Table Of Content@ tre UnitedN ationUs niversit1y9, 80
Printedin Japan
rsBN9 2-808-0',!31-7
tssN0 379-5764
HSDRGPID-20/UNUP-131
ABOUTC RITICALG R OUPSIZE
YonaF riedman
GPIDC onsultant
.gT ED
o4.
f/'
r:a 'lq
,\
$'.2 c1E--6'/ "q
îts-
CONTENTS
| . lntroducti on: Groupsiz e as a Key Variable
I
I l. Groups, Graphsa nd Graphp arameters
L
I | | . Vale nce and Degradatio n il
I V. Factors Affect ing Cri t icaì Groupsiz e {tF)
V. Socíal Consequenceosf ExceedingC riticaì Groupsize l 9
Vl. Critical Groupsizea nd Beyond 23
Vll. Conclusion 28
Thisp aperb y Yona Friedmanw asf irst presenteadt the GPID ll meetingG, enevag, -14J anuary, 197g.
It canb e consideread contributiont o the Visionso f DesirablWe orldsu not n. Methodso f Anaiysis
sub-projectosf the GPIDP roject.
GenevaJ, anuary1 980
JohanG altung
It is beingc irculatedin a pre-publicatiofno rm to elicitc ommentsfr om readersa ndg enerated ialogue
on the subjecta t this stageo f the research.
I NTRODUIC OTN : GROUPI ZSE AS A KEYV ARIA BLE
It is evident for anybodyr eady to think about what he observes, that
nothing in the worìd can groww ithout limits: things growings top doing
so when reaching somed etermineds ize. This fact seemss o evident
that most peopìe (and most scientists) accept it without further
thought, and without posing the questions: at which size, why and by
rvhato peration does an entity stop its growth?
Wew iìl call "critical size" the I imiting size whereg rowths tops and
beyondw hich, if an entity continues to grow, the growinge ntity
suffers importantq uali tative changes. "Critical groupsize"w i l I thus
be the size of a social group characterizedb y certain quaìi ties' over
which size this group can not keep these quaìi ties. Critical group-
size is effectively critical to the good functioning of a group or of
an organization.
Criticaì groupsize might be the numbero ne problemf or humans ocial
life. lf humans ocial organizationsa re considereda s one of the most
important'rtooìst o assures urvivalrrfor the species, it is evident
that, if this tooì changesi ts quality simply by growingb eyonda
characteristic I imiting size, it might lose its efficaciousness,a nd,
insteado f assuring survival of the species, it might becometh e tooì
of its destruction. \^/em ight today be facing this situation.
Howt o be masters of our own survival, if we don't gain knowledgea bout
critical groupsize,a nd this in a stage wheno ur instinctive knowledge
aboutt hese limits is (at least, for a while) practicaìly lost? We
speaka bout "limits to growth" in manyd omains' except that of our own
organizations; perhapst he inverse would be better' and we should
think morea bout "limits to growtho f humano rganizationsrrin order to
understanda nd to avoid other probìemso f growth.
I
I I. GROUPSG, RAPHASN DG RAPHP ARAMETERS
In order to consider criticar groupsizeo therwise than in a super-
ficial manner,i t couìd be useful to start with statin_ai t as a
principle expressingi ts relation to other factors characterizing
organizations. In order to be able to achieve this, we have to ìook
for an appropriate terminoìogy.
I wil I call "group'ra set of individuals in which there exists some
'rrelationrrbetween
sort of any two individuals belongingt o the set.
A person who has no such "relation" to at least one other who in his
turn is reìated to the others, can be considereda person,,cut of qrouD.r,
Let us supposet hat I would like to sketch the imageo f a group. I wilì
first draw all personsb elongingt o ít, then continue by drawing línes
linking any two persons betweenw homI observe an existing relation.
I will thus represent this group by meanso f a figure in which any one
personw ilI be Iinked to any other by at least one "path" passing from
one to the other by the intermediary of other persons.
lf I replace the "mannikins"o f this mapo f the group by points, the
result will be a figure consistingo f points and lines, in which tnere
is at ìeast one path betweena ny pair of points chosena rbitrarilv
(connectedg raph).
t-f----
,
)*
Obviouslyt his graph gives an oversimpiìf ied imageo f a group. Using
this figure wiI I require further explanation.
"relations,"
I wiìl thus reexaminet he concepto f which I represented
by lines in the figure. First I will try to find out whod rew it.
0therwise stated, one has to knoww ho the observer of this group is.
I shouìd point out immediateìyt hat different observersw i I I see
"maps'r
different for the sameg roup. In most cases it will be difficuìt
to find two observers who would attribute the same importancet o the
I'importance"
samer eìation; thus the of relations cannot be observed
without error, and therefore I will not introducerrimportancei"n to my
terminology.
0n the other hand, the mere existence of a direct relation betweent wo
individuals belongingt o a group can be observed. This meanst hat the
"l
existence or the nonexistenceo f a ine" in a graph representing a
group can be agreed on by a larEe numbero f potential observers. I
will thus be content to note the existenceo f a relation.
Another characteristic of such relations cen be observeda nd noted:
I'direction."
their Let me expìain the term.
In observing two persons who "communicate" (and thus a reìation tretween
thenr)w e can generally state that once the communicationi s finished,
one or the other (or both) of the two persons changesh is previous
behaviour. Wew ill say, in such a case, that one of the two persons
(or both of them) received an "influence" from the other one. This
infìuence has a direction, which points from the person who exerts the
influence to the other one who receives it.
I will define'rinfìuence" as a relation betweent wo individuals which
has an observabl e d i rect ion.
In conclusion, a society will thus be representedb y a dírected
connectedg raph, that is, by a figure consisting of points and I ines,
in which there is no point not linked by at least one path to any other
point (whena rrow directions are not considered)a nd in which every
I ine carries an arrow representingt he direction of the influence in
this particular I ink.
This representationo f a society makesi t possible to describe its
structural characteristics by using mapsc onstructedw ith graphsa s
explaineda bove. structural characteristics do not refer to any
measurablesi ze, as I had to exclude,f irst of all, the possibiìity of
knowingi mportance,i ntensities, and so on, of influences in general.
I have thus I imited myself to consideringc ertain topological
properti es of these graphs, i n order to characteri ze soci a ì organi z-
ation. Theset opological properties indicate characteristics impli ed
by the ì inkages chemes,t he paths, and the circuits within such maps
of a group; they correspcndt o those of the influences within a set of
individuals.
To get to the description of such importantc haracteristics I shall
invokea n image:t he imageo f the "social situation" of a particular
personw ithin a group. This social situation will be defi ned by
influences this particular person receives from and gxerts upon the
other memberso f the group. For example, if he exerts an influence
upon four of his neighboursa nd he does not receive any influence from
anybody, he could be considered as having more "power',t han another
person who also exerts four influences but receives two influences
from others.
Thus the social situation of a partícular personw iìl be expressedb y
the difference betweent he sumo f influences startinq with him and the
sumo f influences having him as their endpoint.
In practice, the social situation of a particular personc orresponds
to his "baìanceo f influences." But - and this is important- we
calculate this balancew ithout associatinga ny difference in size with
I'size"
different influences. as vreh avea qreedt hat of influence is not
observabel .
This convention does not mean,o bviousìy, that the two persons linked
by this influence do not attribute to it any value, atrYi mportance
whatever.
VJec an be sure that they do, and even that very probably this importance
wiìl be different for each of the two: the one who exerts the
influence might consider it as important, and the other, who receives
it, might ignore it; or inversely, one might ignore the influenceh e
exerts on others and the ones who receive it might appreciate its
imoortance.
To avoid such ambiguity I had to take as a standard the observation of
someonee xterior to the society observed. But if this observer con-
siders alì direct influencesa s equal, he might observea degradation
of rrindirect" infìuences (influences transmitted by several intermediary
persons),d egradationt hrough errors, omissions,b ecauseo f successive
transmissions( degradationr esulting from what information theory cal ls
"noise").
Wew ill use, in order to describe this degradationo f an indirect
influence becauseo f the necessaryt ransmissions,a simple rule: we
will supposet hat the "intensity" of an influence will decay in inverse
proportion to the numbero f intermediary transmissions necessaryt o
its forwardinq.
\y'ea re now ready to define the social situation of any person within
a group by an observer exterior to this group. lt will be expressed
by the difference betweent he sumo f all influences (direct and in-
direct) exerted by this particuìar personu pona ll other personsw ithin
the group, and the sumo f all influences starting from all other persons
5
in the group and received by him.
In order to do this simplec alcuiation it is sufficient to construcr
therrpath matrix" of the graph mappingt he group in question. 0n the
basis of this graph (cr of this matrix) we can find both sumsn ecessarv
to define the parametero f social sÌtuation,
È
lrq Fa h''tro-
À ltt tnv
lgt?
Ì
a
î
-
c ?&r
F _ y ' pa{o t :.: 34 5
5,'tI CeÀllI alOc oOoOI oOt aoO1oo4oo Xia4t- H :l;l/a^q:5' {tr^il
alo î'"n !ilfir irj.i3:oit'a. l ('ray,") (s^)
;lìi;;rl'i
6l s r r a.l .to
F*Eg
*$$$ffi.o
-Ja/r
3 lo a t- l-srt:9 1" ìF(5 {.5aut cf Sl r r,l0- 1rl ' qlt
,Clll -c9 llo c o. or. cD réll qlriJ f,rÈ/ 4ÈÉ\t@. S1, !'sg.<; r.qrr
afrsre3.sslt* d4 d r qor .J.r. r.t.ll
f[ 11s .r.e.a.blt4 Sg.1.r3.2.1g : è.rt
( llr r' b 'l 5.lu 55jr.'3<3,1ú.-1\cu:o. 31t..r
S( '2.rt-:.rf , e,rr
using this method,w e can obtain not only the social situation of any
person within the group of our example( here containing seven persons)
that we obtained, but also the real I'hierarchy"e stabìi shed (frequently
in a tacit way) within the group. 0bviousìy,t he hierarchyw e find in
the figure is only the one observableb y somebodeyx terior to this
group. lt is quite possible that the híerarchy as observedb y Mr. A
or by Mr. B differs sensibly from thís'robjective" hîerarchy.
Let us suppose,n ow, that one membeor f this group decides to leave it
for sonrer easono f his own. lmmediatelyt he establ ished hierarcny
will be transformeda s a consequencoef his leaving it. Certain persons
remainingw ithin the groupw ill benefit from his defection (their
social situation wiìl be improved)o; thers will be prejudiced. Thus,
if we suppose- in order to keep the exanple simpìe - that all members
of the group want to get a "higher" place in the hierarchy, those who
benefit by the defection of Mr. X can be consideredh is'radversaries,"
6
who are interested in rennving him from the group. 0n the contrary,
those who are prejudiced by the departure of Mr. X wi I I try to keep
"al
him: they are his l ies."
"dependence"
Usingt his simple function, which I call of a particular
person in a group upon the departure of Mr. X, we can construct a
"table of al l iances'rc haracteristic of this qroup.
"Dependence"
can be calculated in a simple way: by obtaining the
difference betweent he social situations of a personb elongingt o the
group before and after the departureo f Mr. X. The first social
situation can be calculated on the graphw hich mapst he group as a
whole (ltr. X included); the seconds ocial situation wiìl be obtained
by calculating on a subgrapho f the sameg raph, è subgraphw hich does
not contai n Mr. X.
1{-7i\
\ / | t
$lb rrî, SuA-Cealqs
rllpptN( Dgpr',ireN(r3E
y''b
| /
Eo-r!,
,
gxcrr.tbD C É<c.urÈfl I Éx4!{J!E) 4 A(l!)6
& n @
W
AI, I.PqBreJl|l..r(A So4t71Y Ara>LUbt4L ; -g,îxc)r.u D$ Ff E,Rcrlrl:à
,){-J / ./
Weo btained thus, by a methods ufficiently simpìe to be used by a ten-
year-old child, a fairly goodd escription of the real structure of a
group. This methodc an be used for any imaginables ociety.
After all these prel iminary explanations I would try to define, for the
sake of exampleo nly, two concepts of group structure, concepts which
"egal "hierarchic"
I consider important: itarian" groupsa nd ones.
"egal
I calI a group itarian" if aìl its memberhs ave the sames ocial
situation. Put otherwise, in an egaìi tarian group the difference
betweent he totality of înfiuences exerted and the totality of infìu-
ences received will be the samef or everybody. Such a group does not
"upper
contain any cìassil influencina the others.
An egali tarian group is thus a possible one. There are a large number
of graphsw hich satisfy this condition (event houghc ertain constraints
restrict this number,c onstraintsw e wiì I investigatel ater on).
Another type of group having great lmportanceb ecausei t is a very
"hierarchicr':
commono ne, is this type of group can De representedb y
"tree"
a (a graph in which betweena ny pair of points there is bur one
path). This group is characterizedb y a "clegressive"h ierarchy of
social situation, starting f rom the ilapex,'( i.e. the person represente,J
by the apex is the rnostp owerfuì, the personsj ust below him are
somewhalte ss powerful, and so on). lts other characteristic is a
"progressive"
hierarchy of aì l iances, progressions tarting from the
samea pex; thus dependenceo, r change in social situation causecib y
the departure of somebodyfr om the group, will be ìess noticeabie for
personsa t the I'bottom'o' f the group than at its "top.',
A 6F.Ò\rP trrlTh ,|. fi,cJP \JrTî
r f,4,0,4a2r91 4p " sÍTrr.(ruÈ€ "htÉRA.Pcti<-"
sThrxrug=
,-', ,/\\..
r7\l'
i/r\.\
I
\I\d
/
J\1 \
we saw before that this methodo f representing a society is basedo n
the exchangeo f influence within a set of indivíduals. The two rypes
of groups sketched above are practicaì ly two schemeso f exchange
diametricail y opposed. In egaii tarian groups, exchangec an start with
any membeor f the group and will certainly reacha ll other members,
perhapss lowly, but surely. 0n the other hand, in hierarchic groups,
there is but one personw ho can start an influence which will reach alì
memberosf the group, and it wiìl do so relatively quickly. As for
the probabili ty that this influence will reach everyone,i t is very low
(becauset he noncooperationo f onìy one person somewherein the group
represents a blockage for a certain numbero f paths).
8
