Decomposition of communication networks

JOURNAL

OF MATHEMATICAL

4, 162- 173 (1967)

PSYCHOLOGY:

Decomposition

of Communication KENNETH

Carnegie

Institute

Networks1

D. MACKENZIE

of Technology,

Pittsburgh,

Pennsylvania

This is a study of the structural relationships among four commonly used networks in small-group experiments. Using graph theory several relationships among these networks are developed. The results demonstrate how these networks can be decomposed into subnetworks. Several behavioral consequences are developed and a rationale is provided for using 5-person groups in experiments. Other consequences indicate several experimental procedures that can he used in the analysis of organizations.

INTRODUCTION

Small-group communication network experiments often employ an apparatus which permits communication between designated pairs of subjects. If it is possible for a given pair to communicate with each other, then this communication channel is said to be open; if not, it is said to be closed. Graphs have been used to describe the open channels, where the vertices of the graph represent the subjects and the edges of the graph represent the open channels. There is a small number of graphs which are used in many experiments. This paper examines the relationships among the four most common graphs and uses these relationships to derive several behavioral consequences and a suggested optimum-sized network. In the small-group literature the four most commonly used networks are called the wheel, chain, circle, and all-channel, which are illustrated in Fig. 1 for a graph of five vertices. They are defined in the next section. [These basic networks, the definitions of R. G. Busacker and T. L. Saaty (1965), are described in graph theoretic terms as the once-connected centered tree, the simple chain progression, the minimal 2-connected graph, and the complete graph of n-vertices. The small-group descriptions are used in this paper.] In this paper, the analysis is restricted to these four connected graphs. Loops and parallel edges are excluded from here because of the nature of the apparatus used in small-group experiments. 1 This research was supported by Ford Foundation Grant l-40055 to the Graduate School of Industrial Administration, Carnegie Institute of Technology, Pittsburgh, Pennsylvania.

162

DECOMPOSITION

x2

x3 WHEEL

CHAIN

FIG. 1.

EDGE

Four

OF

COMMUNICATION

x4 GRAPH

x3 CIRCLE

x3 ALL-

GRAPH

frequently

PARTITIONING

used

graphs

AND

163

NETWORKS

CHANNEL

in small

GRAPHIC

GRAPH

group

x4

x4 GRAPH

experiments.

SUMMATION

Let the graph G be denoted by G ;=; (X; I’) where X is the set of vertices .x, , .A-,,_.., x,, and r is the set of edges, Tjj , between all connected vertices (xi, xj) lor i,,j, .- 1, 2,..., IZ. If G has n vertices, the edges can be given by a square matrix of order IL denoted by /I G ,,, whose entries gtj for i, j, = I, 2,..., n and i fj are defined bY iii

( I if rSj exists E /O otherwise *

(1)

164

MACKENZIE

Note that for an edge rij = rii, but for a digraph the arcs are not, in general, symmetric. For any connected graph G = (X; P) it is possible to partition the edges into T disjoint connected parts PI , r, ,..., I’, where ri n ri = 4 for all i, j = 1, 2 ,..., r and i fj. Associated with each connected part r, C I’ are subsets X, C X containing the vertices in I’, . Each connected part, r, , of the edge partition and associated set of vertices, X, , define the connected s&graph G, = (X, ; r,). For the purposes of this paper, the edge matrix of the connected subgraph G, , denoted 11G, 11is a square matrix of order n. Therefore,

k;l!l Gc II = /I G II .

(2)

Let the connected subgraphs Gi = (Xi ; Pi) and Gj = (Xj ; r,) be the two subgraphs defined by the parts ri and ri of r and their associated sets of points, respectively. Then, the graphic sum2 of Gi and Gi , denoted Gi u Gi , is defined by Gi u Gj = (Xi

u X, ; ri u rj)

(3)

for i,j, = 1,2 ,..., r and where ri n rj = 4 and for each Gi there is at least one subgraph Gj for which X, n Xi # + and Xi C X and Xi C X. The graphic sum of all r parts is denoted by uL=r G, , where

tjj& =G.

(4)

If G = u;=, Gk , then Jj G 11 = Liz, Jj Gk I). Further, by definition of edge partition and graphic summation, if G = (JL+ Gk , then I’, , r, ,..., r, form an edge partition 0f

r.

Whenever an edge partition r, , I’, , . . .. r, is formed from r, the associated subgraphs G, , G, , . . .. G, form a decomposition3 of G = (X; r). Figure 2 shows three decompositions of a 4-vertex all-channel graph. r It should be noted that a subgraph is not the same as a factor since, in general, each subgraph does not span G and is connected. The graph is not assumed to be regular. The graphic sum is not the same as the product of edge-disjoint spanning subgraphs. s Decomposition is different from factorization since the subgraphs are not necessarily factors even though they are line-disjoint. In the special case in which each of the subgraphs of a decomposition spans the graph, decomposition becomes factorization. All factorizations with connected factors are special cases of decomposition. A factor is an (edge-disjoint) spanning subgraph. That is, every vertex of G must be included in every factor and no two factors can have any common edges. Since every vertex is in a factor, alI participants in a network are included along with a few of the channels connecting them, even though in a factor they need not all necessarily be connected. There are many cases in which it is much more interesting behaviorallyto include only connected subsets of the participants

DECOMPOSITION

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lx!:=I:!IiE!lc”;: FIG.

2.

X fern

graphic

SOME

decompositions

of a 4-vertex

DECOMPOSI?‘ION

all-channel

graph.

THEOREMS

Using the definitions given in the preceding section, several decomposition relationships among the four graphs (the wheel, chain, circle, and all-channel) are derived. Throughout the rest of the paper the following conventions are used in labeling the vertices. The vertices are labeled counterclockwise starting from the top of the graph (as oriented in Fig. 1) for the circle and all-channel graphs and from top to bottom for the chain. The central vertex of the wheel graph is labeled X, and the remaining vertices are labeled from left to right (cf. Fig. 1). or subgroups,

cliques,

etc..

in

a study

in certain substantively decomposition

branches of graph as decomposition is used in this

their implications theorists and social

resulting scientists.

of group

behavior

or

to analyze

described in decomposition bemg able to connect all the \Vhik the ided of factorization

the group. The type of subgraph group within the group. ,Ilerely 11) the study of social phenomrnC~.

theory, with its limitations to researchers in the paper rather than factorization. from

the

idea

of

decomposition

it social

how

a subgroup

fits

into

can represent a functioning subparticipants is not very interesting has many fruitful applications is not potentially as interesting sciences. So for these reasons, It is hoped that the theorems and will

be

of

interest

to both

graph

166

MACKENZIE

Following these conventions, follows: For the wheel graph:

the matrices

of the four

connected

graphs

are as

lifi=land

(5)

gij =

For the chain

graph: \ lfor/z-jl gij = I 0 otherwise.

1 where

i, j, = 1, 2 ,..., n;

(6)

For the circle graph: lforli-jj gij =

1 where i, j, = 1, 2 ,..., n ori= 1 andj=n ori=nandj= 1;

(7)

0 otherwise. For the all-channel

graph:

i 1 for i # j for all i, j, = 1, 2 ,..., n; gij = 1 0 for i = j for all i = 1, 2 ,..., n.

(8)

Note that these four graphs of two vertices are identical and a 3-vertex chain is identical to a 3-vertex wheel and a 3-vertex circle is the same as a 3-vertex all-channel. All these graphs are different for n 3 4. THEOREM 1. Any connected graph G = (X; T’) of Y edges can be decomposed into r two-vertex subgraphs Gl , G, ,..., G, . The maximum number of 2-vertex subgraphs is (t), the number of possible edges of a n-vertex all-channel graph. Also, a wheel graph of n-vertices can be decomposed into (n - 1) 2-vertex subgraphs and a circle graph of n-vertices can be decomposed into n 2-vertex subgraphs. Furthermore, a circle graph of n-vertices can be decomposed into k 2-vertex subgraphs and one (n - k t I)-vertex chain subgraph. THEOREM 2. An all-channel graph n - 1 wheel graphs, Gk = (X, ; r,), respectively.

G = (X; r) of H- ver tices is the graphic sum of k = I,2 ,..., n - 1, of n, 11 - I,..., 2 vertices,

DECOMPOSITION

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Proof. Let .q be the central vertex of the wheel of n vertices G, =-: (XI ; F,). Then i G, ;~ is given by (5). Let .x2 be the central vertex of the wheel of n - I vertices G, =-= (-Yz ; P?). Then the elements of ‘I G, !j are given by

gij =

1 ifi - 2 and j =p 3,..., n; i 1 ifj = 2 and i = 3,..., n,: iI 0 otherwise.

Continue until reaching G,,+l ; (X,,_, the elements of ~ G,_, ‘1 are given by

; r,,+,)

where

central

vertex

is s,,+~

Then

Hence, every element of /~G ~;is in one and only one il G, ” and the diagonal elements are all zeros and the off-diagonal elements are all unity. Therefore, ~~~~ 11G, ~1= 1G 11,and the theorem is proven. The minimum number of wheel subgraphs that a n-vertex all-channel can be decomposed into is n - 1, since any smaller decomposition would violate the condition of edge partition. THEOREM 3. An all-channel graph of n-vertices can be decomposed into the ,followinx ,mbyraph.s: one n-vertex circle, two (TV 2)-vertex wheels, (I (n - 3)-vertex wheel, a (n - 4)-vevtes urheel, .. .. n 2-vertex wheel.

Proof. Let the circle subgraph of rz-vcrticcs be given by (7). Let the central vertex of a wheel be s1 Then, because of the circle subgraph, the vertices of the wheel subgraph about .yl are .x~ , .vJ , .x~ ,..., X, , Thus we have a (12 - 2)-vertex wheel on s1 Let a second wheel have .Y~ as its vertex; then, because of the preceding two subgraphs, the vertices of this new wheel subgraph about s? arc .r9 , sq , s5 ,..., s,, , which defines a nem- (~2 ~~ 2)-vertex wheel graph. Now consider a wheel subgraph centered on sR ‘rhe possible new (not yet accounted for) vertices of this wheel Fubgraph arc .pzl, .x-, , . .. . .x,, which deiinc a (rr 3)-vertex wheel. (‘ontinuing counterclockwise until reaching .v,, z , the last wheel subgraph is a 2-vertex wheel. Ever\ tidge has been accounted for once and onlv once bv this procedure. ‘1%~ dccompositluu of‘ all-channel graphs illto circle subgraphs is much morf di.?icuit than into wheel subgraphs or a comb&&n of a circle subgraph and (~2 ~~- 2) wheel subgraphs. If the number of vertices is prime, a verv simple result ohtains; but if n is odd but not prime (composite), the problem is -much more difficult. It is necessarilv complicated because of the way the circle subgraphs return to the same \-ertex in less than rl-edges when the number of edges is not prime and the distance

168

MACKENZIE

between vertices is a greatest common denominator not equal to unity. For example, in a g-vertex all-channel graph one obtains a 9-vertex circle whose edge progression whose edge progression is x1 , X, , xs , X, , xs , x2 , x4 , x2 ,a.., x9 , Xl ; another 1s Xl, x1 ; three 3-vertex circles whose edge progressions are x, , x4, x, , x1, and xa , X6 I X8 9 and another 9-vertex circle whose edge progression x8 , x2 , and x3 , x6 , x9 , x3 ; x5 9 any distance or length of an edge Is xl,x5 > x3 > x4, x3 > x3,x7, x2, x‘j, xl . For progression greater than $(n - 1) no new arcs are used. For example, for n = 9 an edge progression of length 5 gives (x r , xs), (xs , x2), etc. which have already been accounted for in the other circles. In order to prove the result for an odd composite (not prime) number of vertices, it is necessary to know two facts. First, every composite number can be factored uniquely into prime factors. Second, if a = JJi pia is the prime factorization of a number a, where 01~is the multiplicity of the prime number pi , and similarly n = l-Ii P,~E is the prime factorization of n, then the greatest common D enote the greatest common divisor of (a, n) divisor of the pair (a, n) is JJipimrn(ui,*). by g.c.d. (a, n). Let 1, 2,..., m < +(n - 1) be the set of integers less than or equal to &(n - 1). Given this set of integers, let a, , a2 ,..., a, and a, < m be those integers greater than unity for which the fraction n/q is integer not equal to unity and b, , b, ,..., b, be those integers for which the fraction n/b? is not integer and where g.c.d. (by, n) is an integer not equal to unity. For example, if n = 15, then a, = 3, a2 = 5, and b, = 6. Further, let cj be defined such that it is the smallest integer such that (n/bj)cj is a prime factor (not equal to one) of n and b,/c, is prime. For n = 15, b, = 6, and cr = 2, then nc,/b, = v * 2 = 5 and b,/c, = 3. Let p be the number of integer of type ai and bj . With these definitions in mind we can state Theorem 4. THEOREM 4. An all-channel graph of n-vertices, where n 2 3 is an odd number, can be decomposed into +(n - 1) - p circle graphs of n-vertices, a, circles of n/u1 vertices, . .. . n/a, circles of (n/a,) vertices, b,/c, circles of (nc,/b,) vertices,..., b,/c, circles of (nb,/c,) vertices.

Proof. Consider first those integers < +(n - 1) that are neither an a, nor a bj . Then, for the first (unity), define a circle as in (7). The result is a n-vertex circle whose closed edge progression is x1 , x2 ,..., x, , x1 . For the second integer not an a, or by (two), define the n-vertex circle whose edge progression is x1 , x3 , x, , x7,..., x%-r , The two n-vertex circle subgraphs are two parts of an edge x2 > X4,...,%-2, Xl. partition. Similarly, all other integers < i(n - 1) which are neither an ai nor a bi will yield n-vertex circles. There will be exactly &(n - 1) -p of these. Now consider the a, , u2 , .. .. a, , integers for which the fraction n/ai (i = 1, 2,..., 1) is an integer not equal to unity. For a circle starting with x1 , the edge progression of the subgraph is x1 , x1 + a, , x1 + 2q ,..., x1 + (n/q) ai = x1 . The resulting subgraph is an n/ai vertex circle. Likewise for the circle starting at x2, etc., until the final circle starting at

DECOMPOSITION

OF COMMUNICATION

169

NETWORKS

vertex n/ai . Similarly for each ai . Sext consider the b, , b, ,..., b, , integers for which the fraction n/b, ,J’ = 1, 2 ,..., q, is not integer and whose g.c.d. (bj , n) is an integer not equal to unity and the integer cj such that n/b, and bj/cj is prime. Take the number bj and form a circle subgraph beginning with xi The edge progression is x1 + bj , . . . . xi + nbj/cj = x1 which defines a circle of nbj/cj vertices. For the same s1 2 bj start the next circle at xa and form a new subgraph. Continue until starting vertex b,Jcj . In none of the constructions is the same edge ever used more than once and all are used. This proves the theorem. COROLLARY 1. If n 2. 3 is a prime number, all all-channel be decomposed into +(n -~ 1) circle graphs of n-vertices.

graphs

of n-vertices

can

Proof. If n is prime, then the greatest common denominator of any number .< &(n - 1) and n is unity. Hence, p in Theorem 4 is zero. Hence, by Theorem 4, the corollary is proven. THEOREM

the graphic

5. If n > 3, an all-channel graph of n-vertices can be decomposed into sum of one (n - I)-vertex all-channel and one n-vertex zuheel.

Proof. Let the z-vertex the symmetric edges of this The remaining elements in vertices are xa , xa ,..., x, . Corollary 1 and Theorem 4, respectively.

wheel have xi as its central vertex. Then, as in Eq. 5, wheel are the one in the first row and column of // G 1). graph whose 11G 11 define a (n ~ l)- vertex all-channel 5 arc illustrated

for a 5-vertex

all-channel

in Figs. 3 and

COROLLARY 2. If n 1~ 4 and n - 1 is prime, then an all-channel graph G of tl-vertices can be decomposed into an n-zlertex zuheel and i(n ~ 2) circles of (n - I )-vertices.

XI

XI

XI

p&%y);~ X3 ALL-CHANNEL GRAPH

FIG. 3.

x4

x3

x4 “OUTSIDE” CIRCLE GRAPH

An example

of Corollary

1 for the 5-vertex

X5

X2 “INSIDE” CIRCLE GRAPH

all-channel

graph.

170

MACKENZIE

Proof. By Theorem 5, an all-channel graph of n-vertices can be decomposed into a n-vertex wheel and a (z - 1)-vertex all-channel. And, since n - 1 is prime, by Corollary 1, the (n - I)-vertex all-channel can be decomposed into $(n - 2) circle graphs of (n - I)-vertices.

x@zhxH x4

x3 FIVE -VERTEX ALL-CHANNEL GRAPH

FIG.

4.

x2

x3

x4

x5

FIVE-VERTEX WHEEL GRAPH

An example

of Theorem

5 for the 5-vertex

x3

x4 FOUR-VERTEX ALL-CHANNEL GRAPH

all-channel

graph

The preceding results are only a subset of those obtainable through graph-theoretic analysis. The main interest of this paper, however, is not graph theory; but rather the behavioral interpretation of the preceding results. The next section is a discussion of the potential behavioral consequences that derive from the mathematics.

DISCUSSION

An edge partition is used to develop the various results of the preceding sections because it corresponds to subgraphs connected by open channels in small-group experiments. Another possibility is a vertex partition which would focus on the subjects in an experiment. An edge partition and the resulting graphic decomposition is very helpful in the analysis of the organizational structure of the group and the identification of the possible subgroups and even cliques. The next few pages examine the various results in terms of the organization structure and substructures. Theorem 1 states that a connected graph of Y edges can be decomposed into Y 2-vertex subgraphs. While being trivial mathematically, it means that a communications network can be considered as a graphic composition of its dyads. That is, the structure of a small group can be analyzed in terms of each of the dyads. If the analysis were to proceed further to include an analysis of the content of each dyadic communication, the authority, decision, or functional structure can be pieced together from the

DECOMPOSITION

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171

parts. This is a result that has been used in several studies (Faucheux and Mackenzie, 1966; Mackenzie, 1964; Mackenzie and Huysmans, 1965; and Mackenzie and Frazier, 1966). A direct corollary of Theorem 1 is that a wheel graph of n-vertices can be decomposed into (~z -- 1) 2-vertex subgraphs. The central vertex of a wheel graph can be considered as a “leader” in a social situation and the other n - I vertices his “subordinates.” ‘I’he content of the interaction may diAer with the individual subordinate due to division of labor or to other specialization based upon the requirements of the organization and the individual capacities or interests of the subordinates. Another direct corollary of Theorem 1 is that a circle graph of n-vertices can be decomposed into k 2-vertex subgraphs and one (n -~ R + I)-vertex chain. Reversing the proposition, a careful content analysis of a complex organization (one having many levels and subgroups on each level) might reveal the existence, functionally, of a circular flow of interaction among the various parts of the organization by piecing together the interacting dyads and chains. On the other hand, what appears to be a simple circle might be composed of functionally differentiated subgroups which may relay information, solve problems, or serve as coordinators. Theorem 2 states that an all-channel graph of tf-vertices is the graphic sum of n I wheel graphs ranging from a n-vertex wheel down to a 2-vertex wheel. Thus it is logically possible to view an all-channel as a graphic summation of PZ I different wheels. The all-channel is often used to represent a committee where all individuals arc freely interacting. It is usually considered an equalitarian organization form since all participants have equal access to one another. On the other hand a wheel network is considered authoritarian since it places the maximum possible restrictions on communication in a connected network by having only the open channels going to the central person, But if what looks like an all-channel is really a series of wheels, then the network might be considered oppressive rather than democr:itic. In fact, one member of the all-channel could have tr 1 “supervisors.” Conclusions about one network, like an all-channel, being a priori more or less equalitarian thall another do not result from a careful analysis of the topological properties of tht various networks. ‘rheorem 3 states that an all-channel graph of n-vertices can be decomposed int[J a n-vertex circle, ttvo (fl - 2)-vertex wheels and n - 4 other wheels of diminishing number of vertices. Thus an all-channel graph can be decomposed into a circle is that it is connecting all vertices and a series of wheels. One direct implication possible to have wheel subgraphs in an all-channel transmitting one type of message (as in a local problem solving center) within a subwheel and different kinds of message,: between subgroups and yet another set of messages to the overall circle group. Theorem 4 gives the general way that an all-channel can be decomposed into just graph of n-vertices circle graphs. Corollary 1 states that for n-prime, an all-channel can be decomposed into ~(YZ- 1) n-vertex circles, For e?tample, in Fig. 3, the S-vertex

172

MACKENZIE

all-channel is the graphic sum of two S-vertex circles which are labeled the outside and the inside circles. That is, the “star” inside the all-channel unwraps into a 5vertex circle. Hence, there exists the possibility for such a graph of considering the all-channel graph as the superposition of two communications networks. Experiments could be performed to see whether two circle groups, properly arranged, could be considered as one group which in terms of communication channels appears to be an all-channel. Theorem 5 states that any all-channel graph (n > 3) can be decomposed into an n-vertex wheel and an (n - I)-vertex all-channel. This result is particularly interesting to the author because it has been useful in interpreting an experiment he is currently conducting. In this experiment a five-man all-channel functions as if it were two separate networks in the manner described in Theorem 5. In terms of the pattern of communications, each group is an all-channel and yet when subjects are questioned, they always designate one of the subjects as leader. Analysis of the content shows a perfect all-channel among four (excluding the leader) over which the group shares information and a perfect wheel centered on the leader over which the group’s decisions are made. So in this case the possibilities suggested by Theorem 5 actually obtain experimentally (cf. Fig. 4). In a substantive sense, the behavioral consequences of this paper were anticipated by Simon (1962, p. 477) when he stated: “ ...

hierarchies have the property of near-decomposability. Intra-component linkages are generally stronger than in intercomponent linkages. This fact has the effect of separating the high-frequency dynamics of a hierarchy-involving the internal structure of the components-from the low frequency dynamicsinvolving interaction among components.”

The results of this paper suggest methods for identifying the components of an organization. This leads to clearer statements about the total organization (graph) of which the components (subgraphs) are part. Simon’s (1962) concept of a hierarchic system (a system composed of interrelated subsystems which are in turn subsystems, etc.) can be made more precise using the type of analysis presented in this paper. The five theorems and two corollaries establish a few of the structural interrelationships among the wheel, chain, circle, and all-channel graphs. It was noted earlier that all four graphs are identical for a 2-vertex graph, and for a 3-vertex graph the wheel and chain and the circle and all-channel are identical. So that the smallest graph for which all four graphs are different is a 4-vertex graph. Since each graph represents a different and commonly employed communications network, one desirable feature of a communications network is that it permits the different configurations simultaneously. If, however, the size of the network is too large, it becomes difficult to organize and control during an experiment. So, what is the “best” size of a small group? The answer, suggested here, is that the “best’‘-size group is the smallest

DECOMPOSITION

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NETWORKS

(to reduce cost and control problems) that allows the other three networks to operate simultaneously in an all-channel to give maximum network flexibility. Given an all-channel and the similarity of the different graphs for n = 2 and 3, the number must be at least four. A graph G = (X; r) is defined here to be a general n-vertex all-channel graph if it can be decomposed into all of the following subgraphs: (i) one or more wheels of three or more vertices, (ii) one or more circles of three or more vertices, (iii)

one or more chains of four or more vertices.

An all-channel graph of 4-vertices is not general. By Corollary 2 a 5-vertex allchannel can be decomposed into two 5-vertex circles and a five-point circle can be decomposed into a 3-vertex wheel and a 4-vertex chain. Hence, the smallest general all-channel has five vertices. Therefore, by the stated criterion five is the “best” size for a communications network experiment. Any larger number will be general too, but has the disadvantage of increasing the problems of cost and control.

CONCLUSIONS

Using an edge partition and a special form of graphic decomposition, the structural interrelationships among four frequently used communication networks are analyzed. The results of the analysis are used to examine the related behavioral implications and to provide an answer to the problem of the “best’‘-size small group. The analysis, as far as it goes, demonstrates that important behavioral consequences can be derived by viewing group structures in terms of their possible subgroup decomposition. It is very possible for different subgroups within an organization to communicate within the subgroup on one dimension and with other subgroups or the whole organization on other dimensions. The contents of messages are probably more important than the frequency and direction of interaction in understanding the behavior of subjects in an experiment. The mathematical results of this paper give a systematic way of determining the possible subgroups. The four graphs studied in this paper are closely interrelated and that because of these relationships, it is not correct to assert properties about groups merely on the basis of the total structure of the channels of communications.

ACKNOWLEDGMENTS The author wishes comments concerning

to thank Theorem

Fred W. 4 and

Glover of the University the referees for their

very

of California, helpful

Berkeley suggestions.

for

his

174

MACKENZIE REFERENCES

R. G. AND SAATY, T. L. Finite graphs and networks: an introduction with applications. York: McGraw-Hill, 1965. FAUCHEUX, C. AND MACKENZIE, K. D. Task dependency of organizational centrality: its behavioral consequences. Journal of experimental social Psychology, 1966, 2, No. 4, in press. MACKENZIE, K. D. A mathematical theory of organizational structure. Unpublished doctoral dissertation, University of California, Berkeley, 1964. MACKENZIE, K. D. AND FRAZIER, G. D. Applying a model of organization structure to the analysis of a wood products market. Management Sciences, 1966, 12, B340-B352. MACKENZIE, K. D. AND HUYSMANS, J. H. B. M. A formal analysis of a group decision making experiment, Working Paper No. 11, Ford Foundation Project in Organizational Behavior, Graduate School of Industrial Administration, Carnegie Institute of Technology, Pittsburgh, Pennsylvania, March, 1965. SIMON, H. A. The architecture of complexity. Proceedings of the American Philosophtcal Society, 1962, 6, 467-482. BUSACKER,

New

RECEIVED:

August 16, 1965.