R. D. - Home | IMBS

PSYCH(~METRIKA~V0L. 14, No. 1 MARCH, 1949

A M E T H O D O F M A T R I X A N A L Y S I S O F G R O U P S T R U C T U R E

R. DUNCAN LUCE AND ALBERT D. PERRY GRADUATE S T U D ~ T S ~ DEPARTME~qT OF MATHI~CIATICS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Matrix methods may be applied to the analysis of experimental data concerning group structure when these data indicate relationships which can be depicted by line diagrams such as sociograms. One may introduce two concepts, n-chain and clique, which have simple relationships to the powers of certain matrices. Using ,them it is possible to determine the group structure by methods which are both faster and more certain than less systematic methods. This paper describes such a matrix method and applies it to the analysis of practical examples. At several points some unsolved problems in this field are indicated.

1. Introduction In a n u m b e r of b r a n c h e s of t h e social sc iences one e n c o u n t e r s

p rob l ems of the ana l y s i s o f r e l a t i ons h i p s b e t w e e n t h e e l emen t s of a grouP. F r e q u e n t l y the r e su l t s of t hese i n v e s t i g a t i o n s m a y be p r e - sented in d i a g r a m m a t i c f o r m as soc iog rams , o r g a n i z a t i o n cha r t s , flow cha r t s , a n d the like. W h e n the d a t a to be a n a l y z e d a r e such t h a t a d iagTam of th i s t y p e m a y be d r a w n , the ana lys i s a n d p r e s e n t a t i o n of the r e su l t s m a y be g r e a t l y exped i t ed b y u s i n g m a t r i x a lgebra , Th i s p a p e r p r e s e n t s some of the r e su l t s o f a n i n v e s t i g a t i o n of t k i s app l i - ca t ion of m a t r i c e s . I n i t i a l t r i a l s in the d e t e r m i n a t i o n of g r o u p s t r uc - tures ind ica te t h a t t h e m a t r i x m e t h o d is no t only f a s t e r bu t a l so less p rone to e r r o r t h a n m a n u a l i nves t iga t ion .*

The second sec t ion of th i s p a p e r p r e s e n t s c e r t a i n concep t s used in the ana ly s i s and a s soc ia t e s m a t r i c e s w i t h the g r o u p in ques t iom The t h i r d s t a t e s t he r e s u l t s ob t a ined and the f o u r t h g ives i l lus t ra - t ions of t h e i r app l i ca t ion . F ina l ly , sec t ion five con ta in s a m a t h e m a t i - ca l f o r m u l a t i o n o f the t h e o r y and d e r i v a t i o n of the r e su l t s p r e s e n t e d in sec t ion th ree .

2. Definitions 2.01. The t y p e s of r e l a t i o n s h i p s w h i c h th i s m e t h o d will hand l e

a re : m a n a chooses m a n b as a f r i end , m a n a c o m m a n d s m a n b, a sends m e s s a g e s to b, a n d so f o r t h . Since in a g i v e n p r o b l e m we conce rn

*Some of these examples have been worked out by the Research Center for Group Dynamics, Massachusetts Institute of Technology, in conjunction with some of its research.

95

96 PSYCHOMETRIKA

ourselves with one sort of relation, no confusion arises f rom ~eplac- ing the description of the relationship by a symbol : > . Thus, in- stead of "man i chooses man ] as a f r iend," we wr i te " i : > ]." If, on the other hand, man i had not chosen m a n ], we would have wr i t t en " i if: > ] , " using the symbol :~" > to indicate the negation of the rela- t ionship denoted by ~--- > .

2.02. Situations such as mutual choice of f r iends or two-way communicat ion would thus be indicated by i ---~ > ] and ~ > i, or briefly, i < - - > ]. We describe such situations by saying tha t a s y m m e t r y exists between i and ].

2,03. When the choice is not mutual, t ha t is i -~ > 3" or ] ~ > i but not both, we say an a n t i m e t r y exists between i and ].

2.04.* The data to be analyzed are presented in a mat r ix X as follows: the i ,] entry (x~) has the value of 1 if i ~ > ] and the value 0 i f i ¢ > ]. For convenience we place the re&in diagonal terms equal to zero, i.e., x~ ~--- 0 for all i. This convention, i ¢ > i, does not re- s t r ic t the applicability of the method, since there is little significance in such s ta tements as "Jones chooses himself as a f r iend."

Suppose, for example, tha t we had a gToup of four members wi th the following relationships: a ~--- > b, b = > ¢, b ~--~ > d, d ~ > b, c ~ > a, c ~ > b, d ~ > a, and d ~-~ > c. All other possible combinations of a,b,c, and d are related by the symbol ~ > . The X matr ix associated with this group is:

a b c d

b 1 0 0 1 ~ c 1 1 0 0 d 1 1 1 0

2.05. F rom the X mat r ix we extract a symmetric matr ix S having entries s~j determined by s~- = ss~ - - 1 i f x~i ~--- xj~ -~ 1, and otherwise s~j ~ sj~ ~-~ 0 . All the s~mmetrie---s in the group are indicated in the mat r lx S . The S ma t r ix associated with the above X mat r ix is:

0 1 0 0 7

i 1 0 0 1 0 0 0 0 | 0 1 0 0 J

*In the course of the present work it was brought to our attention that in "A matrix approach tc the analysis of sociometric data," Socio,metry, 19~6, 9, 340-347, Elaine Forsyth and Leo Katz have used matrices £o represent sociometric reIations. They considered a three-valued logic ra ther than the present two-valued one, and the operations on the matrices are different from the ones discussed in this paper.

R. DUNCAN LUCE AND ALBERT D. PERI%Y 97

To ~ndicate the i,] entity of the ma t r ix X ~, which is the nth power of X , we shall employ the symbol x~/~k Similarly, the i,] entry of S ~ is s~/~.

2.06. In the group considered above, we had a : > b, b ~- > d , and d : > c as three of the relations. I f the symbol___~ > indicates the relationship "sends messages to," .it appears t ha t a can send a mes- sage to c in three steps, via b and d . We call this three-step path a 3-chain f rom a to c . l~ather than wri t ing out the above sequence of relations, we may omit the symbol : > and simply wri te the 3-chain as a,b,d,c.

In a group involving more elements one might have the 5-chains: a,e,c,b,d,f and a,d,b,c,d,e. We notice tha t the first sequence involves five steps between six elements of the group. The second sequence also involves five steps but only five elements of the group, since the element d appears as both the second and fifth member of the sequence. Thus, al though these two five-step sequences contain different numbers of elements of the group,, they both have six members. Using this concept of membership in a sequence, an n-step sequence has ~+1 members.

These examples of 3-~hains and 5-chains suggest a general definition for a property within the group: an ordered sequence with n + l members, i , a , b , . - . , p , q , ] , is an n-chain f rom i to ] if and only if

i--- > a , a ~ > b , . . . , p ~ > q , q - ~ > ] .

2.07. When two n-chains have the same elements in the same order, i.e., the same members, then they are said to be equal, and otherwise they are distinct. I t is impol ' tant in this definition of equal- i ty tha t it be recognized tha t both .the elements of the group and thei r order in the sequence are considered. The two chains i,],k,l,p and i,p,k,],l are distinct though they contain the same five elements.

2.08. When the same element occurs more than once in an n-chain, the n-chain is said to be redundant. (Thus, in a group of elements any n-chain wi th n grea ter than m is redundant ) . The chains a,b,e,d,b,c and a,c,a,b,d,c,e are, for example, both redundant, for the element b occurs t~dce in the former and the elements a and c both occur twice in the latter. An example of a non-redundant 5-chain is a,d,p,b ,q,] .

2.09. A subset of .the group forms a c,lique provided tha i i t con- sists of three or more members each in the symmetr ic relation to each other member of the subset, and provided fu r the r tha t there can be found no element outside the subset t ha t is in the symmetr ic relation to each of the elements of the subset. The application of this definition to the concept of f r iendship is immediate: i t states t ha t a

98 PSYCHOMETRIKA

set of more than two people form a clique if they are all mutual friends of one another. In addition, the definition specifies that sub- sets of cliques are not cliques, so that in a clique of five friends we shall not say that any three form a clique. Although the word "clique" immediately suggests friendship, the definition is useful in the study of other relationships.

2.10. This definition of clique has two possible weaknesses: first, if each element of the group is related by ~- > to no more than c other elements of the group, then we can detect only cliques with at most c + 1 members; and second, there may exist ~Sthin the group certain tightly knit subgroups which by the omission of a few s>nn- metrics fail to satisfy the definition of a clique but which nonetheless would be termed, non-technically, "cliques." I t may be possible to alleviate these difficulties by the introduction of so called "n-cliques" which comprise the set of n elements which form two distinct n-chains from each element of the set to itself. This requires that the n-chains be redundant with the only recurring element being the end-point and also that all the relations in the n-chains be symmetr/c.

This definition means that the four elements a,b,c, and d form a 4-clique if the 4-chains (for example) a,b,c,d,a and a,d,c,b,a, both exist. These by the definition of n-chain require that the relations

a < = > b, b < - - > c , c < - - > d, d < - - > a

exist, but nothing is said about the relations between a and c, and b and d . The original definition requires, in addition, that

a < = > e and b < = > d

for a,b,e, and d to form a clique of four members. Thus we see that the definition of n-clique considers "circles" of symmetries, but it fails to consider the symmetric "cross" terms that exist between the members of the n-clique. These cross terms will be investigated, however, by determining whether any m of these n-elements form an m-clique.

The usefulness of the definition of n-clique can be judged only af ter experience has been gained in its application. This is not con- veniently possible at present, unfortunately, because the problem of the general determination of redundant n-chains has not been solved (see §5.09).

The most general definition of a elique-like structure including antimetries will not be discussed, for it is believed that this will not be amenable to a concise mathematical formulation.

3. Statement of Results 3.01. In X ~ the entry x~j (~) ~ c if and only if there are c dis-

R. DUNCAN LUCE AND ALBERT D. PERRY 99

tinct n-chains f rom i to j ( for p roof see §5.04). Thus, i f in the fifth power of a mat r ix of data X we find tha t the number 9 occurs in the third row of the seven th column, we may conclude tha t there are 9 distinct 5-chains f rom element 3 to element 7.

3.02. In X ~ the i ~h main diagonal en t ry has the value m if and only if i is in the symmetr ic relation wi th m elements of the group (§5.05). Since by the definition of a clique each element i in a clique of t members mus t be in the symmetr ic relation to each of the t--1 other elements, i t is necessary tha t x~ (:2~ > t--1 for i to b e in a clique of t members. We may not, however, conclude f rom the fac t tha t x~ (~ > t--1 tha t i is necessari ly contained in a clique of t members .

3.03. An element i is contained in a clique if and only i f the i ~ entry of the main diagonal of S ~ is positive (§5.06). The main diagonal te rms of S 3 will be ei ther 0 or even positive numbers in all cases, and when the value of the en t ry is 0 the associated element is not in a clique.

3.04. If, in S 3, t entries of the main diagonal have the value ( t--2) (~--1) and all o ther entries of the main diagonal a re zero, then these t elements fo rm a clique of t members (§5.08). I t also follows f rom the next ~tatement (§3.05) tha t if there is only one clique of t members then these t elements will have a main diagonal value in S s of ( t - -2) ( t - - l ) . The fo rmer s ta tement is, however, the more significant in analysis, for i t is the aim to go f rom the matr ix repre- sentation to the group structure. There is no difficulty in going f rom the s t ructure to the matrices.

3.05. Since by s ta tement 3.03 the main diagonal values of S s are dependent only on the clique s t ruc ture of the group, i t is to be expected tha t a formula relat ing these values and the clique structure is poss ib le . I f an element i is contained in m different cliques each having t~ members, and if there are d~ elements common to the k t~' clique and all the preceding ones, then

s ~ ~3~ - - Y~ { (iv - - 2) (iv - - 1) - - (d~ - - 2 ) (d~ - - I ) ) + 2 V=l

(§5;07). Thus, if we have three cliques: (5,7,9,10), (1,4,9), and (1,2,5,9,11), then d~ -- 0, for there are no preceding cliques; d¢___--~ 1, for only element 9 is common to the second and first cliques; and d~ -- 3, for clique three has the elements 1,5, and 9 common wi~h t h e first two cliques. Subst i tu t ing ~ ~ 4 , t~ - - 3 , ~ ---- 5 , d ~ - 0 , d ¢ - - 1 , and d3 ---- 3 and evaluat ing t I~ - fo rmu-~ for element 9, which is the only one common to all three cliques, we obtain

100 PSYCHOMETRIKA

s,~9 (~) = [ (4 - - 2) (4 - - 1) - - (0 - - 2) (0 - - 1)] + [ ( 3 - - 2 ) ( 3 - - 1 ) - - ( 1 - - 2 ) ( 1 - - 1 ) ] + [ ( 5 - - 2 ) ( 5 - - 1 ) - - ( 3 - - 2 ) ( 3 - - 1 ) ] + 2 - - 18 .

In the evaluation of this formula it is immaterial how the cliques are numbered initially; however, it is essential once the number ing is chosen tha t we be consistent.

3.06. The redundant 2-chains of a mat r ix X are the main diagonal ent r ies of X ~ (§5.09). Thus fo r a mat r ix

I O 1 0 1 0 1 0 0 1 1 0

X = 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0

I I 0 1 1 1 1 2 0 0 0 2

X~__ = 1 2 0 1 0 , 1 2 0 1 0 0 1 1 2 0

with the square

the matr ix of redundant 2-chains is

I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

To obtain the mat r ix of redundant 3-chains we compute the following matrix, in which the symbol R(: ~) s tands for the matr ix of re- d u n d a n t 2-chains:

X R c:2~ + R c 2 ~ X ~ S .

Deleting in this sum the main diagonal and replacing it by the main diagonal of X 3 gives the mat r ix of redundant 3-chains (§5.09). I f the main diagonal of X R ~ + R(=~X - - S is denoted by y(3) and the main diagonal of X C3) by Z (3), then let E ~3) = Z TM - - Y(a) and thus the mat r ix of redundant 3-chains, R (3), is given by

R ~3~ = X R ' ~ + Rc2~X + E TM - - S .

I t has not ye t been possible to develop formulas which will give ,the mat r ix of redundant n-chains fo r n larger than 3. Wha t work tha t has been done in this direction is presented in §5.09.

1%. DUNCAN LUCE AND ALBERT D. PERRY 101

3.07. The several theorems on cliques give a method that to some extent determines the clique s t ructure independent of the res t of the group structure. I t would be desirable to find a simple scheme tha t determines the clique s t ructure directly. Since a certain amount of knowledge in this direction can be obtained f rom S 3 , it is conjec- tured tha t possibly there is a simple formula relat ing clique structure to the numbers in S 3. As yet no such formula has been developed.

In a consideration of this problem, it was questioned whether certain aspects of the s t ruc ture would be lost in the multiplication, which, i f true, might make the discovel~y of the desired formula impossible. The following theorem shows tha t nei ther the clique ~tructure nor any of the proper t ies of S are lost in the matr ix S~: Any real symmetr ic mat r ix has one and only one real symmetr ic n ~ root if n is a positive odd in teger (§5.12). This theorem is somewhat more general than was required, since it does not res t r ic t the entries in the n t~ root to 0 and 1, and since it is t rue for any odd root r a the r than jus t the cube root. (In general the real symmetr ic even roots are not unique.)

This theorem suggests a fu r ther problem to be solved: to find a symmetr ic group s t ructure which will insure the presence of certain prescribed minimum n-chain conditions for odd n . To ca r ry this out it will probably prove necessary to discover a theorem that uses not only the realness and symmet ry of the S mat r ix and i t s powers, but in addition the fac t tha t only the numbers 1 and 0 may be entries in S .

4. Examples 4.01. As the first example, let us compare and analyze the fr iend-

ship s t ructure in the two following hypothetical groups. The matr ices are (where a blank entry indicates a zero) :

l,--i

g, A9

~"

m

I ....

I

I

b~

I

c~

,.,.i

oo

p~

~s

b ~

~b

d~

d)

b~

°.

¢b

l l

I -~

~.~

J f

--

I

-q

e~

-',-1

r j

~.

~

~,-

L~

~

..~

l..~

i,~

~b

t~

t~

°°

~ ~

,~.

I I

I ¸

--

I

i-.L

l-~

l.~

~--~

i ~

. i-

~

• ~w

.

; J

b~

-.I

O0

¢b

b~

,..I

O0

~J

b~

0


1

3 4 5 6 7 8 9

10

(§3.02). Thus, as f a r as symmetr ic relationships are concerned, these men are isolated f rom the group. In the same way we determine tha t 5 and 6 each have jus t one symmetr ic f r iendship relation (s~5 (~) - - s6J ~ -- 1, §3.02) which we determine to be 5 < - - > 6 f rom the S matr ix. TKe remaining elements in S 2 form a ra---fher dense set of quite large numbers, which means, roughly, a t ight ly kni t group.

In the second group, on the other hand, every man has a non-zero main diagonal in S 2. The men 2, 5, 7, 8, and 10 each have a single mutual fniend, which we determine to be: 2 < -~ > 6, 5 < - - > 1 , 7 < -- > 6 , 8 < - - > 9 , and 10 < ~- > 1 . Then since s6J 2 ~ 2 and since we-h-~ve jus t cited 6's two mutua-l-friends, 6 need not b~-~onsidered further. We note tha t the off-diagonal areas of this S ~ mat r ix are not so completely filled as group I, indicat ing tha t the group is not so t ightly bound,

The S 3 matrices indicate the differences in compactness of the structures quite clearly:

I I I 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 1 0

1 4 1 5 14 1 4 1 2 12 2 5 6 4 1 1 4 1 5 1 4 14 1 4 1 2 12 2

1 4 1 4 10 13 8 10 1

2 4 1 1 1 1 4 2 1 4 1 1 1 1

lr 2 3 5

5 4 6

9 1 10 4

1 , 2 2 1 4 1 4 13 1 0 1 0 8 j 2 1 2 1 2 8 10 6 7 1 2

1 4 1 2 1 2 1 2 10 8 7 6 1 1 1

I I J

Since the corresponding main diagonal te rms are non-zero, men 1, 2, 4, 7, 8, and 10 of group I are in cliques (§3.03). These, wi th 3 and 9 which have no symmetries in the group and 5 and 6 which are mutual friends, account for all members of the group. The terms ss8 (~ - - S~.o.~.0 (3~ - - 6 suggest a clique of four members; however, the-6xistence of other mai--n diagonal terms makes it impossible to apply the formula (t--2) ( t - - l ) (§3.04). Invest igat ing in S f irst the elements 1,. 2, and 4 because the i r columns have the largest values in the tenth row, we find tha t elements 1, 2, 4, and 10 form a clique of four mere- bers. In the eighth row the largest entries are in columns 1, 2, and 7, and an investigation reveals tha t 1, 2, 7, and 8 form a cl.ique of four men, which then overlaps the first clique by the men 1 and 2. In row four the largest entries are found in columns 1, 2, and 7. We then find tha t 1, 2, 4, and 7 form a clique of four elements which

104 PSYCHOMETRIKA

overlaps the previous two. All the men contained in cliques have been accounted for a t least once, and a check either with the formula for main diagonal entries (§3.05) or directly in the S mat r ix indicates tha t all the cliques have been discovered. This, coupled wi th what we discovered in S ~, completely determines the symmetr ic s t ruc ture of the first group.

For purposes of qualitative judgznent and a guide to carrying out analysis, we note tha t the first two rows of S 3 present an interest ing summary of the clique structure. The entries sl~ (3) and s2~ (3) have the largest values, next largest are in columns four and seven, and then finally in columns eight and ten. Men 1 and 2 are contained in all three cliques, 4 and 7 are each contained in two cliques, and finally men 8 and 10 are each in only one clique. This indicates tha t the magnitude of the off-diagonal te rms determines to some extent the amount and s t ructural position of the overlap of cliques.

h group II there are only three elements with non-zero main- diagonal entries, all wi th the value 2. This fits the formula ( t--2) ( t - - l ) with t ------ 3 (§3.04). Thus the men 1, 3, and 4 form a clique of three members. Returning to S ~, we see tha t there remains one unaccounted symmet ry each for men 4 and 9, hence 4 < - - > 9 .

In group I, the off-diagonal terms a re large in magn.itude and are quite dense in the array, wi th some rows completely empty or wi th single entries in the S 3 matrix. This indicates a closely knit group wit, h certain men definitely excluded. The S ~ mat r ix for the second group has f ewer entries of a smaller value indicating a tess t ight ly kni t s tructure, bu t i t has no empty rows and only one row wi th a single en t ry ; tha t is, i t has fewer people than gToup I who are not accepted by the group or who do not accept it.

A consideration of the mat~;ix X ~ S will give all the antimetries in the groups and complete the analysis of the structures.

It. is clear tha t this procedure gains s t rength as the complexity of the problem increases, for the analysis of a twenty-element group is little more difficult than tha t of a ten-element group.

4.02. The second example is a communication system compris- ing two-way links between seven stat ions such as might occur in a telephone or telegraph circuit. The number of channels of a given number of steps (i.e., n-chains in the general theory) between any two points and the minimum number of steps required to complete contact between two stat ions will be determined. Suppose the matrix of one-step contacts is:

R. DUNCAN LUCE AND ALBERT D. PERRY 1 0 5

1 2 3 4 5 6 7 _

2 3 4 5 6 7 1

which in this case is also the are given by X~:

1 1 1 - 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1

S matr ix .

1 1 - 3 2 1 3 1 4 2 5 2 6 1 7 0

and the three-step ones by X3:

Then two-step connections

2 3 4 5 6 7

1 1 2 2 1 0 t 2 1 1 1 0 1 1 4 1 1 2 3 1 1 4 3 2 2 ] 1 1 3 4 2 2 j 0 2 2 2 3 2 1 3 2 2 2 4

1 2 3 4 5 6 7 2 4 8 4 4 4 8 4 2 5 3 3 3 3 8 5 4 1 0 1 0 5 5 4 3 1 0 8 9 9 1 1 4 3 1 0 9 8 9 1 1 4 3 5 9 9 6 8 8 3 5 1 1 1 1 8 6

(2) (:2) From the former, the two connections 1 < ~- > 7 and 2 < ~ > 6 can- not be realized because x17 ¢~) ~ x71 ¢2) ~- 0 and x2+ (~> ~ x+~ <2) - - 0 (§3.01). The contacts a re possible i n - ~ r e e steps, however, since~ X 3 is completely filled. Thus two steps a re sufficien~ for most contacts and three steps fo r all.

In dete~Tnining the number of paths between two points it is desirable to eliminate redundan t paths: For two-step communicat ion this is done by dele*ing the main dia~onal of X ~. The remaining te rms represent the number of two-step paths between the stations indicated. The mat r ix of redundancies for three-step communication is given by R ~+~ "-- X R ~2) ~+ R ( ~ > X + E ~> - - S (§3.06), which works out to be:

106 PSYCHOMETRIKA

1 2 3 4 5 6 7

1 2 3 4 5 6 7 2 4 6 6 4 2 5 6 5 4 7 7

7 8 7 6 7 7 7 8 6 7

6 6 5 6 6 7 7 6 6 1

The matr ix of non-redundant three-step communication paths is X 3 ~ R ( ~ ) :

1 2 3 4 5 6 7

1 2 3 4 5 6 7

2 4 4 4 2 J 3 3 3 3 2 3 3 5 5 4 3 3 2 3 4 4 3 3 2 3 4 4 3 5 3 3 2 J 2 3 5 4 4 2

We notice that the three-step all redundant but that there tions. All other combinations ing them.

paths between I and 2 and 2 and 3 are are two-step paths for these combin~- have at least two t~ree-step paths join-

5. Mathematical Theory 5.01 To c a r r y out the following mathematicM formulation and

the proofs of theorems it is convenient to use some of the symbolism and nomenclature of point set theory. As there is some diversity in the literature, the symbols used are:

Sets are either defined by enumeration or by properties of the elements of the set in the form: symbol for the set [symbols used for elements of the set I defining properties of these elements]. When a single element i is treated as a set it will be denoted by (i), otherwise sets will be denoted by upper case Greek letters.

The intersection of (elements common to) two sets P and ¢ is denoted by F • ¢ .

The union of two sets F and ¢ (elements contained in either or both) is denoted by P + ~ . The context will make it clear whether the symbol i+ refers to addition, mat r ix addition, or union.

The inclusion of a set P in another set ¢ (all elements of P are elements of ¢) is denoted by P < ¢ . The negation is P <* ¢ .

If • < 2 , then the complement of ¢ with respect to 2 , ¢% is de,


fined by ¢ + ¢' - - ~= and • • ¢' ----- 0 where 0 is the null set.

The inclusion of ~ single element i in a set ¢ is denoted by i ~ ~ .

For any two elements i and ] of a set 2 and a subset ~9 of 2:

(i) + (j) < ~2 i f and only if i s ~2 and j ~ ~ . (i) + (]) <* t2 implies i e ;2' and /o r ] e ~2'.

The symbol 6i~ = l i f i = j - - 0 i f i=#]

5.02. Consider a finite set 3 of x elements denoted by 1 , 2 , . . . , i , . . . , ] , - . . , x for which ~here is defined a relationship - - > between elements and its negation ~ > having the properties:

1. E i the r i - - > ] or i ~ > ] for all i and ] e ~ .

2. i~=>i . Let a number x~j be associated wi th i and 3" such tha t

- - 1 i f i - - > ] ----0 i f i ¢ > ] .

A matrix X ---- [x~] is formed from the numbers x~s. I t wilt be found useful to denote the i,] entry of the nth power of X , X ~, by x~/~.

A symmetry is said to exist between i and ] i f and only i f i ---- > ] and ] - - > i , in which case we may wri te i ' < = > ] . For the mat r ix X this--requires tha t x~s ---- x~ ----- 1. If, however, e i the r_ / /= > ] and ] ¢= > i or i ¢ > ] and ] = > i then an antimetry is said to exist between i and ] .

The symmetric matrix S associated with the ma t r ix X is defined by S ---- [s,¢] , where

- - 1 i f x ~ j = x j i = l , i.e., i < : > ] . s~j ----- sj, ~__ 0 otherwise.

The i,] en t ry of the n *h power of S is s~/~. 5.03. Definitions:

1. An ordered sequence w~th n + l members, i -- ;~, ?~, --- , 7 , , ?**~ ----- ] , is an n-chain P f rom i to ] i f and only if

i = ~ ' - - > ~'~, ?~-- > ~s, " " , ? ~ = > ~',*÷l'~ ] •

(n) In brief, i - - > j indicates t ha t there exists an n-chain f rom i to ] , which may-EIso be enumerated as i -- 71, ?,2, --- , ~'~, ~'~,÷~ =- ] , or, when no ambigui ty will arise, as i , k , l , . . . , p , q , ] with the order- ing being indicated by the wr i t ten order of the sequence.

108 PSYCHOMETRIKA

2. Two n - c h a i n s F and • a r e equa l i f and only i f the r ta member of F equals the r th member of ~ , i.e., 7~ - - ¢~, for 1 < r < n + 1.

I f this is not true, t h e n / " and • are d i s t i n c t .

3. Each pair of elements 7~ and 7~ of an n-chain with 1 < k < m < n + 1 and 7~ : 7.~ is said to be the r e d u n d a n t p a i r ( k , m ) . A n n-chain is r e d u n d a n t i f and only i f it contains a t least one redundant pair.

4. The elements 1 , 2 , - . . , t (t > 3) form a c l ique 0 o f t members i f and only i f each element of O is symmetr ic wi th each other element of 0 , and .there is no element not in O symmetr ic wi th all elements of O.

This is equivalent .to

x i j = 1 - - &j f o r / , ] = 1 , 2 , . - . , t but not f o r i , ] = 1 , 2 , . - . , t , t + 1 , whatever the (t + 1) ~ element.

5.04. Theorem 1: x ~ (') = c i f and only if there exist c, distinct n-chains f rom i to ] . Proof: By definition of mat r ix multiplication

X~j (n) ~: ~ • . . ~ Xi~X~ . . • XmXcj,

with the summations over n--1 indices. Suppose tha t the indices have been selected such tha t i , k , i , . . . , p , q , 3" is an n-chain f rom i to ] .

Then by definition 1 (§5.03)

~ i k ~ Xkl = ' ' " ~ Xpq- - - Xqi = 1 ,

and if the indices were not so selected then a t least one x~, ~-- 0 . Thus n-chains contribute 1 to the sum and other ordered sequences con- t r ibute 0 . Since the indices take on each possible combination of values jus t once; every dist inct n-chain is represented jus t once. If there are c such n-chains, then there a re a total of c ones in the summation.

5.05. Theorem 2: An element of ~ has a main diagonal value of c in X ~ i f and only i f it is symmetr ic ~' i th c elements of 2 . Proof: Let ~ be the set of ]'s for which i < ~- > ]. By definition

x . (~) = E x~jxj~ + E x ~ x j ~ = ~ ~+ ~,~. j e ~ j e ¢ "

~]~ "- c by theorem 1 (§5.04) and N~ - - 0 because i and ] are not symmetric for ] s ~', so ei ther x~j = 0 or xj~ ~--- 0 or both. Thus i f i is symmetric with c elements of 2 , x~j (~) : c .

R. DUNCAN LUCE AND. ALBERT D. PERRY 109

I f xi~ (2) ~ _ c , then by theorem I there exist c dist inct j ' s such tha t x~j----xs~ ~-__1, i.e., i < ---->__] fo r c ] ' s .

5.06. Theorem 3: An element i is contained in a clique if and only if the i t~ en t ry of the main diagonal of S ~ is positive. Proof : Suppose tha t i is contained in a clique O . By definition

(/)+(k)<~

S e l e c t ] and k s u c h t h a t (]) :+ (k) < O and such t h a t i # ] # k # i . Such elements exist b y the definition of a clique (definition 4, §5.03). I t is t rue by the definition o f a clique and of the ma t r ix S that: s~s ---- sj~ = s¢~ - - s~j : s~ = s~ - - 1 for such ] and k . Thus this choice of ] and k contr ibutes 2 to--~e summation, and because s~j > 0 for all i and ] there are no negat ive contr ibutions to the sum; therefore s~ (~) > 2 > 0 .

Suppose tha t s~ (8) > 0 . Then there exists at least one pa i r of elements of ] and k such tha t s~i = sj~ ---- sk~ ~--- 1 and this implies i < = > 3",3' < - - > k , and k < = > i . I f there are no other elements symmetr ic wi th i , ] , and k then these th ree fo rm a clique. I f there is another element symmetr ic wi th these three, then consider the set of four fo rmed b y adding i t to the previous three. I f there ~s no other element symmetr ic with these four, they fo rm a clique. I f there is, add it to the set and continue the process. Since the set 2 contains only a finite number of elements, the process must terminate giving a clique containing i .

5.07. Theorem 4: I f 1) O~ are cliques of t~ members, 2) the sets

zl~ ~--- O~ • (e~ :+ 0~ + -.. + O~_~) have d~ members , and 3) i is con-

tained in the cliques @~, z - - 1 , 2 , -- . , m , then

s,,(~) = ~ ( (t~ - - 2) (t~ - - I ) - - (d~ - - 2) (d~ - - I ) } :+ 2 .

Proof : By definition

- - ( i ) + ( k ) < Y ~

The set of all the pairs ] , k is the union of the following three mutually exclusive sets:

T~ [ ] , } I there exists v such tha t (]) + (k) < 0 , ; there does not exist ,a such tha t (]) :+ (k) < 0~ , (]) + (k) <" A~]

110 PSYCHOMETRIKA

N~ [ j , k I there does not exist a such tha t (j) + (k) < O,]

T~ [ j , k i there exists a such tha t (j) + (k) < 0~, (j) + (k) <* A~] .

1. For T1 then ei ther

a) (j) + (k) <* O~ for all a . This is not possible because (]) + (k) < or;

b) (j) + (k) < A~ f o r all a . This is not possible because A I = 0 ;

o r e ) (j) + (k) < O~ if and only if (j) + (k) < A ~ f o r a l l a . This is not possible because A~ = 0 . Thus gJ~ is empty.

2. (j) + (k) < T2 impties'suss.~s~ = 0 for SuSjkSkl ~-- 1 implies tha t i ; ] , and k are either a clique or a subset of a clique (by the a rgument of theorem 3), but (j) + (k) < gz implies ] and k are not contained in any clique.

3. T~ gives tha t

( j)+ (k) <q,~

= Z Z Z s~jsi~s~ . V=l (j)+(k)<@ v

(j)+ (k)<*A v

We observe tha t : £2~[j,k 1 (j) + (/~) < O,] = D 2 [ j , k I ( ] ) + (k) < o.~, (y) + (k) < A,] + Q~[S,k i (]) + (k) < O,, (y) + (k) <* Av] and since .% - £2~ = 0, it follows tha t Y~ = Y~ + N or Y~ = N -- ~ .

~h Ft2 fl~ ~3 f~l fte

t?, is the set of all ordered pairs (j) + (k) < O,,. I f i C j C k ¢ i , then s~ = s~k = s~ = 1 , othetavise one of the s~ - - 0 . Since every O, contains t,, elements, there are %-aP~ ordered pairs sa t isfying these

conditions. Thus:

Similarly

z={o (d'- 2) I) ' o . 11 C o m b i n i n g t h e s e ,

since AI--O.

~a [ (tv - - 2 ) (tv - - 1 ) - - ( d v - - 2 ) ( d ~ - - l ) , ~ > l

(iv - - 2) ( t , - - 1) , v - - 1 .

R. DUNCAN LUCE AND ALBERT D. PERRY Iii

S u m m i n g ove r ~, gives

s.~ (~) = ~ ( ( t , - - 2) (t~ - - 1) - - (d~ - - 2) ( & , - 1) } Y = 2

+ (t l - - 2) ( t l - - 1)

~- ~ ( (t~ - - 2) (t~, - - 1) - - (d~ - - 2) (d, - - I ) } + 2. V=l

Since the entries szj (3) are uniquely determined from the entries of S by the laws of matrix multiplication, all valid methods of cal- culating s~ (a) will give the same result. Specifically, in the above formula the numbering of the cliques is immaterial.

Similar formulas to that just deduced may be given for the off- diagonal terms of S 3, but they are considerably more complex, and, to date, they have not been found useful in applications.

5.08. T h e o r e m 5: I f 1) 0 is a set of t m e m b e r s w i t h t > 3 , 2) s~/~) ~ ( t - -2 ) ( t - - l ) f o r i con ta ined in O , and 3) sj/~) = 0 f o r ] con ta ined in O', t h e n O is a clique of t m e m b e r s . P r o o f : T h e r e a re two cases:

1. i < = > 2" f o r all i , 2" s O , t h en 0 is a clique by def in i t ion 4 (§5.03) and t h e o r e m 3 (§5.06) , and i t ,has t m e m b e r s b y p a r t 1 of

the hypothes i s .

2. T h e r e exis t p and q s O such t h a t p and q a r e no t sy mme t r i c . T h e n b y def in i t ion

S~/~) = Z Z s~jsjks~ U)+(~)<o

+ Z E s~jsj~sk~. (j)+(k)<*O

I f s~jsj~sk{ ~- 1 , the e l emen t s i , 2", and k a r e a clique o r a subse t of a etique and thus by hypo thes i s (3) and t h e o r e m 3 (§5.06) t h e y a r e all con ta ined in O; t h e r e f o r e t he second sum ---~ 0 . I n t r o d u c e in 2 sufficient r e l a t i onsh ips p = > q to m a k e 0 a elique { of t me mb e r s . Sinee s{~ > 0 f o r all i and 2", t he i n t r o d u c t i o n of these spq - - 1 m u s t i nc rease the sum by 2 o r more , f o r a t l eas t two add i t iona l 3-chains a r e i n t ro du ced (i,p,q,i and i ,q,p,i) ; hence b y t h e o r e m 4 (§5.07)

S u (~) - - Y. Y~ si~sj~s~i - - 2 = ( t - - 2 ) ( t - - l ) - - 2 (j)+(k)<~

< ( t - -2 ) ( t - - l ) ,

112 PSYCH0~ETRIKA

which is contrary to hypothesis (2). Therefore 0 is a clique of t members.

5.09. Redundancies:

By definition 3 (§5.03) an n-chain is redundant if and only if it contains at least one redundant pair (k,m), where a redundant pair defines two members of the n-chain ?k and ?~ with ?~ -- ~,~ and k < m . I f these ordered subscript pairs (k,m) and the end point pair (i,]) (the latter not necessarily a redundant pair) are considered as sets, then five classes of mutually exclusive redundant n-chains may be defined which include all redundant n-chains:

1. The A~ class: There exists at least one redundant pair (k,~) and it has the property:

( k , m ) • ( i ,]) = o .

2. The B, class: There exists one and only one redundant pair (k,m) and it has the property:

( k , m ) • ( i ,]) = i .

3. The C~ class: There exists one and only one redundant pair (k,m) and it has the property:

( k , m ) • ( i ,]) = ] .

4. The D,~ class: There exist two and only two redundant pairs (k,m) and (p,q) and they have the properties:

( k , m ) • ( i , D = i

(p ,q ) • ( i ,]) = ] .

5. The E, class: There exists one and only one redundant pair (k,m) and it has the property:

( k , m ) • ( i , j ) = ( i , i ) .

If there are t n-chains i ~ ' > ] of the class A, from i to ] , then define a~s(") - - t . From these numbers the matr ix A(") = [a~/~)] is formed. Th-i~ is the matr ix of redundant n-chains--~ the class A , . If R (') is the matrix of redundant n-chains it follows, if analogous definitions are made for matrices of the other four classes, that

R ( ' ) - " A ('° + B ('*) '+ C ('~) + D ( ' ) + E ¢'° .

It follows directly from the definitions and the limitations on n that

R. D U N C A N L U C E A N D A L B E R T D. PERRY 113

R ( z ) r - - 0

A (8) - - 0 .

I t will now be proved t h a t D (3) = S . By the definition of the class D 3 , there exist two and only two redundan t pai rs ( l¢ ,m) and ( p , q ) , and they h a v e the proper t ies :

( k , m ) • ( i , D = i

( p , q ) ( i , ] ) - " ] .

These pairs may define in total ei ther three or four members of the 3-chain ( three members when m ~- p , but no fewer fo r i f / c - - p and m = q then ( k , m ) • ( i , ] ) = ( i , ] ) , which is con t r a ry to the definition of D:~). Suppose m = p , then ei ther i ---- y~ - - ] or i = y~ = ] , which is impossible fo r i ¢ > i by assumption. Thus m ¢ p . Wi th fou r members the re a re two possibilities fo r a r edundan t 3-chain= e i the r i ----- ?~, 7~ - - ] or i ----- ~,~, ~,~ -~ ] . The f o r m e r is impossible by the previous a rgumen t ; thus the only 3-chains o f the class D~ are of the form

i , r ~ , ) , ~ , ] - - i , ] , i , ] ;

tha t is, - - 1 if i < = > ]

d~/8) : 0 otherwise,

Therefore , by the definition of S , we have D (3) = S . I f the matr ices of redundancies up to and including R (~-2) are

known, then we can find A (~') by A (~) ----- X R ( ' ~ 2 ) X .

Proof : By the definition of the class A,~, a r edundan t n-chain of this class has the fo rm

(a) (b) (c)

~ = y l , y~ , ~ , y~ , - - , y,~ , - - , 7~ , ~'~,÷l -= ,

w h e r e a + b + c + 5 = n , k < m , a n d y k = y , ~ .

(~t-2) I t follows f rom the definition tha t p - - 72 > ~ ----- q is a redundan t n--2 chain, and each such dist inct n--2 chain determines no more than one dist inct r edundan t n-chain f rom i to ] . Thus the number of redundant n-chains of type A~ f rom i to ] is the sum over all combinations p - ~2 and q -- y~ fo r the number of r edundan t n : -2 chains f rom p to q, t ha t is,

(p)+(q)<,~

o r

A (~) = X R ( ~ ' 2 ) X .

114 PSYCHOMETRIKA

I f the mat r ix [e~/*)] is defined as

[e , / - ) ] = X R ( , ~ ) X + D oo

then the relations

A ('~) + B (~) ~+ D(~) = R ¢ ~ ) X

A (~) + C TM) + D('O = X R ( ~ x ) E TM) = [~j (x~fcr - - e#.~ ) ]

follow through an enumerat ion of cases and by using similar pa t te rns of p roof to tha t j u s t given.

These various relations permit the specific conclusions:

R(2) = [&ixi/~)] = E(2) R (3) = X R (2) + R ( 2 ) X + E (3) ~ S

and the general result

R(~) = X R ( ~ ) + R ( ~ - I ) X _ X R O ~ 2 ) X

+ EOO - - D(~) .

This la t ter expression is not useful in its present form because D (") has not been expressed in te rms of the matr ices of redundancies up to and including R (--t). This problem of the determinat ion of the mat r ix of redundant n-chains is lef t as an unsolved problem of both theoretical and practical interest .

5.10. Uniqueness:

In certain applications it is desirable to know whether a power of a ma t r ix uniquely determines the matrix. This is not t rue in general, fo r Sylvester 's theorem gives a multiplici ty of n t~ roots of a matrix. The matr ices being considered are ra the r specialized, however, and it is possible tha t some degree of uniqueness may exist.

The following two theorems indicate cer tain sufficient conditions for uniqueness. Since these theorems do not utilize completely the special characteris t ics of the matr ices in this study, i t is probable tha t more appropr ia te theorems can be proved.

5.11. Theorem 6: I f p and q a re posit ive integers, i f two integers a and b can be found such tha t a p ~ b q - - 1 , and if X is a non- s ingular matr ix, then the powers Xp and X~ uniquely determine X . Proof : Suppose tha t there exist two non-singular matr ices X and Y such tha t X~ = YP and Xq = Yq. Then X ~ = Y~ and X ~q = Y~q. Now, fo rm X b q Y = y b q y = y~q÷~ - - y~p, since ap - - b q = 1. Similarly X ~ X = X bq÷~ = X ap. But since X ~p - - Y~ it follows tha t X ~ q X - - X b q Y .


Since X is non-singular, IX~I ¢ 0 , and thus there exists a unique in- verse of X % X -~q, such tha t X ~ X bq = I ; therefore X = Y .

5.12. Theorem 7: I f n is a positive odd integer and S a real symmetric matnix, then there is one and only one real symmetric n ~I' root of S .

Proof: 1. There is one such n t~ root.

Since S is real and s~m~netric there exists a real orthogonat matr ix P such tha t P ' S P = D ( P ' is the transpose of P) is diagonal with real entries d~ which are the characterist ic roots of S .* Assume P is so chosen tha t dl~ ~< d~ < .-. < d ~ . Let B be the diagonal matrix of the real n tj' roots of the elements of D , i.e., b~ - - real ( d ~ ) ~/" ,

so B ~ = D . (1)

Define R = P B P ' . Then R " = S , for

R ~ = ( P B P ' ) ~ ~-- P B ~ P ' = P D P ' =- S .

Since B ~s real and diagonal and P is real and orthogonal, R is real and symmetric.

2. There is only one such n th root.

Suppose there exists a real symmetr ic matr ix /71 not equal to R such tha t R~" = S . Then there exists an orthogonal mat r ix Q such that Q ' R ~ Q -~ T is diagonal in the characterist ic roots of R1, and ordered as before. Consider the n th power of T:

t n - - ! T ' ~ : ( Q R ~ Q ) - - Q R ~ Q ~ - Q ' S Q

: Q ' P D P ' Q = (P 'Q) 'D(P 'Q) T ~ -~- U ' D U ' 7 - (2)

where U is the orthogonal mat r ix P ' Q . Since U' = U -~, T" and D are similar, and hence have the same characterist ic roots . t Because they are diagonal in the characterist ic roots, ordered in the same way, they are equal:

D - - T . . (3) Subst i tut ing (3) in (2)

or

D - - U ' D U

U D " - D U .

*MacDuffee, C. C. Vectors and matrices. Ithaca, N. Y.: The Mathematical Association of America, 1943, pp. 16~-170.

tIbid., p. 113.

116 PSYCHOMETR!KA

By definition of mati ix multiplication this means

u~jd~k = Y~ d . u j ~ . J J

Since D is diagonal, this reduces to

o r

u~k (d,k - - d . ) = 0 .

Since the dkj: are real and n is odd, equation (4) implies

u ~ [ (G~) v~ - - ( d . ) ""] = 0

where the (Ga)~/= are real. Thus by the definition of B ,

U B -~-- B U

o r

By (1) and (3)

( 4 )

B = u ' B u . (5)

T ~ : D : B ,~ ,

but by construction T and B a r e both real diagonal matrices and n is odd, so this implies

T = B .

This substituted in (5) gives

T - - U ' B U - - Q ' P B P ' Q

o r

Q T Q ' ~ - P B P ' .

But Q T Q ' ---- R1 and P B P ' = R by definition ; therefore

R I ~ R .

6. A c k n o w l e d g e m e n t

We wish to acknowledge our indebtedness to Dr. Leon Festinger, Assistant Professor of Psychology, Massachusetts Institute of Te&- nology, for his kindness in directing this research to useful ends, en- couraging the application of this method to practical problems, and providing many constructive criticisms of the work.

Documents

R. D. - Home | IMBS