f3?
THE NUMBER OF COMPLETE CYCLES IN A COMMUNICATION NETWORK *
by
Frank Harary and I a n C Ross
Research Center f o r Group Dynamics
University o f Kichigan
The w r i t i n g of t h i s paper was supported, i n ' part by a grant from the
Rockefeller Foundation.
INTRODUCTION
There are many problems o f t h e o r e t i c a l and p r a c t i c a l i n t e r e s t
associated w i t h communication networks o r , e q u i v a l e n t l y , sociograms.
We wish t o deduce a c r i t e r i o n ' f o r the -existence o f a path i n a network such that a rumor or some item o f information i n i t i a t e d by any
member o f the group w i l l r e t u r n t o him a f t e r having passed through
each of the other persons exactly once. Such a path w i l l be c a l l e d
a complete cycle. We solve t h i s problem by presenting a t h e o r e t i c a l
formula f o r the number o f complete cycles i n any communication network.
Our formula i s based on previous work (5) on the determination
of the number o f redundancies i n sociometric chains. The r e s u l t i n (5)
i s expressed i n terms o f the m a t r i x approach t o the analysis o f sociometric data discussed i n (1) and ( 2 ) .
The question o f the existence o f a complete cycle i n a network 4-s
closely r e l a t e d t o a w e l l known unsolved problem i n the mathematical
theory of graphs (see (3) and (U)). This mathematical problem i s the
determination o f a c r i t e r i o n f o r the existence o f an Hamilton l i n e j
i.e.
, a complete cycle i n a graph. I n (5) we have obtained e x p l i c i t
formulas f o r the matrices of Srstep paths through s * 6. Therefore,
by the nature of the formula presented here, we are enabled t o count
e x p l i c i t l y the number o f complete cycles i n any network of up t o 7
people. T r a n s l a t i n g t h i s observation t o the language o f graph theory,
we see that we can count the number o f Hamilton l i n e s i n any graph,
d i r e c t e d or ordinary, of up t o 7 p o i n t s .
DEFINITIONS MP
NOTATION
As defined i n ( 5 ) , a redundant path i s one i n which a t l e a s t one
person occurs more than once, and a pure path i s one which i s not r e dundant. Let n be the number o f people i n the group. I f a pure path
o f n-1 steps from person A to person B. i s followed by a one-step path
from B t o A, the r e s u l t is" a complete cycle.
Let M be the sociometric n by h matrix of one-step paths i n the
group. Let the elements o f the m a t r i x M be denoted by m^y wherern^j= 1
i f the i ' t h person communicates w i t h the j ' t h person and i s 0 otherwise. We adopt the convention t h a t a person does not communicate t o
himself.
Let P denote, as i n ( 5 ) , t h e m a t r i x of pure s-step paths. Let
g
the elements o f P be
where
0
J
i s the number o f pure s-step
J.
paths from the i ' t h person t o the j ' t h person. I n ( 5 ) , e x p l i c i t f o r mulas were found f o r the matrices P^ through P^ i n terms o f the
Matrix M, and i n a d d i t i o n a method was developed f o r f i n d i n g any P
g
given enough time.
I f Q i s any n by n m a t r i x , then by d(Q). we mean the m a t r i x obtained from Q by leaving the p r i n c i p a l diagonal of Q unchanged and
replacing a l l non-diagonal elements o f Q by 0.
. I f Q and T are n by n matrices, then by Q»T we mean ordinary
matrix m u l t i p l i c a t i o n .
THE THEOREM
Our r e s u l t i s contained i n the f o l l o w i n g theorem:.
The number of complete cycles i n a communication network i s the number appearing i n each o f the p r i n c i p a l diagonal locations o f e i t h e r
of the matrices d(M*P _i) or d ( P _ i *M).
n
n
I t xd.ll be seen during the proof of t h i s theorem t h a t the matr i x M*P _i has i n each o f i t s 'principal diagonal locations the
n
number of complete cycles i n the given "communication network, i . e .
the network whose sociometric .matrix i s M.. The two matrices d(M-P _^)
n
and d(P _-j_*M) are then shown to be i d e n t i c a l .
n
Proof ; Using the n a t a t i o n o f the preceding s e c t i o n and the usual
d e f i n i t i o n o f m a t r i x m u l t i p l i c a t i o n , we obtain f o r the 1,1 element o f
the product m a t r i x M » P
the sum:
n - 1
n ^ p f t D + - ^ P j f t * * ™l, Pfe
(1).
1
3
1 ) +
• ' *
+
\ATl
1]
However, by our convention t h a t the p r i n c i p a l diagonal of the sociometric m a t r i x M consists e n t i r e l y o f zeros, i t follows that ] _ ]_
m
13
0
and hence the f i r s t term o f the above sum vanishes.
Consider now the second term, m ^ P ^ D ^
T h g
n u m b e r
p
£ _
n
1 }
.
g
p r e c i s e l y the number of pure paths from the 2nd person t o the 1st
person passing through a l l o f the other persons o f the n-person
group..
Hence the second term, m ^ p j ^ I ^
1
i s the number o f complete
cycles i n which the 1st person communicates t o the 2nd person.
S i m i l a r l y , the next term, i ^ P ^ i ^ *
m
1
i s the number o f complete
cycles i n which the 1st person communicates t o t h e 3rd person, e t c .
Therefore the e n t i r e sum (1) i s the number of complete cycles
i n which the 1st person communicates t o any other person, i . e . i n
which the 1st-person i s involved. Thus t h i s sum (1) must be t h e t o t a l
number o f complete cycles i n the given communication network.
The remaining elements o f the p r i n c i p a l diagonal o f the product
m a t r i x M*P _i are a l l equal- t o t h i s f i r s t element, since each o f them
n
i s also the number o f complete cycles. The same cycles are merely
counted from d i f f e r e n t s t a r t i n g points.An e n t i r e l y analogous argument shows that the p r i n c i p a l diagonal
of P _i'M i s the same as that o f M«P _-^.
n
n
ILLUSTRATIONS
We s h a l l now give three examples o f graphs or communication
networks and apply the above theorem t o t h e i r matrices.
\
\
V
Fig.
1
\
I ^
/
Fig. 2
\
Fig. 3
The f i r s t two o f these graphs are ordinary graphs and therefore
have symmetric matrices. The t h i r d graph i s d i r e c t e d and i t s matrix
i s not symmetric.
Let the matrices of these three graphs be M^, M^, M^, respecti v e l y . Then we have:
0
0
0
1
1
Mi
0 0 1
0 0 1
0 0 1
110
110
1
1
1
0
0
O i
10
11
O i
10
l
1
0
l
0
0
1
I
0
1
1
0
0
1
0
Mr
0
1
0
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
Let R be the matrix o f redundant paths o f length s. The matrix
g
P of length s may be obtained by appropriately subtracting R from
s
Q
the matrix o f a l l paths of length- s which i s the s'th power o f M,
i . e . M . I f Q, T are n by n matrices, l e t Q x T denote elementwise
s
m u l t i p l i c a t i o n , as i n (5), and l e t Q' denote the transpose o f Q.
Then i t was shown i n (5) t h a t ;
R = M-d(M ) + d(M )-M - M x M '
£
2
3
= M-d(M3) + d(M3)-M + M -d(M ) + d(M )-M + M<d(M )-M
2
2
- M x M ' - 2M x M' x M
2
2
s
2
- M-(M x M')-M
2
The formulas f o r R^ and R^ are much longer and appear i n (5).
Let P ^ , P j ^ , P ^ be the matrices o f pure paths of length U
corresponding t o
Mg, M3 r e s p e c t i v e l y . A s t r a i g h t f o r w a r d calcu-
l a t i o n shows t h a t the matrices M^'P^-), M 'P^ ), MyP^3) have main
2
2
diagonals consisting e n t i r e l y o f O's, U's, and 3's r e s p e c t i v e l y .
Thus the network o f f i g u r e 1 has no complete cycles, as can be v e r i f i e d by inspection. The ordinary graph of f i g u r e 2 has four complete
cycles because the d i r e c t i o n of the cycle i s taken i n t o consideration
by t h i s theorem. I f d i r e c t i o n i s ignored, t h i s graph has two complete
cycles. F i n a l l y , the t h i r d network has three complete cycles.
REFERENCES
Festinger, Leon. The analysis o f sociograms using m a t r i x algebra,
Human Relations, 19k9 2, 153-158.
9
Forsyth, Elaine and Katz, Leo. A m a t r i x approach t o the analysis
of sociometric data, Sociometry, 19U6, 9, 3UO-3U7.
Harary, Frank and Norman, Robert Z. Group structures and communi c a t i o n networks i n terms o f the mathematical theory of graphs,
i n process o f p u b l i c a t i o n .
Konig, D. Theorie der endlichen und unendlichen Graphen, Leipzig,
1936,
'•
Ross, Ian C. and Harary, Frank. On the determination of redundancies i n sociometric chains, Psychometrika, 1952, 17* 2.