization.

An organization of a social group is a complex made up of
many interactions among all the pairs of individuals comprising
the group. One way of describing the organization of the group
is by specifying the relationships between each pair of individuals.
The relationship between a pair of individuals can be represented
as a many-dimensional vector, in which each component is the strength
of the relationship for a particular activity (category), measurable
on some scale. The hope for simplicity and economy in a scientific
description of the group rests on the possibility that most of the
information is contained in relatively few facets of the group
organization; perhaps, only one.

In particular, the simplest model of a group organization is
a one—dimensional model with a binary relation between pairs of in—
dividuals which is not necessarily reflexive and takes only the values
zero and one in each direction. Although the model is superficially
simple, it is by no means empty, for it is the exact model used in
sociometric investigaiions. The journal Sociometry, which has been
in publication for some seventeen years, has contained papers by

sociometrists and others on the investigation of group organization

 

2|"This mark was sponsored by the Office of Naval Research.

2

using this simple model. We shall see shortly that this same model
is also useful for representing communication networks and in other

re ar—sociometric investi gati ons .

0:2. Sociometric investigations.
Moreno [2.3], in 1954, invented, as a technique for the study

of interpersonal relationships, the well—known sociometric test,
which is still the basic tool used today. In the words of Moreno
[17,p. 15], "The sociometric test consists in an individual choosing
his associates for any group of which he is or might become a member."
The test is applied to a well-defined social group (hereafter the
word group shall mean social goup) as follows?

Each member of a gnoup of n infividuals is asked to

choose a ”number, specified or not, of the n-l others

with whom he would prefer to be associated in a par-

ticular single activity. In addition, each member

may be asked to name those with whom he does not wish

to associate, and/or he may make selections, separ-

ately, for more than one activity.
We shall only be concerned with the test as applied to groups for
the purpose of eliciting positive responses for one activity. Typical
questions asked for eliciting positive choices are!

With whcm do you wish to sit? I

With whom do you wish to work?

There are two common forms employed for exhibiting the collected 1

data. The first, the directed gaph on a points (sociogran), has been

explored extensively by many sociometrists, most of whose publications-
appear in the journal. Sociometry. The sociogram is essentially des-
criptive in character. The second form, proposed by Forsyth and Katz
[l3], is that of a matrix of choices, C . (cij): n x n, where Icij is

a representation of the response (choice) from individual i; to indi-

vidual 1. The simplest representation is c. - l or 0 according as

1:]
i does or does not choose 1. The principal diagonal elements of C,

cii’ are usually taken to be zero. If rejection is also present,

013 may equal -1 whenever individual _i_ rejects individual 1. If
graduated choices are permitted, the possible values of cij are further
complicated.

One of the problems in group organization theory is the con—
struction of summary indices which attempt to measure various aspects
of the organization of the whole group. In connection with this
problem, it is essential to study, under some suitable null distribu-
tion, the probability distributions of the indices used. We shall in-
vestigate, under the simplest mathemtical model for group organization
theory, the appropriate sample spaces to which should be referred these
probability distribution problems. This model corresponds exactly
to the experimental technique proposed by Moreno in his sociometric
test 8111 will be defined in more detail in the preliminaries of

Part I.

0:5. Review of work on related problems.
Luce and Perry [95], in 1949, used the same matrix formulation

as preposed by Forsyth and Katz but restricted the entries to only

4

zeros and ones. Luce and Perry were interested in more complex con—
figurations. In particular, they were interested in two concepts,

k-chain and clique. A k-chain from individual i to individual 1k +1

1
is an ordered sequence with k+l members i1, 12, 15’ ..., ik-l’ 1k, 11*].

such that i chooses i , i

1 2 chooses i3, ..., ik-l chooses ik, and

2
ik chooses ik+1' A 2133?}; is a subset (proper or not) of the group
such that each individual in the subset chooses all the others in

the subset. There is a fairly extensive literature on the problems
involving mcre complex configurations. However, we will not be con-
cerned with these problems.

A communication network can be represented by a netrix of zeros
and ones. Shimbel [35’] uses such a matrix representation with unit
entries on the principal diagonal. He gives, ccrrectly, the number
of communication networks involving 11 points (individuals) as 2n(n—1).
We shall see later that this agrees with our expression for the nunber
of directed graphs on a points. Primarily, he is concerned with pgwpegs
wgatrg in order to answer questions on the connectivity of the
mtwork. Christie, Luce and Macy [3 ] have done a good deal of ex-
perimental work on communication problems in very mall groups. A
ratrer extensive bibliography on comurfication networks ard its related
problans appears at the end of their report.

The structure of animal societies has been studied by Rapoport
[33, 33 ,39] and Landau [94,21,33]. The problem of "peck order“ or
"peck right" in a flock of birds is represented by a~mtrix of ones

and. zeros. ‘However, the "peck right" relation is dominant, 1.6., if

animal i peeks animal 1 then 1 cannot peck i_.. Thus, their results
are of no use to us since we do not want to rule out the possibility
of tvm individuals choosing each other.

Thrall [40] considered the ranking of social organizations on
the basis of their common members. We will not be concerned with
this type of investigation.

A monogaph on Graph Theory as a Mathematical Model in Social

 

Science, by Norman am Harary [’91, gives a review of the previous
work" done on numbers of graphs. Pdlya [3:] gave explicitly the numbers
of trees2 and rooted treesfi, and, implicitly, the nunber of ordinary
graphs on _n points. Otter [30] gave simpler, purely combinatorial
methods for counting trees and rooted trees and Harary and Uhlenbeck
[l6] gave the numbers of Husimi trees4. Also, Harary and Norman {/4}
have given the number of directed graphs on n points and 3 lines.
Davis [ 9] defines and gives a method far counting the nunbers of
various subclasses of directed graphs on n points. The problem we
shall face is essentially different from all of these and requires

quite different methods of enumeration.

 

2A tree is an ordinary yam with a path between every pair of
its points, i.e., a connected graph, and which has no cycles.

5.4 rooted tree is a tree in which there is a distinguished point.

4A Husimi tree is a connected graph in which no line lies on

more than one cycle.

0:4.g Previous work on the present_problem.

The application of chance (probability) distributions to socio-
metric data was, first, suggested by Moreno and Jennings [39], in
1958. They took deviations from chance as a reference base far the
measurements of social configurations. In their words, [39,P 9],

"It appeared that the most logical ground for establishing such a
reference could be secured by ascertaining the characteristics of
typical configurations produced by chance balloting far a similar size
population with a like number of Choices." The theoretical computation
of the desired.probabilities was turned.over to Lazarsfeld. In Part

II on applications we shall give the specific history of attempts to
obtain.the distributions of the several sociometric variables whose
exact distributions are obtained. I

Bronfenbrenner [I ,p. 9] , in 1945, attempted "to develop an
absolute criterion - a frame of reference against which the various
phenomena of sociometric choice may be projected.but which is itself
independent of“these phenomena." Thus, what Bronfenbrenner had in
mind was a common universe of discourse for all the sociometric vari-
ables. This is the so—called "constant frame of reference" problem
in sociometry. Bronfenbrennerelaborated on the general technique,
proposed by Moreno and Jennings, of taking deviations from chame as
a reference base. He developed.some expressions and techniques for
determining the probability of occu‘rence of the major sociometric
variables. However, for the most part, the results obtained were in-
correct.

Edwards [M], in 1948, surveyed the published work on the

application of deviations from chance to the problems in sociometrics.
She suggested some modifications and pointed out certain errors in
Bronfenbrenner's work. However, Edwards did not question whetker it
was correct to assume that all sociometric variables should have the
same mtiverse of discourse.

In 1950, Criswell [6] pointed out that indices in different
experimental settings do have different neanings, and that it was
ridiculous to believe that all the variables considered in sociometric
investigations would have the same frame of reference. Criswell
[6, p. 107], furtrer, pointed out the need for the "use of the ap-
propriate chance distribution as a reference base for developing a
score expressing a structural aspect of the group as a whole".

We note, finally, that all of the se investigations considered
probability distribution problems only for tie case where all of the
individuals make the same number of choices. In Part 1, general
methods are obtained which give the distributions of certain sociometric
variables for the less restrictive case in which individuals are free

to make any number of choices.

0:5. Statement of problem and 4913.11 of attack.

 

The basic problems of the one-dimensional theory of group or-
ganization as a complex of irreflexive binary relationships between
the pairs of individuals are

(i) the investigation of the appropriate universe of discourse,

(ii) the determination of the null. distribution for each

proposed index of the group structure, and

(iii) the development of simple, reasonably exact methods

for use by field investigators.

In Part I, we consider the simplest mathematical model for
group organization and develop a decomposition of the total sample
space which simplifies the fir st kind of problem. Also, in Part I,
we develop the machimry of counting methods whi ch makes it possible
to obtain many of the null distributions exactly. In Part II, appli-
cations are given to classes of unsolved probability distribution
problems of group organization theory. Finally, in Part III, simple
approximate distributions are suggested for certain of the variables

treated in Part II.

PART I

THEORY

1:1. Preliminaries.

We mall be concerned with finite groups of _n individuals. The
organization of such a goup for a single activity can be thought of
as a configuration of the connections among all pairs of individuals.
The connections between pairs are not necessarily reflexive and in
the simplest case take in each direction only two values, 0 and 1. Such
a configuration can be represented by a matrix C or by a linear directed
graph G on n points (see Figure 1). C is the n x n matrix (013) With
C13 - 1 if a connection exists from individual i to individual 1, other-
wise cij . 0. We adopt the usual convention that cii
Let the points of the graph G be P1, P2, ..., Pn' ‘ Then, in the graph,
a connection existing fran individual _i; to individual i is represented

'0f01'33-llo

by a directed line from Pi to Pj’ Pi——'>Pj' The absence of a. connection

from individual i to individual 1 corresponds to no line from Pi to P3.

Obviously, c .- 1 (or 0) if and only if a directed line exists (or

1:)

Cbesn't exist) from Pi to P3. Hence, the yaphical representation is

isomorphic with the matrix representation.

Weletr. Ilic
.1 j

be the 1th column total of C. Of course, r

ij be the ith row total of C and s

z...

312.:

i is exactly the nunber of

lines issuing from the point Pi’ and sJ is the number of lines terminating

on the point P . Moreover, Z ri - Z 8;) - t, the total nunber of
1 J

J

10
directed lines. Finally, let the vectors .1; and g, with elements ri
and 31, respectively, be the two n—part, non-negative, ordered partitions
of t which represent, respectively, the marginal row and column totals

of 0.

FIGURE 1

Two representations of organizational configurations on 6 points

I

 

 

 

 

 

P 2

1 2 3 4 5 6

F— ._

1 0 l l 0 1 1

2 1 O O 0 O 1

5 0 O 0 O l 1

4 0 0 0 O O 1

P6 < A P 4

\ 5 1 c 1 o o 1

\ 6 1 c 1 o o 0
p5 .. n

(a) (b)
graphical rqiresentation matrix representation

Unless otherwise noted,all graphs will be on n points and
linearly directed (n-graphs), and all matrices will be n x n with 0's
on the principal diagonal and composed of 1's and 0's elsewhere.

1 2. A decomposition of the space of all n-gaphs (or n x n matrices).

We consider the decomposition of the space of all n—graphs in a
way which seems best adapted to the problems which arise in the investi-

gaticns in group organization theory.

Let us introduce the following notations:

Sample space, the space of all. n—graphs (or n x n
matrices, of the type defined).

[A
Iv
u

First-order subspace, the space of all n-graphs with
fixed number of lines, 3, (t - O, 1, 2, ..., n(n-l-)).

"‘I
iv

8
A
H

V

Second-order subspace, the space Of all n-g'aphs with
fixed I; (t is necessarily fixed).

a
2‘3

Second-order subspace, dual to «3(a).

Third-order subspace, the space of all n—graphs with
fixed 5 and g (t is necessarily fixed).

8
A
H
b
m

v

u

A diagram of the relationships among these spaces muld mpear

as follows?
I
:1 \ "
\ ,l h, ”(‘.3)
\\w(;) or 0(2) \\ ‘s‘
\ x t
:1, / __ ..., ...,. __ _____, __
/
/
/
/

We have a decomposition of [1 into subspaces of successively
higher order. First, ['1 is partitioned disjointly and exhaustively
into a sum of first-order subspaces, flt’ according to

CO

1 2.1 [1 - 1 [it -

Next, there is a decomposition of __ by a disjmctive partitioning into

21.,
a sum of second-order subspaces, w(;) or 00(3). according ‘30

1:2.2 [it - 2:). Mg) - 1 Mg) .

where the summation is over all n-part vectors 1; (or g) with elements

r:l (or 83) subject to O fri fn—l (or O f s fn-l) and

3

Z r - Z s - t. Finally, we have a partitioning of «9(3) and

90(5) into sums of disjoint third-order subspaces 00(3, g), according

to

132.3(3) 00(1‘) ' Z— “)(E: E): and
(g)

132.5(1)) VO(§) " (Z). “(5: §):
3

where the summation is the same as for 1:2.2. We summarize the fore-

going in Table I.

13

TABLE I

A DEOOLLPOSITION OF T1113 SPACE OF ALL n-GRAPHS

 

 

 

_ Space Decomposition _
[2 _Q - 2} at
f). 52+, - Z we) - 2%)
(g) (a)
wok“) w(1;) - 2M5, g)
' (g)
we) «0(3) - Z we, 2,)

(g)

 

 

Double and triple disjoint decompositions are also indicated. For

example ,

We conclude with a lemna and some remarks on the equivalence

under pemutation of certain subspaces. A matrix C A is eguivalent

 

under permutation to the matrix C]... if and onlsr if there exists a per-
mutation matrix P such that C, - PC/A P'. Then, the graphs G ,. and Cy, ,
corresponding to the matrices C ,\ and Cl“ , are equivalent under alias,

or renaming of the points, in the sense that

14.

Pl (____> P2 P1 P
is equivalent to I ,
1’5é """5 P4 P5 P4
6, G»-

when P in G, is renamed P P in G, is renamed P

3, 2 1’ azndPsanGA lS

 

 

 

 

 

 

 

 

1

renamed P2. In the matrix representation,

"' 1 V‘ 1 P ‘ ' '1

O l O O O O l O 0 O l O O 1 0 O

1 0 0 0 - l O O O 0 0 l l O O 1 O

l 0 0 l O l O O l O O O l O 0 O

O O l O O O O l O l O O O O O l

L .J _ i _. .1 .. J °

CA P 0,. P'

We now proceed to the lemma showing the equivalence under permutation
of the subspaces u(1;, s) and w(P£, P).

mm 1.1. _I_i; the vectors 5 and g are any two; n-part, non-no ative,

 

 

ordered partitions 932 t, then the transformation PCP' is _a_._ one-to—one

 

 

lugging 9;; «Ma, §)9_I_1_ MP3. Pg)-
PRCDF: By definition w(£, g) contains all distinct matrices
having 1; and .3. as marginal totals. Let Ck 8, «0(5, 3), k = l, 2, ...,

n(;, g), and consider PCkP' - Bk' Multiplication on the left of ck by

P permutes the rows of Ck and multiplication on the right of Ck by P'

permutes the corresponding columns of Ck. Thus, multiplying Ck on the

.—

15

left by P am on the right by P' simultaneously permutes corresponding
rows and columns. The effect on the marginal vectors 5; and s of Ck
of the pre—multiplication by P and the post-multiplication by P' is
that the vectors 1; and g are simultaneously permuted. Hence, Bk has

row totals P; and column totals Pg, and
(l) Bk 5 «.0035, Pg).

Next, Ck )1 Ch implies that out) i cash) for at least one pair

1:) :1 _
(i: 3). NO", the transformation PCP' . B takes 0:95) into by; and
l l
(h) (h) (k) be) r t .
on into 131131. Thus, bi131 { 11:51 or a least one pair (11, :11).
Hence,
(2) CkfchfikaBh’

so that there are at least as many B's as C's.

Finally, similar arguments show that

(3) Bh 5 ”(Pg, Pg) 11:? Ch - P'BhP ,
and Ch 5 w(;, g) by definition. Combining (l), (2) and (3) completes
the proof.

We give, next, a useful corollary. Consider I, any subset of
{i} - $1, 2, ..., n} , and Pa), any permutation matrix affecting only

the components r1 and/or 8i of (g, g) which have suffixes in 1. Thus,

16

if r = d for i a I, we have that g is identical to 13(1)? This gives

1
us
COROLIARY 1.1. Let ; 31d 2 133 the same; 32 _in the preceding 1222.
_The_p_, _if r1 I d f_o_r_ '1 t, I, t_he_ transformation P(I)CP(I)" is g one—to-
flmamin 2.1: we, .8.) 2.2 we, P(I)§)'
We remark that this corollary together with standard methods of
enumerating permutations will permit the calculations to be materially

abridged in certain cases.‘

1:3. Enumeration of n-graphs (or n x n matrices).

For the enumeration of points in the various Spaces we will find
it more convenient to use the matrix fornmletion. Thus, the number of
matrices (graphs) in [2 is the number of ways in Which n(n-l) positions
may be specified as either zero or one. By elementary considerations,
the number of dLstinct ways this can be done is
1:3.1 q - 23ml) .

For matrices in [it we have _t ones to distribute over n(n-l)
places and the number of ways this can be accomplished is the number

of ways of specifying a partiCular t of the n(n—l). Therefore, the
nunber of matrices (graphs) in [It is given by

1:5.2 '11-. - (“1:”) ,

17

a
where( b) , b 5 a, is the binomial coefficient al/[b3(a—b)1]. As is

well-known, Z(n(:-l)) - 2n(n-1) .
t

In the enumeration of matrices in u)(£) we have, for each

1, r1 ones to distribute over n-l places. This can be done independently,

for each i, in (r21) ways and thus the number of matrices (graphs) in

w(;) is given by
n
1:5.3 «(5) = I"! (”’1) .
By a similar argument, the number of matrices in ”(3) is given by

1:3.4 Y\(3) I if (til-1') .

The number of distinct matrices (graphs) in «Kg, 53). n(g, g).
cannot be handled by any well-known counting methods. In the following
section, we establish a theorem giving this number. This theoren will

canplete the enumeraticns needed for the sample space decomposition

of the previous section.

1:4. The number of matrices (graphs) in 90%, Q.
We first consider a relevant bipartitional function. P.V.
Sukhatme [3 8’] considered, among others, the problem of finding the

nunber, Mg, .3), of possible ways in which the cells of a (J x 0’

\

18

matrix can be filled with zeros and ones so that
i) the column totals from left to right form the
parts of the partition 2 in some fixed order, and
ii) the row totals from top to bottom form the parts
of the partition ; in some fixed order.
He pointed.out that.A(§, ;) - A(§, g). It is also evident that
this number is not altered by addition of a sufficient number of
rows or columns of zeros to make tie matrix square. We may, there-
fore, take n - max( Q, a); then Sukhatme's matrices include those
corresponding to our matrices in od(£, g) as well as those which
violate the condition that only zeros appear on the principal (flagonal.
This led to the idea of deleting from.A(£, g) the number of matrices
having one or more l's on the principal diagonal. The main theoran
of this section expresses 7((3, g) as a linear combination of the known
A(3, 3). This was first accomplished.by a process of alternate inp
clusion and exclusion. However, the theorem and proof as given here
is a considerable simplification of the original argument.

The definitions of n(g, g) and Mg, 9.) are completed with the
conventions that both vanish identically if any component of the par-
titions is negative and that both are linearly additive over the par-
titions, e.g., ‘1[(r;, g) 0 (g', g')] .' V‘(;, g) * '\(£', g').

Sukhatme [as] in 1958 gave tables of mm) for P,Q partitions
of t for t - 1, 2, ..., 8. These are identical, for weights t - 1,
2, ..., 6, with tables given by Mackiahon [as] in 1915. MacMahon was

interested, among other things, in a different formulation of the same

5'

19

problan which Sukhatzne later considered. He showed that the re-
quired solution is obtained from the coefficimts in the expansion
of products of elementary (unitary) symmetric functions in terms of
monomial symmetric functions. The unitary symmetries are defined by

.k. z. ....k

i<j<. ..<k 1 3

whereas the monomial symmetries are typified by

“113172

P1 P P2 P8 PS)
(1311132200013: 8) 'Z-(Xil leoooxhzz fig 00.x“ SXV

-where the surmation takes place over all suffixes i, j, ..., v which
are difierent. Recently, David and Kendall [7 ] recomputed and em-
tended these tables through t I 12. Their tables were collated with
those published by Cayley [2.] in 1857 for weights t I l, 2, ..., 10
andbyDurfee [91in1882fort I12.

Sukhatme's paper [39, p. 587] gave, also, an algorithm for germ-
putation of a single A(P.Q) of am wei git and David and Kendall gave
a systematic method for partially extending their tables to those of
successive higher weights. Thus, implicitly, tie problem of evaluating
the number of graphs in was, a) is completely solved when we express
n(g, g) in terms of Mg, g).

We shall require operators 5: (i I 1, 2, ..., n; j I 1, 2, ...,n)
defined by

20

134.1 62(1‘1’ 00., ri, coo, rn; 81, 0.0, Sj, 000, Sn)
- (1‘1, 000, 1.1.1, 0.0, rns 31, 000, sj-l, coo, Sn).

The effect of the operation Si is to replace the double partition
(r, g) of ‘_t. by a cbuble partition of (tr-l) with the 1th part of a
and the 1th part of g each reduced by unity. If we take (J2 +52)

(1;, g) s 52(5, .3) + 3::(5, g), the obvious commutativity of the 5's

.2
serves to define every sum of monomials of the form 2, [E (51;) j']
(‘0 ’

as an operator on (a, 2). We note that the number of non—trivial terms
in the application of such an operator to (g, g) is necessarily finite

. m+k 3114-1.:
for finite .2 since both AHSi) (g, 3)] a c and ‘1[(Ji) (g, 5)] a o

for k > o and m - min(ri, sj)° Finally, we observe that any identity
in the Mg; g') and n(g", g”) is unaffected by application of these
operators to the partitions involved. Any operation on an A or '1 is
to be interpreted as an operation on partitions. Thus, the operator
filters through A or 71 .

mm 1-2. litre—_ers r2214. 2: Jimmie-£3,302:

negative, ordered partitions 23 _t then

:1
1:4.2 A(r, g I VliT—Rl + 5:)(r, g); .

21

PRNF‘: Every matrix belonging to the set enumerated by A(£,g)

has either no principal diagonal elements different from zero, or

one specific principal diagonal element different from zero, or two

specific such elements, etc.

This exhaustive disjunction of the set

gives subsets each isomorphic with a set enumerated by an n(r‘w ,Is_.. )

where the vectors fiend .3... are formed from 5 and .s_ by reduction by

unity of each component corresponding to a specified principal diagonal

element different from zero.

Then,

 

 

 

 

Thus, the matrices of the type

(ri-l )

22

A(r,s) - 11(r,s) + Zetfigen + ; uni when

+ 000 + [81 5e 000 Sn (r 8)] ,
Y 1 2 n I’-
and, rewriting the riglt-hand member,

n i
Mpg) “flare +81)(§.§)Z .

We next establish inverses for certain operators, as in

_—* _—

£19222
-1 2 3
134.3 (1 +531) ' 1 - S: + (82) "' (£3) + Coo .

’1 i 3’1
moor“: (1 + 52) (1 + Si)(g,g) -= (1 + 51) [(11.2) + (I; - 21.

i .. 93)], where 2.1 md 2;} as vectors with unit 1th and 1th componarts,

respectively, ard all others zero. Next,
(1 +Si)-1[(;.g) 4' (z - 131. g - 13.3)]
- (as) - (2’21, 21,) + (£421: r223) -
* (2'31: r23) - (Hui. r223) *

" (39.3,) s

25

-1
ani ( 1 + $2) is a left inverse. A similar proof shows that

-1
(1 + 51) is a right inverse.

Repeated application of 1:4.5 to both members of 1:4.2 gives
imediately the main theoren.

‘IHEOIEM 1.1. I}; 213 vectors 1; Ed. 3 5.33 E 1'9. n-part, non-

 

matrices _in “Kg, 3) 1122s these vectors _ag marginal totals i3 aven
1'1

n --1
1:4.4 v((1.5 g) I Ail? (1 + 5:) (g, 9.)] .

As a remark in application of Theorem 1.1, and to a lesser extent in
application of Lemma 1.2, we note that the series in 1:4.5 with i I 3
may be terminated at the 21th power, for each i, where 2i I min(ri,si),

since the functions Mg... ,3“) evaluated beyond this point all vanish.

Thus, in every case, 7(5, g) is evaluated by additions and subtractions

Of finitely many A(£¢ $324 )0

Some examples may be given to illustrate the application of
Theorem 1.1 and to show the order of magnitude of the numbers, 1(93).
Suppose we wish to compute the number of directed graphs on four points
where the numbers of lines in and out of the points are, respectively,
(2,5), (1,2), (2,1) and (2,1); i.e., we wish to canpute n(2,1,2,2,35,2,1,l).

The remark following the theorm gives us

24

w(2,1,2,2,5,2,1,1)
2
- 1%[1 - 53:1 + (51) n1 - 3:1[1 - 5:1[1 - 5:1(2.1.2,2;5.2,1.1)§.

Operating as indicated on the right-hand member of the equation

above, we have

2
7(2!192:235:23191) ' A(25,1;3,2,1 )

2
..A(22,1 32,12) - 1142535,?) - 2a(22,12;3,2,1)

+ 21(22,1;2,15) + 21(2,13;22,1) + 2A(22,l;3,12) + A(2.1353.2)

2
- A(2,1 33,1) - M14522) .- 2A(2,12;2,12) — M22314) - 2A(2,12;2,12)

+ M13524) + 1(1332,1) + 2A(2,1;13)

H
2 2
" AU. 31):

where tie terms on the right are, by lines, of weight 7, 6, ..., 2,
respectively. Note that the parts of the partitions in “(15, g) are
pairwise ordered, as also they are (necessarily) in the right-hand
menber of the equation preceding the one above. However, in the
evaluation of a single Mg“ ,g,‘ ), this is not necessary and some
combining of terms occurs. Srfldratme's tables for t I 7, 6, ..., 2

 

* I It
(1111, n22, ...) is conventional partition notation designating

s1 nl's, 112 112's, etc., in the partition.

25

give the required numbers. The sum of the 12 positive terms is 94,
the 12 negative terms total 91 and n(2,1,2,2;5,2,1,1) I 5. The
three graphs, in matrix representation and graph form are given in
Figure 2, below.

FIGURE 2

Representations of points inw(2,1,2,2,55,2,l,1)

 

 

 

 

 

 

1 2 5 4 1 2 5 4 1 2 3 4
r- - P - '- '-
1 0 0 1 1 1 0 1 1 O 1 0 1 O 1
2 1 O O O 2 1 O O O 2 1 0 0 0
5 1 1 O O 5 1 0 O 1 5 1 1 0 0
4 b1 1 O 0‘ , 4 *1 1 O (U , 4-1 0 1 21
P€_____P2 Pie—6P PQ.____)P
0, 5

5 P4 P5————9P4 Pee—P4

Obviously, in this case, the three graphs or matrices could
have been exhibited directly and the method seems cumbersome. By
way of contrast, 16 positive and 16 negative terms give
V1(1,1,1,l,1,1,1,1;3,2,1,1,1,0,0,0) I 4846 - 5705 I 1145; it is
no longer feasible to exhibit the distinct directed graphs on these

‘

26

eight points:
Table II summarizes the enumeration of n—graphs in the sample

space decomposition of Section 1:2.

TABLE II

THE ENIMERATION 111 TE SAL'IPIE SPACE DECCEPOSITION

 

 

 

Space Enumeration
f) Yl' 21101-1)
fl. '1. ' (“‘2‘”)
n n—l
w(;;) \(5) T?(ri)
n n—l
n gt.)
< ) < ) Agfic gi>'l< )(
“7 :5 3’2 ' *‘ 5’3
5- 1 1 i

 

 

Obviously, the following relations, corresponding to 1:2.1,
1:2.2 and 1:2.5, hold?

1:4.5 *1 - Z; qt
Z

1:4.6 "Lo -

l:4.7(a) ‘10:) I Z. “(5,2) , and

1:4,7(b) ”1(a) - Z. naps) .

Double and triple disjoint summations are also implied. For ex-

ample,

According to lemma 1.1 and the corollary following the lemma,
certain of the subspaces w(;, 3) fall into equivalence classes under

permutation. In particular, for the case r I d (i I l, 2, ..., n),

1
i.e., 1; I (d, d, ..., d) I (dn), we have, according to the corollary

to Lemna 1.1, that the structure for any g.‘ I (81:: , 82.< , ..., 8114. )

is independent of the order of the components 5 in 3* . Standard

14:

methods of enumerating permutations enable us to give the nunber of

n n
points (n-graphs) in U(;) corresponding to fix " (O 0,13,..“1: k),

‘0.

28

where the superscripts n1 indicate the number of integers i in

k k
ngith Zn. I'nand.Z:Ln,'-t,a$
i=0 1 1-0 1

1:4o8 ( n ) ' n(dn30n0,1nl,.oo,k%) 3
no’n1"“’nk

 

n
where ( ) is the multinomial coefficient equal to
110,111, 000,1}k
n!
F O
‘ I nil

In practice, the case :-1 - l (i . l, 2, ..., n), does occur
and for his reason it seems desirable to give a different and
simpler method of computing nun; 9 than by Theorem 1.1. For
this method, we shall need a simple combinatorial property of multi-
nomial coefficientso

IEMMA 1.4. m 3 E22 _o_f_'_ k positive integers, 81’82’"°’8k’

k
such that Z s:j " n, them-the multinomial coefficient
3'1

n k n—l

1 13"'

{11:1-3 .
whereu " a
""'"",ij Oifi/j

~~b

O D

29

 

 

 

PROOF:
1‘ “'1 a (11-1): (n-l)!
g ,M , (s am My WWI-~89
0.. i ij .0.
... (n-l):
+ + 31:82x000(8k"1fl

31(n-1) 3‘82 (11-1) 2". o o"'8k(n"'1)£

w
810 32:. O 0 8k}

 

 

n: , n
. :3 10008 I . .
31 2 k 81,82’...’8k

We now proceed to the counting theoran for the special case, r1 '- 1;

THEOREM 1.2. E r1 - l (i - 1’ 2, 000’ 11) 3113.21: VBC‘bOl‘

1 {a 1 for J - 1,400.,k
hag cogonents 83 such that 83 _ 0 for J _ k+l,...,~n £13

2: a -n. 3112:
3'1

’ n _ €n-2)‘ _ ——4—(27
1:4.10 ’10- 331,...,sn) 51'°°8k “21:3...” (11-2)

k ‘1:
- {-1) m 1 9
(n—2)

where ad - £5181 ...si i_§__t_h_e_1th elementary
“'— 11<12<...<13 1 21;)

30

(b) - n(m"1)eo 0(m"’b+l)o

metric function and m
PROOF: First, according to the corollary following lemma 1.1,

we observe that arranging the components of the vector g in any

specified order is no restriction. Second, according to Theorem 1.1

and the remark following the theorem, we have
n -l n .
hung) - A iTer(l-+S:) (fig); - ASP"? (1-g:)(1n,§)}

k i k i i
- A(ln,§) - Z 5131. A(1n,§) + Z. 511 512 A(1n,§)

- (-1)k Z. 5:: 511‘ M1 Ne) .
her-net

Next, we observe that A(ln;sl,...,sn ) is just the multinomial co-

Sr), which may be written( n s) since

efficient<n
81’ O O. , k

$1,000,3

5 = ... '- 5n = 0. Therefore,

k+l
n(lnfi) -( n )— Z ( n. )
31”"3Sk Jl‘l ...’si-uijl’...

k n-k
+ooo(-l)k Z. ( )
090,31 uj ’00. e

Jl<ooe<jk 9000‘“ 13k

.0.

p

31

By lemma 1.4, the first two terms of the right-hand member are

equal, and, hence,

k n—2
(1n; '
Y1 g) 1 (000’s. ,0..)

 

< -u. -u,
3l 32 1 ijl 132
k n-k
k
-...(-1> Z3 ( )
j <ooo< ...,s.-u -...-u ’00. o
l k :1. 1:11 13k
This expression is factored to give
1:
k 8i 31 31
Yul-113%) " __‘___T(n-2): 2 s s. - ilqzqsigr—l 2 3
$1000.31: il<12 1]. 12 T
k 8182...3k
+ooo("l) IE_2 3
(11-2) )
or, finally,

81.00081: W (11-2) .-
We complete this section with an example of the decomposition
in the qaecial case, r - 1. Table III enumerates the directed

1
grth (matrices) in the decomposition of tle second-order subspace,

32

8
VJ(1 ), into third—order subspaces, w(18, '8'“ ), where the par-

n n n
titions g“ are of the form (0 0,1 1,...,k k). Column 2 of Table

III gives the number of vectors equivalent under permutation to
g“ , i.e., the number of permutations of 8 objects in is sets of

which the first contains 11

O’ the second n1,..., and the 15th nk,

where 3% n1 - 8. Column 3 gives the number of matrices in
was,“ ) for a fixed g“ and is computed from 1:4.10 of Theorem
1:2. Finally, column 4 is the product of columns 2 and 3 and
gives the number of matrices in 90(18) for which the column totals
g are equivalent to us" .

Note that

8
) - 5,764,801

:3
H
O)
V
I
in
A
H's)
V
I
A
I-‘d

checks with

“(18) ' %n(18’§) " Z (110,011,111!) W183}?

(fix)

THE ENUMERATEON OF’POINTS IN THE 8

DECOMPOSITION OF oJ(l ) INTO SUBSPACES, oo(1

15153 III

:fia.)

 

 

“k

n
. O
3‘ (o ’00.,k )

(

(

8

n0,...,nk) ‘(lasg‘ )

 

(06,1,7)
(06,2,6)
(06,5,5)
(06,5)
(05,12,6)
(05,1,2,5)
(05,1,5,4)
(05,22,4)
(05,2,52)
(04,15,5)
(04,13,2,4)
(04,12,352)
(04,1,22,5)
(04,24)
(05,14,4)

5 5
(0 ,1

2 5
(05,1 ,2 )

52.5)

(03,15,5)
(02,14,22)
(0,15,2)
(13)

 

8 ) 8
110,...,nk 1(1 ;§¢)
56 1
55 6
56 15
28 20
168 12
356 46
536 85
168 130
168 180
280 93
840 264
420 566
840 562 .
70 864
280 536
1,120 1,145
560 1,758
168 2, 525
420 5,578
56 7,284
1 14,855

 

 

56

556

840

560
2,016
15,456
28,560
21,840
50,240
26,040
221,760
155,720
472,080
60,480
150,080
1,280,160
984,480
590,600
1,502,760
407,904
14,855

 

Total 5,764,801

 

33

54

1.5. Probability distributions.

We have discussed in some detail the structure and a decom-
position of the space of all directed n-graphs, the sample space,
[1. Consider a. probability measure, P, defined on [2 or a sub—
space of £1 . Together, [2 (or the subspace of [1 ) and P form
the probability space. In particular, we shall be concerned with
unifom probability treasures on fl , the first—order subspaces flt’
and the second-order subspaces w(£)[ w(§)].

Next, consider a random variable X whose danain of definition
is [2 or a subspace of [1 . The random variable is a single
valued, measure preserving mapping from its domain to its range
space. In particular, we shall be concerned with random variables
whose range space is a subset of the real line. Then, the probability
measure of a particular value of a random variable X is the proba-
bility measure of the inverse image under X. Hence, if the approp—
riate domain has a uniform probability measure on it, the probability
measule of the inverse image under! is obtained by dividing the
mnber of points in the disjoint subsets which were mapped under X
into the particular value on the real line by the total nunber in
the domain.

A large class of random variables treated in the social-
psychological investigations of group organization have as their
appropriate domain either [2 , the first-order subspaces [1 t’
or the second-order subspaces w(;)[ w(§)]. The context of the
particular investigation will determine the appropriate space.

55

Furthermore, many of these randcm variables have the property
that their values are constant over all the points of each subset
«21(5.‘ ,3“ ). Any random variable whose values depend on a, g
and 3 only has this property. A member of the class of random

variables just described shall be said to have Structure L. Thus,

 

we see that the disjoint and exhaustive decomposition of ['2
considered in Section 1:2 is directly related to Structure L.

The null distribution of a random variable x is the proba-
bility measure induced in the range space of X by the uniform
probability measure over the appropriate domain of X. Then, if
the random variable X has Structure L, we can construct ( if not
expediently) the probability distribution of I using the counting
methods of Sections 1:5 and 1:4.

A number of indices (which we shall discuss in Part II) have
been described in the literature, e.g., indices of group cleavage,
expansiveness, and integration. These indices, being functions of
random variables, are them selves random variables. The value of
each index is a number on the real line and is uniquely determined
by the configuration of the group, 1.6., the index assigns to each
point in the sample space a unique number on the real line. Thus,
if the index has Structure L, the probability distribution of the

index can be obtained.

1:6. Remarks.

The methods outlined give us the exact pmbability distri-

56

butions arising in the null case for a broad class of random
variables. It is unfortunate that, in some instances, the methods
are computationally cumbersome. This is true, even, for moderate
sized groups. In these cases, it is desirable to use simple ap-
proximations. However, it is important that the exact distribution
be available in order to validate any proposed appronmate distri-
butions.

The theory developed above is based on the very simplest
model of a group organization, i.e., a one-dimensional model with
a binary relation between pairs of individuals which is not
necessarily reflexive and takes only the values 0 and 1 in each
direction. As yet no generalization has been made to models with
relations which take more than two values nor to multidimensional

problems .

PART II

APPLICATIONS

2:1. Preliminaries.
In this part we shall examine various distributions of
random variables having Structure L and present examples of some
of the indices from the literature. We hope to cover every possible
situation, but, of course, not-every index. It is not relevant nor
is it within the scope of this thesis to consider the sociological
and/or psychological aspects of the indices. For a list of the
studies in the general area of index analysis see, for example,
the bibliography in LeadershiLand Isolation by H. H. Jennings [I7].
Through the remainder we shall consider only null cases in
which each graph in the appropriate sample space is equally likely,
i.e., a uniform probability distribution over the sample space.
In certain summations, to be encountered, we shall find it
convenient to introduce the following indicator functioni To a
set A we assign a numerical single-valued function I A of x, to be
called the indicator of A and defined by IA(x) I l or 0 according

asxaAorfo.

2:2. Distributions of indices on expansiveness of a social group.
One measure of “Group Expannveness”, equal to the total

number of choices made by group divided by size of group, is given

by Iocmis and Proctorii‘lyj in a contribution to Research Methods in

58

Social Relatims. In our notation the index is E I —z— . A more
appropriate name for this index might be Total or Gross Expansiveness.
The distribution problem, in the null case, is easily solved.
Clearly, the appropriate sample space is [1 and our random variable,
the number of distinct n—graphs with 3 lines, is constant over the
subsets [It in the disjoint and exhaustive decomposition of [2.
Tom, our random variable has Structure L and, according to Section

1:5 and the counting formulas of Section 1:3, the required probabilities

are given by
h (”I”)

2‘24 PW‘R) ‘— " m °
2

This is an example of a random variable with constant values on
Q, n 51.

An obvious extension of the notion of Total or Gross Expan-
siveness is that of Relative Variability of Expansiveness, i.e., the
“spread” of the numbers ri (i I l, 2, ..., n) of outgoing connections
for a fixed total nunber _t_. In this context the appropriate sample
space is fit.

As a neasure of Relative Variability of Expansiveness, one
miglt take as an index, the ratio of the standard deviation of
the ri's to their man, 3.

We proceed to establish the formula giving the probabilities
necessary to construct the probability distribution. First, w(;)

is ”equivalent under pernmtaticn" with (0(3). Next, the number of

I.

II.

59

permutations in which 3 objects can be cdivi&d into .15 sets of which

the first contains no, the second n1, etc. , is given by the multi-

ncmial coefficient

 

(a )‘ nofllfiim’ikj’
O’nl’ ’“k

Thus, the probability, in the null case, of obtaining a t—line
graph with vector 2: equivalent under permutation to r' I

n0 n1 nk k k
(0 ,1 ,oee,k ), £- ni .nand Z in .t, is gmby
i-O

i=0 i
. 1
2:2.2 P(r'°t) -(n n ) 332 “- ’1‘ Eva) .
9 ”1: 7t """“""!'n0....nk W

0’n1"'" t

This is an example of a random variable having constant values on

2:15. Distribution of the number of isolates.
An isolate is an individual rqaresented in the graph ty :1

point Pi with no terminating lines and in the matrix by a column

of all zeros, i'.e., I O in the vector 3. Individual 4 is such

Si
an isolate in Figure 5.

40

FIGURE 5

Two representations of organizational configuration
on four points

1 2 5 4
r. ._
Ple—9P2 10 1 l 0
2 1 o o
31 o o o

P P
5 4 41 o o g

 

 

(a) (b)

Lazarsfeld, in a contribution to a paper by Moreno and
Jennings [17], in 1958, gave the expected (mean) number of isolates
for the case r:L - d (i. - l, 2, ..., n) as

n-l
n[ (n-d-1)/(n-1)] ,

but he made no attempt to mlve the distribution problem. Bronfen-
brenner [ I ], in 1945, gave (without proof) an incorrect version of

the distribution function for the case r - d (i - 1, 2, ..., n),

i
which he claimed was "developed deductively and chedced by anpirical
methods.“ Edwards [l0], in 1948, gave correctly, for the same case,

the probability of the maadmnn possible number of isolates,

2

n ) @351)th

n-l-d

5.1 P(n - l - (1 isolates) - ( n
K“)
, d

The exact probability distribution for this case was obtained
by Katz [:9], in 1950. The probability of exactly _15 isolates

is given by
n-d-l 3
. . _ k'tl

j n-;) -n

m s -(§)(";") (”3-1) (:1) .
Using the methods developed in Part I we can now extend

this result to the general case where the 1th individual has r

i
outging connections, the r1 being not necessarily equal.

In most contexts, our sample space is w(;) and, in the
null case, we desire the number of graphs having a specified nunber
of points with no terminating lines, i.e., a qwcified number of
zeros in the vector g. Our random variable X, the number of zero
sj's, is constant over the subsets w(;, g) in the decomposition of
w(;), i.e., has Structure L. Hence, according to Section 1:5 and
the counting formulas of Sections 1:3 and 1:4, file probability of

exactly 5 isolates is given by

Z: (s) (ms)
(:2) ‘k ‘ '1 " "

 

2:3.5 P(x - 1:35) -
4(2)

n --1
Z IAk(g)A31;T (1+ g1) (3,3)}

(g)
136:)

 

42

where Ak is the set of oo(£,§) such that the vectors 3 have exactly
)5 components equal to zero.

Thus, the chance (probability) distribution of any index (pro-
posed for the study of group structure) which depends only on the
number of isolates in the group can now be constructed. One such inp
dex, equal to the reciprocal of the number of isolates, is given by
Leonie and.Proctor [39] as a.measure of "Group IntegrationF.

Using Table III (p.33) and 2:3.3, we have obtained in Table IV
the exact chance distribution for the number of isolates in a group
of eight individuals, each making one choice (one outgoing connection).
This was checked against the exact distribution obtained from Katz'
formula, 2:3.2.

TABLE IV

PROBABILITY DISTRIBUTlON FOR THE NIMBER OF ISOLATES
IN A CROUP OF EICHT INDIVIDUALS, EACH MG WE CHOICE

 

 

 

k P [exactly k isolates]
0 .002 573
1 .070 758
2 - .528 455
5 .418 875
4 .162 052
5 .017 019
6 .000 311

 

 

43

Finally, we remark that in some contexts the appropriate
ample space might be the larger space [it’ However, our counting
methods of Sections 1:5 and 1:4 will still give us the required
probabilities necessary to construct the probability distribution.
In this case, the pmbability of exactly _l_c_ isolates is given by
g) E3 1%(3) nope)

Wt

 

2:3.4 P(XIkgt) I
.. 1 Z. Z. I (mirth (1+gi)-l( 3)}
n(n—l) ( a l i E’. ’

where the set Ak is the same as before.

Thea are examples of rmdom variables with constant values

on Mr, g) in oa(r) or (it.

2:4. Distribution of maximum 8 .

J
In youp organization theory, individuals who have the maldmun

8:) are sonetimes called stars. The problems of determining whether

a group cmtains a star or stars might be handled by consideration
. of the conditional distribution of the vector g of C, given the vector

5. The distributions of the maximun s the two maximum 83’ etc.,

3,
could then be used for determining stars.

lhe distribution problem, in tie null case, for maximum 5;)

is analogous to the one for isolates. Our random variable, max 3:],

44

is constant over W(;, g), i.e., has Structure L. The appropriate
sanple Space is 90(3) or [2t depending on the context of the par-
ticular investigation. Thus, the probabilities that max 3 I k are

3
given for the two situations by

g) IBk(§)‘1(g.§)

”\(27

 

2:4.1(a) P(max $5 = kgr) I

 

n -l
g). IK(§)A§.TJ(1+S:) (9.55)]
. a

n n-l
1;“ r)
and

2:4.l(b) P(max s '3 ht) "

 

l

(Mil-1)) ’

 

respectively, where Bk is the set of w(£, g) sudl that the vectors
g have the masdmun component . ‘ equal to 10.

These are, again, examples of random variables with constant
values on n(g, g) in 50(5) or fit.

Using Table III (p.33) and 2:4.1(a) we have obtained in Table v

45
the cumulative probability distribution function for max 83. in a

group of eight individuals, each making one choice.

TABLE V

CMUIATIVE PROBABILITY DISTRIBUTION FOR MAX 8
IN A GROUP OF EIGHT INDIVIUJULIS, EACH MAKING ONE CHOICE

+

 

0k P[max s4 5 1:)
J

.(D2 573

NH

.515 275
.918 897
.992 258
.999 582
.999 990

dorms-o:

1.000 000

 

 

2:5. Remarks.
The major classical problems not having random variables with
Structure L are
(i) the distribution of the number of mutuals, and
(ii) the problem of cleavage in a g‘oup.
A mutual is a pair of individuals id such that a connection eldsts
from _i_. to l as well as from 1 to _i_._, i.e., symmetrically placed 1's
in the natrix C. Cleavage is measured by the relative intensities

46

of in-group and out—group connections in the situation where we
have a group divided into subgroups by some external factor, e.g.,
by sex, or by nationality, or by management-labor. These problems
will not be considered in this thesis.

We have presented nethods for obtaining the exact distributions
of a large class of random variables. In the cases of certain of
these, e.g., the number of isolates and of the maximum 53’ the com-
putations necessary to construct the distributions for even moderate
sized groups are lengthy. For this reason we shall, in the next
part, suggest simple approximate distributions for these variables.
Even though the exact distribution mar not be used it is important

that it be. known in order to validate any suggested approximations.

PART III

APPROXIMATE DISTRIBUTIONS

5 :1. Preliminaries .

 

In Part II, two of the exact distributions considered, namely,
distribution of number of isolates and distribution of maximun Sj’
are extremely tedious to compute using the counting methods of Part
I. From a field investigator's point of view, the practical problems
of investigation require simple, easily-computed distributions.
Therefore, with this in mind, we shall suggest for each of the above-
mentioned variables a suitable approximate (histribution. Also, we
shall propose an index which seems reasonably suited for measuring
the phenomenon of "concentration of choice" and suggest an easily-
computed approximate distribution for it. Further, we shall make
certain computational checks of each of the approximations against
the exact distributions. These checks are not to be construed as
answering all the field investigator's questions pertaining to the
use of the approximations but, only, as giving an indication as to
the relatiw orders of magnitude of the approximations. It will, of
course, be necessary for field investigators to examine more closely
and in greater detail the suggested approximation with reference to
the size of the youp, the numbers of choices made by the individuals,
etc. It is beyond the scope of this thesis to consider in any detail

these latter problems.

48

5:2. Approximate distribution of number of isolates.
Katz [M], for the case ri I d (i I l, 2, ..., n) gives the

limiting distribution as Poisson with P [exactly k isolates]

k n-l

..A
A , where )~ I n(l - L) . However, for moderate
n91

‘ e k!

 

values of g, Katz points out that the approximation is not too good.
On the other hand, he shows that a binomial approximation is quite
good. We shall modify the formulas given by Katz to include the
general case where all the ri are not necessarily equal.

First, we shall find it convenient to give a different form of

the exact distribution than that obtained in Section 2:3. Let r2.L

3
be the number of choices made by individual 13.. Then, by a straight—
forward generalization of the exact distribution given by Katz for the
case ri I d (i I l, 2, ..., n), we have that the probability of exactly

E isolates for the general case is given by

max(n—l-ri )

 

 

3 5 k+h h
5:2.1 pm a, (--l) (k) sh ,
(r. )
h (n-h) 1: n (nPh-1)(rij)
‘where sh - 25; 'T—[' (r '1—1— ( y 3 the

. ._ 1'
j‘l (n_1) lJ J—h+l (UPI) ij

suzmation being over the (2) distinct combinations of _h ij's at
a time.

Kata points out that Fréchet has shown that the factorial

49

moments are given by

Hence, rewriting S as

 

 

h

n
(r. )
(n—h)hTT (n-h-l) 1;) h

i=1 1

511' n (r.) iii—LINN ’
TT(n-1) 1;) 3
321

the summation being the same as before, we have that the mean is

 

(r.)
(n-1)TT(n-2) 1
i
5:2.2 mean I E a Z l .
TT(n-1)(ri) 1 n-l-ri
i

For the case r1 I d for all i, Katz gives the mean as

a n-l
mean I x I n(l - 5% . We obtain from Katz' formula for the

mean the following expression

_ n , n-l
5:2.8 mean I x I —_—I (d

(n-1)n_ ’

50

t
where d I n-1-d. Rewriting 3:2.2, we have

5:2.4 _I n TT ')(—1"‘ 1 J 9
" T'w-‘IIS-fi n {-3)

t
where ri I n—l-ri. Letting G and H stand for the geometric and

9
harmonic means of the ri's, respectively, we have

5:2.5 x '—-—I-;'I T 0

When the variation in the ri's is not too great the geometric and

harmonic means are reasonably close together. Hence, the following
approximation is suggested for '55 :
n - l

n,
' (n-l)n-l G °

Ml)

53206

Note that for the case ri I d for all _i_, we rave G I H and 5:2.6 is
exact.
For computing the binomial probabilities, pi I b(i;m,p), Katz

uses the following two formulas to obtain m and p:

n—l

 

5:2.7 mp I i‘n(l - é) , and
n—2
. ' - _1. . 9:1 .. d
3.2.8 In 1 - ( n )[1 (xi-27(n-1-d) ] .

51

We remark that m is not necessarily an integer. Rewriting 5:2.7

I
and 5:2.8 in terms of d , we have

n , n-l
5:2.9 mp I ...—1:1 ((1 ) , and
(n-1)
l ' n-2
5:2.10 3}— = 1 - “'1’ n_ (34;) .
n(n—2) d

!
Next, we equate 5:2.3 and 3:2.6 obtaining an estimate for d

as

A:

$32.11 d ‘3 G 0

Note thatd' IGif riIdforalli. _

Finally, substituting 5:2,11 in 3:2.9 and in 3:2.10, we obtain
the formulas necessary for computing the binomial probabilities re-
quired for the binomial approximation to the exact distribution.

The formulas are
n Gnr-l

5:2.12 mp I——-—-I , and

(11-1)”-

.. (n-l)n-1 (gr-2

1
3:2.15 —- I l .
m n(n—2)n- G

As an example, we consider the distribution of the number of

52

isolates over the sample space w(l,l,2,2). For this small group
it is easier to obtain the exact distribution by examining directly
the 81 sample points (matrices) instead of by the method given in
Section 2:5. The probabilities were checked by using formula 5:2.1.
These values appear in the second column of Table VI.

For the computation of the approximate probabilities ,we obtain
p I .55856 from 5:2.12 and m I 1.16927 from 5:2.15. We then compute
the binomial probabilities, pi I b(i;m,p), i I O,1,2,...,([m]+1),
where [m] is the greatest integer in m (in this case, 1) using
pO I (l—p)m and pill/pi I (m-i)p/(i+l) (1-p) as used by Katz. These
approximate probabilities appear in the third column. It will be
seen that the fit is not too good. However, the size of the group
is very small. Since the computations of the exact probabilities for
a larger group are extremely lengthy, an example of larger magnitude

was not undertaken.

TABIE VI
COILPARIEBN OF THE EXACT AND APPROXBJATE
DISTRIBUTIOI‘TS OF TIE I‘I'LIIBER OF ISOLATES

FOR n I 4, r; I (l,1,2,2)

 

 

 

k P [k l(exact) pk (approx. )
0 .56790 .59521
1 .41975 .38870

2 .01255 .01609

 

 

 

55

5:5. Approximate distribution of maximum sj.

Since the variable under consideration in this section is ex-
pressed in terms of the column totals, 33., of the matrix C, it will
be desirable to examine the asymptotic distribution of the random
variable sj.

Iet ri I d for i I l, 2, ..., n. Then, assuming the <_i_ choices

. . . d
are distributed uniformly over the n-l places, i.e., Phi-3‘1] " 5:1 .9

the random variable 53 follows, for each 1 (j I l, 2, ..., n), the

binomial distribution
n-l k n-l-k
P[Sj=k5 n-l,p] a ( k ) P (1‘13) 9

Where p 5% .
Next, consider the limiting form of the binomial distribution
as p «9 0 subject to the condition that (n—l)p I d. Under these

conditions

It
P[stk3 n-l,p]~ em(1 41-5- 3

see, for example, Cramer [5’ , p. 205]. This is known as the "Poisson
approximation to the binomial distribution". Feller [ H , p. lll],
shows that the error is of the order of magnitude dz/(n-l).

Thus, the limiting form for the probability of SJ. 5, k (jIl,2,...,n)

is given by

k s
_ 3

5:5.1 P[s QC] I Z e d —-£1-1-- .
J 8 8O Sj.

3

If the s3 (3' I l, 2, ..., n) are independent, the limiting form for

the probability of the maximun s S k is

J

k 8:) n
5:5.2 P[max s S k] I Z emC1 -<-i-r .
j sj=0 sj.

Of course, the 53's are not independent since they have one constraint,

namely, i s. I t. However, 5:5.2 satisfies the requirement that
i=1 3

we have a simple, easily-computed approximation to the exact distri-

bution. Hence, we suggest the use of 3:5.2 to obtain the probabilities

necessary to construct the approximate cumulative distribution of

maximun 83.

As an example, we consider the cumulative probability distribution
over the sample space to (18). The exact distribution is obtained from
Table V (p. 45). These values appear in the second'column of Table VII.
The approximate probabilities are obtained from 5:5.2, using tables on
Poisson's Elgonential Binomial Limit by E.C. Molina [17]. These ap-

proximations appear in column three. The fit is not too good at the

low end but improves at the high end in which we are mainly interested.

55

TABLE VII
COEPARISON OF THE EXACT AND APPROXIMATE
'CUMULATIVE DISTRIBUTIONS 0F MAXIMUM s

5
FOR n I 8, 2; I (18)

 

 

 

k P[max sigcg (exact)] P[max 81$; (approx.)]
1 .0025? .08588
2 .51527 .51188
5 .91890 .85782
4 .99224 .97109
5 .99958 .99526
6 .99999 .99954
7 1.00000 .99992

 

 

 

If the rics are not constant, we take, for each i_, P[cijal]

1‘

I expected value I n-I . Then, for each 1, the Poisson approximation

 

takes the following form:

k
..}.3
5:5.5 P[s:j I k] I e 1.3L 9
Where Aj equals the total of the n-l expected values of the cij's
(t-r )

(i=l,;oo,j”1,j+l,ooo’n), 1080, A3 a Fur-fi- . Eve, then, suggest

56

the use of

_ k _ 83
5:5.4 P[max 8:] 5k] I 7T2: 9 jig—1'} ’
’ :i

jj‘o

as an approximation to the exact distribution. Further, if the var-

(t—r

iability of the ri's is not large, we might take >2] I A I (n 1

for all 1, where I" I mean of the ri's. Then 5:5.4 becomes

n
n k s.j
5:5.5 P[max s. S k] .I Z e”x Afi—E .
J sjao Sjo

We shall not consider any check of the suggested approximation for

the case ri not all constant.

5:4. Concentration of choice.

 

In studying the phenomenon of concentration of expressed choices
on relatively few individuals and/or of sparsity of choices going to
certain others, it seems reasonable to use as test criterion the
variance of numbers of choices received, suitably standardized. Ac—

tually, for the case r I d for all i with the one restriction,

i

n
2 r, - t, the sj's in their limiting form make up a Poisson series.
i=1

Hence, we shall use Fisher's [l2 , p. 58] "index of dispersion"

57

calculated by means of the formula

n 2
3:4.1 I =--}_- f_(s -‘s') ,
c s jIl j
._ 1 n
where s I mean I T 2: $3.. The sun, 10’ is large if either or
i=1

both of the previously mentioned effects occur in an appreciable
manner.

It has been shown by Sukhatme [36] and others that the index
of dispersion, calculated as in 5:4.1, follows the ordinary X2
distribution with n-1 degrees of freedom. Tables for this distribution
are readily available, for example, Cramér [ b’ , p. 559].

_ Sukhatme [37] has done some empirical work on the fit of the

7(2 approximation for smallish g and small values of the Poisson para-
meter, ) . His results (show that the agreement is "tolerably good",
even when A I l, for n I 15. In fact the fit is not too bad for

>s I 1, n I 10. However, in our situation we have a twofold approxi-
mation; first, the Poisson approximation and, second, the X2 approxi-
mation. Hence, it is necessary to make a further check of the ap-
proximation to the exact distribution. ‘

Some empirical work indicates that a X2 approximation with n—l
degrees of freedom, corrected for continuity as suggested by Cochran
[‘1‘ , p. 552], gives a good fit in our case. As an example, consider
a group of eiglt individuals, each making one choice. In colunn 2 of

Table VIII we have the values of the index, computed from 5:4.1, for

58

each gg in column 1. The exact cumulative probabilities are found
from the data compiled in Table III and are entered in column 5. The
approximate probabilities appear in column 4 and were obtained from
Tables for Statisticians and Biometricians [3? , p. 26]. This check
indicates that the approximation is reasonably good even for quite
small groups making small numbers of choices. Thus, we are in a

position to recommend the use of the above test criterion.

TABIE VIII

59

COLTPARISON OF EXACT AND APPROMEATE CmIUIATIVE DISTRIBUTIONS

OF 11 xx OF CONCEI-I'I'RATION OF CHOICE FOR 11 = 8, r; - (18)

 

 

 

g“ I; P[ Ics I; 5 (exact) ] P[Ics I5} (approx.) ]
(7,1) 42 1.00000 1.00000
(6,2) 52 .99999 1.00000
(6,1?) 50 .99995 .99995
(5,3) 26 .99958 .99978
(4,4) 24 .99944 .99924
(5,2,1) .22 .99934 .99829
(5,15) 20 .99666 .99625
(4,5,1) 18 .99214 .99181
(4,22) 16 .98719 .98260
(52,2),(4,2,1?) 14 .98540 .96400
(52,12),(4,14) 12 .95968 .92789
(5,22,1) 10 .88699 .86158
(24),(5,2,15) 8 .80510 .74754
(25,12),(3,15) 6 .57254 .57112
(22,14) 4 .55401 .54004
(2,16) 2 .07335 .11500
(18) 0 .00257 .00517

 

 

 

 

SUMMARY

We have been concerned with the one-dimensional theory of

group organization as a complex of irreflexive binary relationships,

taking values '0 and 1, between the pairs of individuals. The problems

considered in connection with this theory were

(i)

(ii)

(iii)

The

1.

2.

3.

4.

the investigation of the appropriate universe or universes
of discourse,
the determination of the null distributions for certain
proposed indices of tie group structure, and
the development of simple, reasonably exact methods for
use by field investigators.
results obtained were:
a decomposition of the. total sample space was given which
clarifies the choosing of the appropriate universe of
discourse,
the machinery for counting the nunber of distinct points
in each of the subspaces in the decomposition of the total
sample space was developed,
the structure of random variables whose null distributions
are possible to obtain by using the developed counting
methods was expounded,
applications were given to classes of unsolved probability
distribution problens of group organization theory; in
particular, the null distributions were given for (a) in-

dices on expansiveness of a social group, (b) the number

61

of isolates in a group, and (c) the maximun SJ. in a grow,
5. simple, easily-computed, approximate distributions were
obtained for the number of isolates and for the maximun 53’

6. a test criterion for concentration of choice was proposed

and a simple, approximate distribution suggested for it.

The first kind of problem was solved by the first result in the
sense that it is now possible to examine the varicus sociometric
variables in the general framework of the decomposition of the total
sample space given here and, thereby, determine their appropriate
universes of discourse.

A combination of results 2, 3 and 4 solved the second problem.
The machinery of counting methods made it possible to obtain the
necessary probabilities for construction of the null distributions.
These probabilities were obtained by dividing the number of points in
the disjoint subsets in the framework of the decomposition of the
total sample space by the total number of points in the appropriate
universe of discourse. The discussion on the structure of random
variables enabled one to tell immediately whether a particular socio-
metric variable fell into one of the classes of random variables whose
null distributions can be obtained using tlre theory developed here.
The applications illustrated the above results.

The third kind of problem is partially solved by the fifth and
sixth results. This particular problem covers such a wide range of

possible investigatic us which might be undertaken that it was not

feasible to consider it in any great detail. Thus, we have given

62

only an indication of the type of solution one might expect. In
particular, result 5 contains one aspect of this problan in the
sense that approximate distributions were given and checked for two
sociometric variables found in the literature. The other aspect of
the third problem is indicated in the sixth result in that we pro-
posed a test criterion not found in the literature and.suggested
and checked an approximate method for obtaining its null distrir
bution.

Related problems considered in this research investigation
which led to this thesis include the null distribution of the number
of mutuals and the problem of cleavage in a group. These problens
were not considered here since neither of them has the appropriate
structure in the general framework of this thesis. However, it
should be pointed out that an iterative method of application of
the counting1meihods developed here enables one to solve both of
these problans.

The unsolved problems in connection'with this research are

(i) definitions of nonpnull cases,

(ii) multidimensional problems,

(iii) relaxation of scale restriction,
(iv) extension of existing tables of the bipartitional
functions, A( g, g), and compilation of tables of
the functions, 1(g, g), and
(v0 more work on the development of simple, approximate

methods for use by field investigators.

63

In connection with the first of the unsolved problems, it
would be necessary for the social—psychologists to define what they
consider to be a suitable non-null case before any theoretical in-
vestigation should be undertaken.

As for the second problem, some exploratory work has been
done but few results are as yet known.

The relaxation of the scale restriction means, for one thing,
that we would have a matrix with elements not all 0's and 1's, and/or
a graph with strengths. The graphical representation becomes very
awkward and, for all practical purposes, the matrix representation
would have to be used. Also, any relaxation of scale means that the
distributions of random variables are virtually impossible to obtain.

Extension of existing tables of the bipartitional ftmctions
Mg, 3), would be desirable since the method for counting the nmnber
of locally restricted directed graphs involves these functions. Some
time has been spent on trying to arrange the computations necessary
for extending these tables in such a way that actual extension might
be carried out by some kind of large scale computational machinery.
The results were not very successful. However, we are able to express
in fairly compact form the numbers corresponding to sociometric situ-
ations in which each individual in the group makes precisely one
choice. Even more important for our use is the compilation of tables
of the functions, YUE’ 5). Of course, this problem depends to a
large extent on the success or failure of the preceding one. The table

of A's has a double entry for any weight, being entered for the

64

non—ordered partitions whereas the 1r{table would be the same kind
except the order of the partitions is now important. Wherefore,

the tabulation of the ’1's has the added complication that there are
many more entries for each weight since there are many'more ordered
partitions than partitions for any given weight. Nothing tractable
has been found for this latter problem except for the special case
where each individual makes only one choice.

The developing and checking of simple, approximate distributions
is, as we mentioned earlier, an extremely broad problem. It is one
where close cooperation between the field investigators and the
theorists would be desirable.

Before finishing this fliesis it should be pointed out that the
general.methods developed here may be applied to the theory cf‘communi—
cation networks and to other nearbsociometric problems.

Finally, the following two published papers by Katz and Powell
[19,20]:

(a) "A proposed.index of the conformity of one sociometric

measurement to another,"

(b) "The number of locally restricted directed graphs,"

include some of the results in this thesis.

l.

2.

5.

4.

5.

6.

7.

8.

9.

10 .

ll.

12.

14.

BIBLIOGRAPHY

Bronfenbrenner, U. "The measurement of sociometric status,
structure and development," Sociometry Lionog'aphs, No. 6,
Beacon House , New York, 1945.

Cayley, A. "A memoir on the symmetric functions of the roots
of an equation," Philos. Trans. Roy. Soc. London. Ser. A,
V01. 147 (1857), pp. 489’4990

Christie, L.S., R.D. Luce, and J. Macy, Jr. "Communication
and learning in task-oriented groups ," Technjial Report No.
_2_5_l_._, Research Laboratory of Electronics, I.I.T., 1952.

 

Cochran, W.G. "The x2 test of goodness of fit," Ann. Math.
Stat., Vol. 25 (1952), pp. 515-545.

Cramér, H. Mathematical Methods of Statistics, Princeton
University Press, 1946.

Criswell, J .H. "Notes on the constant frame of reference
problem," Sociometry, Vol. 15 (1950), pp. 915-107.

David, F.N. and 111.6. Kendall. "Tables of syrmnetric functions.
Parts II and III," Biometrika, Vol. 58 (1951), pp. 455-462.

Davis, R.L. "The number of structures of finite relations,"
Proc. Amer. Math. Soc., Vol. 4 (1955), pp. 486-495.

 

Durfee, 1.“{.P. "Tables of the symmetric functions of the twelfthic,"
Amer. J. Lfath., Vol. 5 (1882), pp. 46-61.

 

Edwards, D.S. "The constant frame of reference problem in
sociometry," Sociometry, Vol. 11 (1948), pp. 572-579.

 

Feller, W. An Introduction to Probability Theory and Its Appli—
cations, John ‘filey and Sons, New York, 1950.

Fisher, R.A. Statistical Methods for Research Workers, 2d ed.
Oliver and Boyd, Londcn, 1928. a

Forsyth, E. and L. Katz. "A matrix approach to the analysis of
sociometric data: Preliminary report," Sociometry, Vol. 9 (1946),
pp. 540—547. .

 

Harary, F. "The number of graphs and directed graphs," (Abstract),
Bull. Amer. Hath. Soc., Vol. 58 (1952), p. 664.

 

15.

16.

17.

18.

19.

20.

21.

22.

35-

24.

25.

26.

27.

66

Harary, F. and R.Z. Norman. "Graph theory as a.mathematical
model in social science," Research Center for Group Dynamics
Monograph, No. 2, Edwards Brothers, Inc., Ann Arbor, 1955.

 

 

Harary, F. and G.E. Uhlenbeck. "On the number of Husimi trees,
I," Proc. Nat. Acad. Sci. U.S.A., Vol. 59 (1955), pp. 515-522.

Jennings, H.H. Leadership and Isolation, 2d ed., Longmans,
Green and Co., New York, 1950.

 

Katz, L. "The distribution of the nunber of isolates in a
social group," Ann. Math. Stat., Vol. 25 (1952), pp. 271-276.

Katz, L. and J.H. Powell. "A proposed index of the conformity
of one sociometric measurement to another," Psychometrica,
Vol. 18 (1955), pp. 249-256.

 

. "The number of locally restricted

 

directedgraphszfi Proc. Amery Math. Soc., Vol. 5 (1954),
pp. 621-62 6.

 

Landau, H.G. "On dominance relations and the structure of
animal societies: I. Effect of inherent characteristics,"
Bull. Math. Biophysics, vol. 15 (1951), pp. 1—19.

 

. "On dominance relations and the structure of

 

animal societies: II. Some effects of possible social factors,"
Bullt Math. Biophysics, Vol. 15 (1951), pp. 245-262.

 

. "On dominance relations and.the structure of
animal societies: III. The condition for a score structure,"
Bull. Hath. Biophysics, Vol. 15 (1955), pp. 145-148.

 

 

Ioomis, C.P. and C.H. Proctor. "Analysis of Sociometric Data,"
Research Methods in Social Relations, Part 2, Chap. 17, The
Dryden Press, New York, 1951.

Luce, R. Duncan and Albert D. Perry. "A method of matrix
analysis of group structure," Psychometrika, Vol. 14 (1949),
pp. 95-116.

 

LIacLIahon, P.A. Combinatory Analysis, Vol. I, Cambridge Uni-
versity Press, 1915.

Holina, E.C. Poisson's Exponential Binomial Limit, D. Van
Nostrand Co., 1942.

67

28. Koreno, J.L. 'Who Shall Survive2, A New Approach to :he Problem
of Human Interrelations, Beacon House, New'York, 1954.

 

29. Moreno, J.L. and.H.H. Jennings. "Sociometric statistics of
social configurations,“ Sociometryguonographs, No. 5, Beacon
House, New York, 1945.

50. Otter, R. "The number of trees," Ann. cf’Math., Vol. 49 (1948),
pp. 585-599.

 

51. Palya, G. "Kombinatorische Anzahlbestimmungen ffir Gruppen,
Graphen und chemische Verbindungen," Acta.Hath., Vol. 68 (1957)
pp. 145-2540

52. Rapoport, A. "Outline of a probabilistic approach to animal
sociology: 1," Bull. Math. Biophysics, Vol. 11 (1949),
pp. 185-196.

 

55. . "Outline of a probabilistic approach to animal
sociOlogy: II," Bull. Kath. Biophysics, V01. 11 (1949),
pp. 275-281.

 

54. . "Outline of a probabilistic approach to animal
sociology: III," Bull. Hath. Biophysics, Vol. 12 (1950),
pp 0 7-17.

 

 

55. Shimbel, A. "Applications of matrix algebra to communication
nets," Bull. Math. Biophysics, V01. 15 (1951), pp. 165—178.

 

56. Sukhatme, P.V. "The problem of k samples for Poisson population,"
Proc. Nat. Inst. Sci. India, V01. 5 (1957), p. 297.

 

 

 

57. . "On the distribution of )02 in samples of the
Poisson series," J. Roy. Stat. Soc. (Suppl.), V01. 5 (1958),
pp. 75-79. '

58. . "On bipartitional functions," Philos. Trans.

 

 

Roy. Soc. London. Ser. A, Vol. 257 (1958), pp. 575-409.

59. Tables for Statisticians and.Biometricians, Part I, 2d ed.
Cambridge University Press, 1924.

40. Thrall, RtM. "Ranking of community organizations," Report No.
M-965, R—l, Engineering Research Institute, U. of Mich., 1955.

 

ulwwurulmlu

3