burt positions in networks

8/13/2019 Burt Positions in Networks

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Positions in NetworksAuthor(s): Ronald S. BurtSource: Social Forces, Vol. 55, No. 1 (Sep., 1976), pp. 93-122Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2577097.

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PositionsinNetworks*RON A L D S. B U R T, University f California,BerkeleyABSTRACTThe existence of an actor as a set of asymmetric relations to and from every actor in anetwork of relations is specified as the position of the actor in the network. Conditions ofstrong versus weak structuralequivalence of actor positions in a network are defined.Network structure s characterized n terms of structurallynonequivalent, ointly occupied,network positions located in the observed network. The social distances of actors fromnetworkpositions are specified as unobservedvariables n structural quationmodels in orderto extend the analysis of networks into the etiology and consequences of networkstructure.

We are each nested in a cacophony of relations with other actors in society. Theserelations serve to define our existence in society. We are who we are as a function ofour relations to and from other actors in society. With the growth of technology andits concomitant division of labor, the determination of actors in society as a functionof their relations with other actors is likely to increase rather than decrease. Theproblem for the social scientist then becomes one of conceptualizing the patterns ofrelations between an actor and the social system in which he exists in a manneroptimally suited to explanation.

Within the total set of all relations which link an actor to other actors in asocial system, there are subsets of similar relations. There are economic relationslinking the actor to specific other actors. There are relations of friendship, relationsof kinship, and relations of status. There are political relations linking the actor toother actors. The list has no end. Each of these types of relations among actors in asocial system serves to define a network of relations among the actors. This paperelaborates a conceptualization of networks of relations among actors in a systemwhich simultaneously captures the basic characteristics of the structure in anobserved network of relations and easily lends itself to the investigation of theetiology and consequences of that structure through the use of structural equationmodels.

The central idea in the conceptualization is that of a position in a network,the specified set of relations to and from each actor in a system. The extent to whichtwo actors jointly occupy the same network position is treated as the social distancebetween them and is specified as a scalar value. Two or more actors jointly occupythe same network position when they have similar relations to and from each actorin the network. Actors jointly occupying the same network position are discussed as

*Workon this paper was supportedby a grantfrom the National Science Foundation GS-405-OOX),grantfromthe Director'sFund at NORC anda fellowship from the NationalInstituteof MentalHealth(3-5690-43-3453). Portionsof thispaperwerepresentedat theEighthWorldCongressof the InternationalSociological Association in the communityresearchsession. I appreciatecomments by J. S. Coleman,T. N. Clark, J. A. Davis, and E. 0. Laumannon previousdiscussions of the ideaspresentedhereand thegenerosity of E. 0. Laumann and F. U. Pappi in making their data on elite actors in a Germancommunityavailable to illustrate he ideas in the paper.93

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94 I Social Forces I vol. 55:1, september1976being structurallyequivalent in Section 2 and definitions of strong versus weakequivalenceare specified then illustrated.Based on the discussionin Sections 1 and2, Section 3 proposes thatthe structuren an observednetworkexists as a patternofrelationsamong M + 1 sets of actorswho togetherform the system of actorsbeingconsidered.The M sets of actorsare each composed of multiple actorswho jointlyoccupy structurallyequivalent positions in the network. Actors in different setsoccupy structurallynonequivalentpositions. The M jointly occupiedpositions existas fourelementary ypes of networkpositions outlinedin Section 3. The remainingset is a residual category of actors among whom there is no network positionoccupied by more than two actors. Sections 2 and 3 describe, primarily,thestructuren an observednetworkof relations.Section 4 moves ahead by specifyingnetwork positions in structural quation models as unobservedvariables n ordertoinvestigatethe etiology and consequencesof occupying differenttypes of positionsin differenttypes of networks.

SOCIALDISTANCEAND ACTORSAS POSITIONS N NETWORKSGiven a systemcomposedof N actorsandone or morenetworksof relations amongthe actors, there are two perspectivesfrom which the intensityof relationbetweentwo actorswithin a single network can be viewed: (1) from the perspectiveof thetwo actors as a dyad which is only secondarilyassociatedwith the overall network,or (2) fromtheperspectiveof the two actors as elementsof the overall network.Theformer can be discussed in terms of an asymmetric individualdistance from oneactor to another.The lattercan be discussedin termsof a symmetric ocial distancebetween the two actors in terms of theirrespectivepositions in the network.Let Idijbe the individual distancefrom actor i to actor] whereIdij can beoperationalizedas any of a varietyof measuresof the directed,asymmetricrelationfrom i towardj qua individual entities such as: (1) a measure of the differencebetween a profileof characteristics f i (e.g., i's values, socioeconomiccharacteris-tics, or aspirations),andi's perceptionof actorj's profile on the same characteris-tics;' (2) a measureof the co-occurrenceof actors and] in archivalrecordswherejinitiates action while i is the object of action;2(3) the presence or absence of asociometric choice link from actor i to actor j;3 (4) the minimum number ofsociometric choice links requiredfor actor i to reach actorj;4 (5) the minimumweighted sociometric choice link distance requiredfor actor i to reach actor ;5(6)the normalizedminimumnumberof sociometricchoice links required or actorto reachactor . 6 Although the six operationalizations f individualdistancedo notmeasure the same qualitative dea, they all reflectan intensityof relationship romone actortowardanotheras a dyad, a pair,of actors. The otherN - 2 actorsin thenetworkareignored except in the last three operationalizationswhich consideronlyas many other actors as are necessary to complete a chain of sociometricchoicelinksfrom actori to actor .7When two actors exist in a network with several additional actors, an



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96 I Social Forces I vol. 55:1, september1976STRUCTURALLYQUIVALENT OSITIONS N A NETWORKActorj's position in a network s definedbyIDj*, the (2N by 1) vector of individualdistances from actorj to each of the N actors and from each of the N actors to actorj. Given thatactors have different relationswith one anotheras a functionof theirinterests n each others' resources such as finances, prestige, and charm, it is to beexpected that positions of actors in a network will be differentially similarto oneanother.Twoactors occupy the same positionin a networkwhen they have the samerelations to and from each actor in the network. Such a pair of actors can bediscussed as occupying structurally equivalent positions in the network. Moregenerally, a set of actors can be discussed as occupying structurally equivalentnetworkpositions when their relations with all actors in the networkare identicalso that:IDi* = IDj* = IDk*, (2a)which means that actors i, j and k occupy structurally equivalent positions in anetworkwhen the social distancesamongthemequal zero:dij = dik = d3k = 0. (2b)Since it requiresthateveryelementof IDj*and IDj*be identical if actors i andj areto be discussed as structurally equivalent, equation (2) is a definition of strongstructuralequivalence.When dealing with actual networks of relations, the strong definition ofequivalence has little utility since thereare likely to be minordifferencesbetweenstructurally quivalent positions due to sampling variability,errorsof observation,and/or theoreticallytrivial differencesbetween actors. For appliedresearch there-fore, it is convenient to relax equation (2) to a weak definition of structuralequivalence.A weakdefinitionof structuralequivalence of actor positions asks thatdij, the social distance between actors i and j, be less than some criterion distancebased on the distances among structurally nonequivalentpositions in orderforactors i and j to be treated as occupying structurallyequivalentpositions. In otherwords, actors i andj occupy structurallyequivalentpositions in a networkunder adefinitionof weak equivalence when:IDj* - IDj*, (3a)which can be stated asdij < a, (3b)where alpha is a criterion distance based on the distances among the structurallynonequivalentpositions in a network.The social distances among actors in struc-



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Positions in Networks I 97turally equivalentpositions under the definition of weak equivalence is less thansome arbitrarycriterion, alpha 12

While the specification of a value of alpha for all networks is arbitrary,useful approachto determining alpha for particularnetworks is througha hierar-chical clusteringalgorithm.A hierarchicalclusteranalysiswill presenta successionof alternative, ncreasing,values of alpha.The firstvalueof alphawill be zero. Theactors who occupy structurallyequivalent positions when alpha equals zero areequivalentunder the definitionof strong equivalence given in equation (2). Sub-sequent values of alpha are then obtained by combining individual positions ofactorsinto subsetsof positionswhich are moresimilarto eachotherthantheyaretoother actor positions. This process continues by gradually allowing greater andgreater social distances between actors clustered together as being structurallyequivalent.Under the finalvalue of alphafor a network,all actorswill be clusteredtogether into a single network position-even though they are separatedby con-siderablesocial distances. Somewherebetween the extreme of a definitionof strongequivalence and the extreme of defining every position as equivalent, will be areasonable partitioningof the actorpositionsinto M + 1 sets of positions;M struc-turally nonequivalentnetworkpositions each jointly occupied by multiple, struc-turallyequivalent,actors and one residualgroupof actorpositionseachoccupied byno more than two actors.Each of the M structurallynonequivalentnetworkpositionsjointly occupied by multipleactors will appear in a hierarchical clusteranalysis asa set of actor positions which are clustered together as equivalentwhen alpha isclose to the definitionof strong equivalence-i.e., demandingminimal differencebetween ID* vectorsclustered together as equivalent-andwhich are not clusteredwith otherpositions of actors until alpha is extremely weak-i.e., allowing almostany difference betweenID* vectors.As an illustrationof the ideas of strong and weak structuralequivalence,Figures 1, 2 and 3 present analyses of the networks of economic exchange, socialexchange and information-seeking among 45 community decision-makers (cf.,Laumannand Pappi, a: Figures 3, 4 and 5 respectively). An (N by N) matrix ofsocial distances as estimated from equation (1) using the operationalizationofindividualdistanceoutlined in Appendix A was inputinto Johnson's connectednessmethod of hierarchicalcluster analysis which is presentedin partA of the figuresand input into Roskam and Lingoes' smallest space analysis in two dimensionswhich is presented n partB of the figures. The location of networkpositions jointlyoccupied by multiple actors is based solely upon the hierarchicalcluster analysis.The smallest space analysis is presented n order to give the reader a feel for thespatial distributionof actors in two dimensions and for comparison purposes withthe original Laumannand Pappi discussion. Part B references actor positions bytheirsequentialnumberas is done in partA of the figuresas well as by the influencerank of the actor as presented n theLaumannandPappi figures. AppendixB detailsthe expected differences which will occur in a nonmetricanalysis of social distancesas opposedto individualdistancesusing Figures 1, 2 and3 versusFigures 3, 4 and5in LaumannandPappi (a) as an example.



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98 / Social Forces / vol. 55:1, september1976

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Positions in Networks /103

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104 I Social Forces I vol. 55:1, september1976Each row in the partA of Figures 1, 2 and 3 corresponds o a differentvalueof alpha given on the left-hand side of the figure. Positions of actors in a networkwhich are clustered together as being equivalent at a given level of alpha are soindicated by Xs connecting them (e.g., the positions of actors 35 and 42 in theinformationseeking network are to be considered equivalent when alpha is greaterthan 1.0).The first value of alpha is zero. No positions in any of the three networksareequivalentunder the definitionof strong equivalence. In each of the figures,the lastandhighest value of alphaallows all networkpositionsto be treatedas equivalent.Following the above discussion, the M structurallynonequivalentpositions jointlyoccupied by multiple actors will appear n partA of the figures as moundsof Xs

separatedby sharpdips or valleys between mounds. Such a condition will evidencea clustering together of actors when alpha is close to the definition of strongequivalence and the lack of additional actors positions entering the cluster untilalphais so large as to allow considerablesocial distances among actors clusteredtogether as weakly equivalent. For example in the network of economic exchangerelations (Figure 1) actors 2 and 39 are clustered together when alpha is 1.0 andother actors are clusteredwith them until alpha equals 1.5. After that, no actorisspecifiedas structurally quivalentto the set composedof actors32, 6, 2, 39 and44untilalpha equals2.5-almost the largestvalue given to alphafor the network. Themound of Xs in the columns associated with these actors indicates that these fiveactors ointly occupy a positionin the networkwhich is structurallyuniquefrom thepositions of other actors and so has been identified as position E1 in the analysis.Actors jointly occupying the M structurallyunique positions in each networkareindicatedbeneaththeir columns in the hierarchical luster analysis and arecircledinthe smallest space analysis (actors in residual category are unlabelled). The fivestructurallyunique positions in the networkof economic exchange relations(i.e.,M = 5) arereferenced as E1, E2, E3, E4 andE5 (e.g., the actorsjointly occupyingposition E2are9, 12, 18, 20 and41). The fourstructurallyunique positions locatedin the network of social exchange relations are referenced as SI, S2, S3 and S4(i.e., M = 4). The three structurallyunique positions located in the network ofinformation eeking relations are referencedas 1I, I2 and I3 (i.e., M = 3).

CHARACTERIZINGHESTRUCTUREIN AN OBSERVEDNETWORKOF RELATIONSThe observed position of an actor in a networkexists as the patternof individualdistancesto and from actors in a system, i.e., as a particular orm of interaction,given a single reasonfor interaction, i.e., given a particular ontentof interaction(see Simmel, 40-57, foran elaborationof the content-formdistinction).Empirically,structure n a networkexists as N observed positions in the network; ID1, ID2,

ID.. At a higher level of abstraction,idiographic positions (positions whose



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Positions in Networks I 105Table 1. STRUCTUREINTHE OBSERVEDNETWORKSPORTRAYEDAS THEPATTERNOF RELATIONSAMONGM+1 STRUCTURALLYNONEQUIVALENT ETS OF ACTORS*

Economic Exchange Network (Figure 1)E E E E E Residual Total1 2 3 4 5

E 87 0 0 13 0 0 100% (15 citations)E 0 0 0 0 0 0 100% (O0citations)E 12 4 56 8 0 20 100% (25 citations)E 0 0 0 90 0 10 100% (21 citations)E 5 0 5 55 13 22 100% (22 citations)5

Residual 22 0 17 39 0 22 100% (18 citations)Total 21 1 18 42 3 16 100% (101 citations)Social Exchange Network (Figure 2)

S S S S Residual Total1 2 3 4S 81 5 0 0 14 100% (21 citations)S 11 78 0 0 11 100% (35 citations)S 11 0 72 0 17 100% (18 citations)S 11 22 0 33 34 100% ( 9 citations)4

Residual 37 23 3 0 37 100% (35 citations)Total 37 32 12 3 16 100% (118 citations)Information Seeking Network (Figure 3)

I I I Residual Total1 2 3I 97 3 0 0 100% (31 citations)I 33 67 0 0 100% ( 9 citations)I 72 3 24 1 100% (76 citations)3

Residual 0 0 0 0 100% ( 0 citations)Total 76 8 15 1 100% (116 citations)

*Cell entries are proportion of citations made by actors in row to ac-tors in column position. Total number of citations made by actors in rowis at right.

particularmixtureof form andcontentof interactionare of special significanceforthe system of actorsbeing considered)and nomotheticpositions (positionsdefinedsolely as forms of interaction)present alternative perspectives on structure n anetwork.As a first cut, those positions which are jointly occupied by multiple,structurallyequivalent actors are of special significance within a network. Thedefinitionof weak structuralequivalence provides a means of aggregatingthe Nobserved positions of actors in a system into M + 1 structurallynonequivalentsubsets of actors. Idiographically,basic characteristicsof empiricalstructure n anetworkare preservedas M specific patternsof relationsto andfrom each of the Nactors(idiographicpositions)anda conglomerationof variouspatternsof relationsassociatedwith the actors comprisingthe residualcategoryof actors. The decision



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106 I Social Forces I vol. 55:1, september1976to treatpositions in the residual category as idiographicwill be based on conceptualgrounds since it has no clear resolution on empirical grounds. The structuren anobserved network of relations can then be characterizedas the patternof relationsamong the M + 1 sets of actors. Such representationgof the structurein thenetworksanalyzed in Figures 1, 2, and 3 are presented n Table 1. Table1shows thedistributionof sociometric choices made by actorsin each subset over the M + 1structurallynonequivalentsubsets of actors.Forexample, the networkof economicexchange relations is presented n Table 1 as a patternof relationsamongsix sets ofactors-five sets of actors where each set jointly occupies a unique idiographicpositionin the networkandone residualset of actors.The actorsin positionE1give87 percentof their sociometric choices to other actors in the same networkposition.The remaining 13 percentof their sociometric choices are given to actorsoccupyingposition E4. This characterization f the structure n a network s similarto the ideaof a blockmodel of a network (see White et al.) with the exception that actors arenot forced into one or another structurallyunique block. Each actor occupies hisown networkposition. Thatposition has finite social distancesfromeach of the Mstructurallyunique idiographicpositionsin the network.Just as each of the M idiographicpositions in a networkis reflectedin theempirical position of an actoras a function of the actor's social distancesto actorsjointly occupying each idiographicposition, so theactor's empiricalposition reflectspure forms of interaction, i.e., nomothetic positions. Viewing the M structurallynonequivalent, jointly occupied, idiographic positions as basic elements of thestructuren a network,one can, as an alternative o the characterizationn Table1,discuss the idiographicpositions in termsof the pure forms of interaction o whicheach is most similar.Nomothetically,basic characteristics f empiricalstructuren anetworkcan then be preservedas M types of pure forms of interaction.Table2 presents an exploratory ypology of four forms of interaction.Eachcell of the typology representsa differentnomotheticposition in a network. Thepositions are classified according o the prominenceof actors within a network zeroversus non-zeroproportionsof sociometric choices made by actorsin the networkwhich are given to actorsjointly occupying a position) and the tendencyfor actorsto only initiate interactionwith actorsto whom they are structurally quivalent (halfor over versus under half of the choices made by actors jointly occupying anidiographic position are given to other actors occupying the same position). Thecolumns distinguish between actors in a position who receive few or no choicesfrom the system versus those that receive a nonnegligible proportionof thosechoices. The rows in Table2 distinguishbetweenactors n a positionwho give mostof theircitationsto other actorsin the position versus those who give most of theircitationsto actorsoccupingdifferentpositionsfromtheir own. The fournomotheticpositions in Table 2 are not intended to be exhaustive of the range of possibleempiricalpositions, however, they do representbasic types of networkpositions aspureformsof interaction.The isolate positionexists as a set of actorswho give most of their citations



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Positions in Networks I 107Table 2. FOUR ELEMENTARY YPES OF NETWORKPOSITIONSAND THEPATTERNSOF RELATIONSWHICHDEFINETHEM

Proportion of networkchoices given to the

position*

Ratio of position ISOLATES PRIMARYchoices given to >. POSITIONthe position overtotal number ofchoices made bythe positiont C D.5 SYCOPHANT BROKER

ILLUSTRATIVE PATTERNS OF RELATIONStA B C D RESIDUAL

ISOLATE A 0? 0 0 0 0PRIMARY B 0 high 0 0 0SYCOPHANT C 0 high low 0 0?BROKER D 0 high 0 low o?

RESIDUAL 0 high low high 0?TOTAL 0 high low high 0?

*Proportion of all choices made by actors in the network which are given tothe actors jointly occupying the network position being classified.tNumber of choices made by actors jointly occupying the network position be-ing classified to actors within the position over the total number of choicesmade by actors occupying the position.

tCell entries refer to the proportion of sociometric choices from actors oc-cupying the row position which are given to actors occupying the column positionas is presented in Table 1.

?No predictions are made for these cells. Although they aid in the descrip-tion of network structure, they have not been used in this exploratory classifi-cation of types of positions in a network.

to actorswithin the set and receive no citationsfrom actors in the system who arenot in the set, i.e., who occupy nonequivalentpositions in the network.Such a setof actorscould be a subsystemwithinthe overall system of N actors or could be agroupof actorseach of whom is an isolate withinthe system (this latterpossibilityassumesthat at least one isolate madea sociometricchoice to himself or to an actoroccupyingthe isolateposition). PositionE2in Figure 1 andTable1 is anexampleofan isolate position.A primaryposition in a networkexists as a set of actors who give most oftheir sociometriccitationsto actors with whom they are structurally quivalentandwho receive a nonnegligibleproportionof the totalnumberof sociometriccitationsmade by the N actors n thenetwork.Using sociometriccitationsas a measureof theinterestof the citor towardthe citee, primarypositionsarecomposedof actorswhohave greater nterestin actors within their position than in actors occupying non-



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108 I Social Forces I vol. 55:1, september1976equivalentpositions although they are the object of nonnegligibleinterest from theoverall system of N actors as represented n the networkbeing analyzed. Severalexamples of primarypositions are illustrated n Table 1; E, E3, E4, SI, S2, S3, IIandto some extentI2.The remaining two types of nomotheticpositions distinguished n Table2(cells C andD) areusually grouped ogetherwithprimarypositionswhen a networkis analyzedin termsof individualdistances thatare forced to be symmetric.Thesetwo types of positionsarecharacteristically ccupiedby actorswho cite prestigiousactors in primary positions but do not have their choices reciprocated.Thesepositions, labelled sycophant and broker in Table2, areoccupiedby theusualhangers-onassociated with prestigiousor wealthyactors.13 The term sycophantis perhapstoo harshhere, however, it emphasizesthe idea of sociometric choicesbeing given to actorsoutside the position andlack of choices being given to actorsin the position by actors outside the position. A brokerposition in a network willalso be jointly occupied by actorswho give most of theircitationsto nonequivalentactors, however, this position differs from the sycophant position in that actorsinthe brokerpositionreceive a nonnegligible proportionof citationsfrom actors n theoverallnetwork.While actorsin a brokerpositionarethe objectof a nonnegligibleproportionof the interestof actorsin the systembeing considered,actorsin a syco-phantposition are not.14Although there areno examplesof brokerpositionsin thenetworkspresented n Table1, thereare two examplesof sycophantpositions;E5 andI3. The actorsin positionE5 give 55 percentof their sociometric citationswithin thenetwork of economic exchange relations to actors occupying position E4. Only13 percentof theircitations aregiven to other actorsin position E5 andno one frompositionE4cites an actorin position E5. Within the overall system, only 3 percentof the available citations are given to actors in position E5. Actors in position I3give 72 percentof their citations to actorsin the primaryposition I1,while no onein positions I1, or I2 gives information-seekingsociometric citations to actorsoccupying I3.Insummary, tructure n a networkof observedrelationscan be characterizedeither as a patternof relations among M structurallynonequivalent, idiographicpositionsanda residualcategoryof nonequivalentempiricalpositions of actors as isdone in Table 1, or it can be characterized n terms of the reflectionof forms ofinteraction,nomotheticpositions, in the idiographicpositions.15 Under the lattercharacterization,the networks in Table 1 would be described as follows: thenetworkof economic exchange relations consists of three primarypositions, oneisolate position, one sycophant position and a residual category of actors, thenetworkof socialexchangerelationsconsistsof threeprimarypositions,a sycophantposition and a residualcategoryof actors, and the network of information-seekingrelations consists of two primary positions, one sycophant position and a singleisolate. The characterizationn termsof idiographicpositions as given in Table1 isthe more accurate,however, the characterization n terms of nomothetic positionsemphasizesthe formsof interactionn a network.The choice between thealternative



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Positions in Networks I 109characterizationscan only be made based on the purposes of a given researchproject.

POSITIONSN NETWORKSAS UNOBSERVEDVARIABLESIN STRUCTURAL QUATIONSMore important han their utility as a means of characterizing he structure n anobserved network of relations, the differentiationof distinct types of networkpositions sets the stage for making statementsabout the expected antecedents andconsequences of actors occupying variousnetworkpositions. To the extent that aparticularnetworkplays a significant partin some social phenomenon, occupancyof different ypes of positions withinthe networkshould be associatedwith differentantecedents and consequences. Related to the questions why particularnetworkstructures houldbe observedgiven specific conditionsin a system of actorsis theproblemof measurement rror n relationaldata. (See Holland and Leinhardt,c, fora detailed discussion of potentialconsequences for a networkanalysis of error inrelational data.) All three problems (the etiology of network structure,the con-sequences of network structureand the measurementerrorusually associatedwithdata) can be rigorously investigated using the ideas presented in the previoussections.Let Dnnbe the (N by N) symmetricmatrixof social distances among the Nactorsin a network. Column of the matrix,Dj, is a (N by 1) vector of the socialdistancesof each actor from the networkposition occupiedby actorj. The set of Ndifferentcolumnvectorsin Dnncanbe partitioned nto M + 1sets correspondingothe M + 1 structurallynonequivalent sets of actors located in the hierarchicalclusteranalysis.Eachof the vectorsof social distancesto actorswithin one of theMsets of structurally quivalent actorscan now serve as an indicator of the truevector of social distances separatingeach actor from the jointly occupied position.The idea here is that there is an unobserved, true vector of social distancesseparatingeach actor from a jointly occupied position in the network and thatthisunobservedvector is responsible for the observed variation n each of the indicatorvectors of social distances. Just as socioeconomic status canbe conceptualizedas anunobserved variable with level of completed education and dollars of incomeserving as indicator variables, the sycophant position in the network of socialexchangerelations, S4 in Table 1, can be conceptualizedas an unobservedvariablewiththepositionsof actors 11, 21 and45 serving as indicatorvariables.In algebraicform, this suggeststhreeepistemic statements inkingthe unobservedvariableto itsindicators where all variables are expressedas deviationscores):D = (6 I1,S4)S4 + Wll, (4a)D21= (621,S4)S4 + W21, (4b)D4= (645,S, S4 + W45, (4c)



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110 I Social Forces I vol. 55:1, september1976where WI11,W21and W45 are vectors of errorcomponents in the observed socialdistancesfrom actors 11, 21 and45 respectively, OiLS4) is a covariancebetweentheunobservedpositionS4 andthe observedposition of actor i, andS4 iS the vector oftrue social distances separating each of the N actors from the unobservedsycophantposition.A (3 by 3) observedvariance-covariancematrix,S, can be estimatedfromthe three observed vectors:D1I, D21and D45. The matrix S can then be approxi-matedby a predictedvariance-covariancematrix, X, which is basedon six unknownparameters ssociatedwith equation(4) (the variance of S4 is assumedstandardizedin equation5 in orderto makethe equation dentified):

S X =821 [02S4](611621645) +0

W21 ? (5a)L545J L0 0 021

or, expressedin matrixnotation:S = X = zA[p] Al + 02, (Sb)where the subscriptsto the delta coefficients, 6, have been abbreviated o includeonly the indicatornumbersince they all contain S4, 'F2S4 iS the variance-herestandardized-of the vector of true social distances separatingactors in thenetwork rom positionS4, and 02wll, 02w21 02,45 are variancesof the error coresin the vectors WI,, W21, W45. Equation(5) is the basic factor analysis model forwhich a varietyof estimationproceduresareeasily available(see e.g., Mulaik). 6The confirmatoryactoranalyticmodel has the sameform as the unrestrictedmodel in equation(5), except that specificunknownparametersn Av,0 or ? canbeforced to equal a priori values. (See e.g., Joreskog, a; Lawley and Maxwell;Mulaik.) Joreskog (b) extends the multiple factor, confirmatoryfactor analyticmodel such that causal inferencescan be drawnamongunobservedvariablesbasedon the observed covariancesamongthe indicatorvariables. (See JoreskogandvanThillo for the computerprogramassociatedwiththe model and Burt,c; Wertset al.;Alwin and Tessler for applied discussion of the model.) Assuming a populationmultivariatenormal distributionon the indicators,maximum-likelihoodestimatesare obtained for unknown parametersin the model and a chi-square statistic isroutinely availablewhich assesses the significanceof the differencebetweenS andX and is distributedwith degreesof freedomequalto the numberof overidentifyingrestrictionson the model being estimated.17As an illustrationof the investigationof the antecedentsand consequencesof aspectsof networkstructurevia structural quationmodels, consider the diagramin Figure4. In Figure 4, squaresenclose unobservedvariablesrepresenting dio-graphic positions (labeled with the nomothetic position to which each is mostsimilar) and circles enclose the observed positions of actors. E5 is the sycophantposition in the observed networkof economic exchange relations. S4 is the syco-



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112 / Social Forces / vol. 55:1, september1976phantpositionin the observednetworkof social exchange relations.I1, I2 and13 arethe M structurallyunique, jointly occupied network positions in the network ofinformation seeking relations. The observed positions of actors 3, 19 and 38 inthe network of economic exchange relationsare specified as indicatorsof E5 (seeFigure 1). The observedpositions of actors 11, 21 and45 in the networkof socialexchange relations are specified as indicators of S4 (see Figure 2). Similarly,observedpositions in the networkof informationseeking relations are specified asindicatorsof their respective networkpositions (see Figure 3).Figure 4 illustratesthe idea of specifying jointly occupied networkposi-tions as unobserved variablesin structural quationmodels. Its generalizability snegligible since the structural equations determining S4, I, 12 and 13 have notheoretical rationale,'8 and the model (a restricted factor analysis with the factordispersion matrix unconstrained) has to be further restrictedin order to insureunique unstandardizedparameterestimates of other than the errorvariances (thestandardized stimates are unique). As a way of thinking, however, Figure4 dealswith an interestingphenomenon and one that is difficult to analyze using moretraditional conceptualizations. It is hypothesized that being a sycophant in thenetwork of economic exchange relations leads an actor to occupy a sycophantposition in the network of social exchange relations. Occupying a sycophantposition in either the network of social or the network of economic exchangerelations then affects the type of position the actor occupies in the network ofinformationseeking relations.Table 3 presents the correlations and standarddeviations from which theunknownparameters n Figure 4 have been estimated. Table4 presentsthe standard-ized and unstandardizedparameterestimates for cPand A.19 The chi-square ap-proximation ndicates the adequate it of the hypothesized model to these data.The unobservednetworkpositionsin Figure4 measure he social distance ofan actorfrom each position in the metric of social circles where a distance of onesocial circle from actor is defined as the numberof sociometricchoices away fromactor actorj must be in orderfor the probabilityof actor not initiating nteractionwithj to equal 1.0 (see Appendix A). Accordingly,the path coefficient,P,s leadingfrom unobserved position E to unobserved position S measures the expectednumber of social circles of change in an actor's social distance from S which willoccur as a result of an increaseof one social circle in the social distance of the actorfrom position E. For example, if an actor in the network of economic exchangerelationsmanipulatedhis exchange relations such that his position in the networkmoved one social circle away from the sycophantposition, E5, then other thingsequal, proceeding as if the coefficients in Figure 4 are unique, he would simul-taneously move .37 social circles away from the position jointly occupied bysycophants in the network of social exchange relations (Ps4ee = .37). In the in-formationseeking network, he would move .28 social circles toward the primaryposition I, (Pile5= -.28) he would remain relatively the same social distancefromposition '2 (Pi2e. = .05) and he would move .38 social circles awayfrom the



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Positions in Networks / 113Table 3. CORRELATIONSAND STANDARDDEVIATIONSFORTHE OBSERVED POSITIONSIN THE EXCHANGENETWORKSAND INFORMATION EEKINGNETWORKWHICHARE SPECIFIEDASINDICATORVARIABLES NFIGURE4*

D D D D D D D D D D D D3 19 38 11 21 45 4 6 35 42 8 30D 1.003D .76 1.0019D .83 .82 1.0038D .31 .27 .33 1.0011D .30 .28 .32 .97 1.00D .35 .29 .39 .90 .92 1.0045D4 -.34 -.33 -.36 -.41 -.44 -.48 1.00D6 -.30 -.30 -.30 -.31 -.33 -.36 .89 1.00D35 .09 .05 .08 .09 .10 .14 -.56 -.49 1.00D .06 .02 .06 .08 .08 .13 -.54 -.46 .96 1.0042D .47 .34 .39 .40 .43 .45 -.87 -.72 .35 .34 1.00D .31 .24 .26 .32 .35 .37 -.77 -.62 .29 .28 .86 1.0030

standarddeviations .83 .79 .74 .89 .90 .91 1.16 .77 .76 .79 .90 .76*Positions D3, D and D38 are from the economic exchange network, D1, D21 and D45 are from the

social exchange network, and D4, D6, D35, D42, D and D30 are from the information seeking network.

823sycophantposition 13 (Pi3e5 = .38).2 In a similar manner,the path coefficientsleading from the sycophantposition in the network of social exchange relationstothe three ointly occupied positionsI1, I2 and13 canbe interpretedn terms of socialcircles.

COMMENTSUsing the idea of a position in a network,I have sought to extend the analysis ofnetworkstructure nto areasof investigationwhich heretoforehave been difficult tohandle with more traditionalperspectives. Several advantagesderive from theperspectiveoutlined n thepreviousdiscussions.First,multidimensional symmetricrelational measures are expressed in a symmetricscalar of social distance whichincorporatesrelations toward and from actors so that easily available cluster andfactoranalytic algorithmscan be used to analyze asymmetricrelationaldata(Sec-tion 1). Second, the conceptof structural quivalenceis weakenedin order o allowfor the possibility of measurementerrorin the observed positions of structurallyequivalent actors (Section 2). Third, network structure s characterizedby therelations among M + 1 structurallynonequivalentsets of actors in a network-whereonly M sets of actors ointly occupy networkpositions-so that each actor isdiscussed in terms of his social distance from the M jointly occupied networkpositions rather hanforcing himto occupyone of them, i.e., rather hanpartitioning



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114 I Social Forces I vol. 55:1, september1976Table 4. VARIANCE-COVARIANCEMATRIX,D,AND FACTORLOADINGMATRIX,A, FOR THE MODELIN FIGURE4*

E5 S4 1 2 31.04 (.34) (-.38) (.08) (.44)

.39 1.22 (-.44) (.10) (.43)) = -.43 -.54 1.21 (-.56) (-.88)

.08 .11 -.63 1.03 (.36)L53 .57 -1.14 .43 1.40 J

.72 (.88) .0 .0 .0 .0

.67 (.87) .0 .0 .0 .0

.69 (.94) .0 .0 .0 .0

.0 .79 (.97) .0 .0 .0

.0 .81 (1.0) .0 .0 .0A= .0 .76 (.92) .0 .0 .0.0 .0 1.06 (1.0) .0 .0

.0 .0 .62 (.89) .0 .0

.0 .0 .0 .75 (.99) .0

.0 .0 .0 .75 (.97) .0

.0 .0 .0 .0 .75 (.99)

.0 .0 .0 .0 .56 (.87)*Standardized estimates are given in parentheses.

the network into nonoverlapping cliques (Section 3). Fourth, network structureis characterizedby the elementary types of jointly occupied positions of which itis composed so that the etiology and consequences of occupying different typesof positions in networks on concomitant social phenomena can be investigated(Section 3). And finally,networkpositionsare specifiedas unobservedvariables nstructuralequation models so that the flexibility, mathematicalsimplicity andprecisionof structural quation models can be broughtto bearon questionsof theetiology and consequencesof network structure, o analyzethe applicabilityof theusual no errors-in-variablesassumption in network analysis, and to assess theadequacyof alternativehypothesizednetwork structures s descriptionsof observednetworksof relations (Section4).In order to simplify the exposition here, a single network of relationshasbeen assumed within a system of actors. Multiple network systems of actors areaddressed elsewhere as a simple expansion of the conceptualizationgiven here(Burt, 1976f). A computerprogram, STRUCTURE,is availablein FORTRANIVfor the IBM 370 thatinputseither a sociometricchoice matrix as analyzedhere ormultiple networks of relations defined on any other grounds and outputs socialdistances, a hierarchical cluster analysis of empirical positions of actors in anetworkor acrossmultiplenetworks,and various sociometric ndices specifiedherein footnote fifteen and in Lin. Card mages are outputso as to be used in analysesofantecedentsandconsequencesof anactor's empiricalpositionin a systemof actors.



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Positions in Networks / 115NOTES1. Let Gik be a vector of actor i's scores on K variables which characterize ome domain of attributesover which actor i is to be compared o other actors. Let GiZkbe actor 's perceptionof actorj's scores onthe sameK variables.Then a composite score of the differencebetween actor and actorj-as perceivedby i-can be given as the Euclidean distance between the vectors Gik and Gijk (See Cronbach andGleser's discussionof weighting profile differences).2. Burt (b) and Burt and Lin discuss the use of content analysis of archivalrecordsas a means of generat-ing relationaldata over time which is based on the idea that the relations between actors in a system arereflectedin the joint occurrenceof actors in descriptionsof events occurring n the system.3. This is the standard ociometric data generatedby questions such as: Who are yourbest friends? ,With whom do you most often discuss communityaffairs? , From whom do you most often purchasegoods? , etc.4. Stemming from the work of Festinger,Luce and Luce and Perry,matrixmultiplication s often used todetermine he smallest numberof sociometric choice links requiredby actor i to reach actor]. If Z is an(N by N) matrixof binary sociometricchoice data (fromfootnote3), thenthe matrixof shortestpossiblechoice link distances among actors can be foundthrough he sum Z* = Z + Z2 + . . . + ZN 1, wherenonzeroelements in the ith power of the original matrixare set equal to i, each successive power of theoriginal matrix has deleted from it any elements which were nonzero in lower powersof the matrixandzii* alwaysremainsequal to zero. The longestchainof choice linksconnectingtwo actors n the networkequals q choice links when Zq+lhas no nonzeroelements which were nonzero in lower powers of thematrix.5. In regard to the shortest numberof sociometric choice links connecting actor i with actorj (z*ij infootnote 4), Katz (b) suggests that as choice links become further emoved from actor , they should haveless significance for him than do choice links close to him. Katz suggests inserting a scale factor in thecomputationof Z* which decreases as Z is raised to increasing powers. Hubbell outlines a weightingscheme which weighs choices from prestigious actors more heavily than choices from relativelyinsignificantactors.6. See for example, the measureof individualdistance outlined in Appendix A as the probabilityof actori notinitiating nteractionwith actorj. This operationalizations theone which is used in the forthcomingnumerical llustrations.7. Even whenIdijis analyzed by a nonmetricclusteringalgorithm or bending data to a specified numberof dimensions (e.g., Lingoes; Roskam and Lingoes; Young) the analysis will overlook the structuralaspectof social distancedue to the reliance by these methods on the criterionof monotonicitywhich isonly concerned with the distance between pairs of actors-even though the entire (N by N) matrixisbeing analyzed. The inabilityof methods of analysis based on the criterionof monotonicity to considerindirect inkages among actors is succinctly discussed by Coleman.8. Equation (1) makes two major assumptionsabout the relationsin a network.First, it is assumedthatthe individual distance of an actor from himself is zero, (Idii = 0). When this is not the case, thenequation (1) can be given as:d*u = V[(IDi* - IDj*)'(IDi* - IDj*)], which is quite different fromequation (1). In order for actorsi andj to have zero social distance in a networkwhere {Idii}4 0, theymust have zero differences between theirrespectiverelationswith each other actor in the networkandthey must relate to each other in the same manner hatthey relate to themselves. Equation (1) is a lessgeneralstatementof the same idea. If it is assumedthe {Idii1= 0, thendijwill only equalzero if actorsandj have zero individualdistances between them and identicalrelations o other actors n the network.Ihave subtractedID2j and ID2j from the equation in (1) since these terms appeartwice in d*ij when{Idii}= 0. A second assumption made in equation (1) is that all relations among actors should beweighted equally. This assumptioncan be deleted if the equation s complicatedstill further.Let W be a(2N by 2N) diagonal matrixwhere {wii} = {Wi+NAi+NI the weight of the importanceof actor i relativeto the other actors in the network (e.g., prestige scores, power scores, etc.). The generalizedform ofequation (1) given abovecan be furthergeneralized o takeinto accountthe differential mportanceof theactors n the networkas d*i*j= V [(IDi* - IDj*)' (W)(IDi* - IDj*)]. Inorder o weightdifferent ypesof relationsmore heavily than others(e.g., to give null relationsmoreimportanceovernonzerorelations)all 2N diagonal elements can vary accordingto the difference term they weight. For example if zeroindividual distances were to be weighted as twice as importantas nonzero relations, the diagonalelements n Wcould be defined as {Wkkl = 2 if Idik orIdJk, Idki orIdkjequal zero, otherwise t equals 1.9. Katz(a), in a discussion of the general utilityof treatingsociometricchoice datavia matrixmethods,suggeststhe cosine of the angle between the choice vectors of two actors as a measureof theirsimilarity.This measure s the correlationbetween the rows in ID,, when individualdistance is operationalizedas



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116 / Social Forces / vol. 55:1, september1976given in footnote 3. If the vector is extendedto be defined in terms of relations from other actors as wellas to other actors, then Katz's idea could be applied here as the cosine between IDi* and IDj* as asymmetricmeasure of the social distance between actors i andj. I have instead selected the Euclideandistancein equation (1) because it does not force the linearity assumptionon the quantification f socialdistance.10. In a similar sense the idea of individual distances being transitive or balanced is invariant overresearchapplications (see e.g., Davis, a, b; Hallinan;Heider; Holland andLeinhardt,a, b). The socialdistancebetween actors i and can be seen as a measure of the dissonanceor imbalance in the overallrelationbetween actors i andj within a specific network.dijwill increase as a function of the differencein relationsbetween actors andj and all actors n the networkbecome different. Unfortunately,much ofthe work on transitivity within a triad does not consider the three person group in terms of the actorslinked to each of the persons in a triad (see Flament;and Hallinan and McFarland or exceptions to thisstatement).In relation to the discussions of transitivity, quation (1) considersall N persons in a networkin order to expand the distance between persons who are in intransitivetriads or tetrads of N-persongroups.11. The definition of strong equivalence is taken from the basic concept of equivalence in typology (seee.g., Kelley, 9-10) discussed in sociology by Holland and Leinhardt b) and Lorrainand White. Lorrainand White(63) suggest: Objects a, b of a category C are structurally quivalent f, for any morphismMand any object x of C, aMx if and only if bMx, and xMa if and only if xMb. In other words, a isstructurally quivalent to b if a relates to every object x of C in exactly the same way as b does. Amorphism s any direct or indirect relationbetween two objects or actors. A category is a set of objectsand a set of morphisms.Inequation (2) the category over which two actorscan be structurally quivalentconsists of the set of N actors as represented n networks of some type of relations and the morphismof individual distance (aMx = Idax, bmx = IdbX, xMa = Idxa, xMb = IdXb). In a related perspective,Holland and Leinhardt (b, 110-1) specify structuralequivalence of actors in an M-clique when theM-clique is situated within a largernetwork of actors. The utility of stating structuralequivalence interms of the scalar in equation (2) is twofold: (1) its greatersimplicity, and(2) its facile generalization oa weak criterionof structural quivalence as given in equation (3).12. Referencethe differencebetween two actors' relationsto otheractors n the networkas structuraldistance. This standsin contrastto individualdistances between the two actorsthemselves. Admittingthe possibility of weak structuralequivalence brings in problems of interpretingthe conditions ofindividualand structuraldistance under which actorswill be deemedequivalent. The only time that thevalues of individualandstructural istanceresponsiblefor an observed socialdistancecan be determinedfrom the observed social distancewith certainty s when dj equals zero. In such a case, all componentdistances have to be zero. In equation (3), however, an individualonly knows thatif dj is less than thecriterion distance then structuraland individual distances are also less than the criterion. Isolates willhave zero structuraldistance between them but individual distancesof 1.0 separating hem so that thesocial distancesamongisolates will equal 1.41. This means that the isolates in a networkwill be weaklystructurally quivalentwhenever the criterionalphais greater han 1.41 (e.g., in the economic exchangenetworkin Figure 3, the isolates are simultaneouslyclustered together as position E2 as soon as thecriterion ncreasesfrom less than 1.4 to less than 1.5). In short, special attentionneeds to be paidto thedegree to which a chosen nonmetric solution has collapsed otherwise meaningful distances amongactors. This caution extends to the clustering algorithmsmentionedby White et al. in order to locateblocks of actors in a network-particularly in the case of solutions which are a lean fit (SeeBreigeret al.).13. The idea of sets of actorsbeing definedby their common reference to actorsor events externalto theset-as are the actors jointly occupying sycophantor brokerpositions as discussed here-has beendiscussed under various labels such as; quasi-groups (Ginsberg, 40), latent groups (Olson, 50),levelling coalitions (Thodenvan Velzen, 241-2) and latentcorporateactors (Burt, b).14. Boissevain (147-69) provides an interesting discussion of some of the expected behaviors andcharacteristics f actorsoccupyingwhat is here described as a brokerpositionin a network.15. This line of reasoningcan be continuedto express the extent to which the empirical position of anactor reflects each of the four nomotheticpositions in Table 2. Define two scalarsdescribing aspects ofactor j's empirical position in a network: Aj = the prominence of actor j within the network =(N - >N J1 d/) IN = 1.0 when all actorshave maximumprobability f initiating nteractionwithactorj;Bj = the tendencyfor actorj to initiate interactiononly with actorsstructurally quivalentto himself =[ Ni=1IdjidjJ / [Ni=1dji] = 1.0 when actorj only has maximumprobabilityof initiatinginteractionwith those actors who have zero social distance fromhim, where Idijand dijarerespectivelyindividual



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Positions in Networks / 117(AppendixA) and social (equation 1) distancefrom actor to actorj. The extent to which actorj occupiesa primary position within the network can then be given as; (A) (Bj). The extent to which actorjoccupies a sycophantpositionin the networkcanbegivenas; (1 - Aj) (1 - Bj). The extent to which actorjoccupies an isolate position is given as; (1 - Aj) (Bj). Finally, the extent to which actor occupies abrokerposition is given as; (Aj) (1 - Bj). Each of the indices equals 1.0 at its maximumandzero at itsminimum. Since they are expressed without referenceto particular elations between particularactors,these indices can be used as crude indicatorsof structural orms reflectedin the empirical positions ofactors n separatenetworksand separatesystems.16. MacRae and Bonacich discuss cliques in networksin terms of a factor analytic model. The presentsituation differs from theirs in two respects. First, use of social distances, equation (1), allowsasymmetric relational data to be expressed in the form of a symmetric scalar so thateasily availablecluster analyticandfactor analytic algorithmscan be used to analyzethe structuren a network.Second,while both MacRae and Bonacich explore possible methods of applying the factor analytic model tolocate cliques, the present approach s using a nonmetric, nonlinear cluster analytic model to locatestructurallynonequivalentpositions and then using a confirmatory actor analytic model to assess theadequacy of the hypothesized structure. This second difference provides a method of assessing thesignificanceof the lack of fitof a hypothesizedclique structure r nonequivalentpositionalstructureo anobserved networkas well as enablinganinvestigator o specify antecedentsandconsequencesof networkstructuren structural quationmodels (see Figure4).17. The chi-squarestatisticis given as a functionof the discrepancybetween S and l;X2= (N -1) [lInI ; I + trace(z-1 S)- In I51- r]where r equals the total numberof observedvariables n a hypothesized structural quationmodel.

Since the social distance function estimatesdj by takinginto accountotherobservations,however,therewill be fewer than N independentobservationsunderlying he chi-square tatisticand the estimationof S.This, in combinationwith the usual small N in networkanalysis, means that the chi-squarestatistic s avery rough approximationand should only be takenas an indication of substantivesignificancewhen itsvalue has a high probabilityof occurringor an extremelylow probabilityof occurring.18. I have in mind here the specification of the determinantsof change in patternsof relations amongactors in a network. How do jointly occupied, structurallynonequivalentpositions come to exist in anetwork? Explanations along this line have been idiographic due to the emphasis in past networkanalyses on description. However, this promises to be a most fruitful area for investigationas severalrecent works emphasize. Jointly occupied positions could arise as a function of causal relationsamongnetworksbased on differenttypes of relations (e.g., Laumann t al.) as a function of social backgrounds(e.g., Barton et al.; Gans; Laumann;LaumannandPappi,b), as a function of changes in environmentalconditionswhich affect the system of actors (e.g., Allen; Burt, d, b, e; StanworthandGiddens)or as afunction of actors purposively exploiting their positions in exchange networksin order to realize goals(e.g., Boorman;Burt, d, e; Granovetter, , b).19. The structural oefficients among the unobservedvariables in Figure 4 are regression coefficientswhich can be estimatedfrom the variance-covariancematrix(Dusing ordinary east squares since themodel specifies no paths equal to zero. These estimateswill be equivalentto those in Figure4 which aremaximum likelihood estimates generated from a confirmatory actor analytic model (programmedasLISREL, Joreskog andvan Thillo). Since the emphasisin the presentdiscussion is on network structurerather than structuralequation models, I have avoided the statistical and algebraic complicationsinvolved in the general estimationandspecificationof coefficients amongunobservednetworkpositionsbeyond the bivariatecovariances in equation (5). The reader interestedin more detailed treatment sreferredto Joreskog (b) for a general discussion or to Burt (a, c), Werts et al., and Alwin and Tesslerfor applieddiscussion.20. I have ignoredthe second-order ffects of thecompoundpathcoefficientsmeasuringchangein posi-tion in the information-seekingnetworkresulting romchangein the socialexchangenetwork.Change nE5 leads to change in S4 which in turn leads to change in II, I2 and I3. For example, an increaseof onesocial circle of distance in E5would lead to an increase of (Pi3e5 + Ps4e. Pi34) = .38 + .13 = .51 socialcircles of distanceaway fromposition I3 instead of the .38 reported n the text.



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118 / Social Forces / vol. 55:1, september 1976APPENDIXA: AN OPERATIONALIZATIONF INDIVIDUALDISTANCEAS THEPROBABILITY F ACTOR NOT INITIATING NTERACTIONWITHACTORJIn the interestof stating ndividual distancein a metricwhich is interpretable cross differenttypes of networks (e.g., networksof friendship relations, of economic exchange relations,etc.) it would be convenient to have a measureof the probability of actor i not initiatinginteractionwith actorj as an operationalization f Idijto be substituted n equation (1) of thetext. In other words, if Idijequals 1.0 in a network of economic exchangerelations,it showsthat actor i will not seek economic goods fromactorj as the network s currentlyarranged.The operationalization iven here is based on the minimum numberof choice links requiredby actor i to reach actorj (zij* in note 4). It has the advantageof a metricinterpretation nterms of thecircleof actors linked to actor within the networkandhasthereforebeen usedinthe text illustrations.The operationalizationof Idij as the above described probabilisticmeasure requiresthree pieces of information: 1) conditionsleadingIdijto equal its minimum,(2) conditionsleadingIdijto equal its maximum,and(3) a functionrelatingchangein Idijto changein zi.Actor i will have a circle of actors with whom he has a nonzero probabilityof initiatinginteraction cf. Kadushin's discussion of the concept of an actor's social circle). Within thissocial circle, actor has thehighest probabilityof interactingwithhimself. Idii = 0. There isa close to zero probabilityof interactionbeing initiatedby actor i toward actors separatedfrom him by an infinitezw*(e.g., isolates in the network or persons in separatesystems ofactors).Idijcan be treatedas having reached ts maximumwhen zu* is larger hanthe largestfinitezij*for actori. Let jjbe the largest finitezij*linking actor to any actor n the network.Since values of zij* which are larger than jj imply a zero probabilityof actor i initiatinginteractionwith actorj, jprovides a boundary or the social circle aroundactor . The valueof 4wan be discussed as the rangeof actori's social circle. Actori would not be expectedtoinitiate interactionwith actorsoutside his social circle unless he needs unusualresourcesnotreadily available from actors within his circle (e.g., see Granovetter, ; Lee;Milgram;White,for discussionof the small worldphenomenonand the use of weak links to realize unusualinterests). Given the above anchorings or the maximum and minimumvaluesof Idjjrelativeto zw*the rateof changein Idij in relation to change in zi.* can be drawn fromresearchonhuman perception. Findings in psychophysics demonstrate he limited ability of persons toretaininformation past a boundary evel of stimulation and inverse ability to discriminatedifferences in levels of stimulation as the overall level of stimulation approachestheboundary evel (see e.g., Haber;Simon). The level of stimulationhere is the proportionofactorsin his social circle with whom actori will initiate interaction.It is to be expectedthatactori would have little difference in Idijfor the few actorswho are close to him relativetothe other actorsin the network, that the changein Idijwould increase with changein z* as afunctionof the numberof actors at a distancez*, and that actor wouldhave littledifferencein Idij for the actorswho are near the boundaryof his social circle andbeyond. Let the Nvalues of zw*be rankordered romlowest to highest for actor i: Changein Idijwould thenbea function of the cumulativenumberof actors n i's social circle who areless thanor equaltoZi.*choice links awayfromi:

,if i =j,cumulativefrequencyof zIj* f *Id13 [(numberof actors n social circle of actor i) + 1]' ifz*1 j1, if z* > (U.

IdWwill be low (i.e., reflect a high probability of i initiating interaction with j) whenZ*ij s low relativeto the otherZik* linking actor i to actorsin the network. The importantcharacteristic f the individualdistance between actori andj-in termsof theprobabilityof iinitiating nteractionwithj-is not the absolute numberof choice links requiredby i to reach



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Positions in Networks / 119j, but rather he locationof actorj in actor i's social circle. The decision to interactwithactorj can be seen as a marginal consumptiondecision. The commodity being consumed here isthe initiation of interactionwith a proportionof the actorsin actor i's social circle. Fromaninitial point of Idij= .5, increases in zw*result in smaller and smallerincreases in Id1juntilIdijreachesits maximumof 1.0. From the same point, decreases in zw*result in smaller andsmallerdecreasesin Idij until Idij reaches its minimumat 0. What this says is that as Zw*approaches ts limits at eitherthe boundaryor center of actor i's social circle, the marginalpropensityof actori to initiateinteractionwith actorj declines.APPENDIXB: CLUSTERSOF ACTORS N NONMETRICCLUSTERING LGORITHMSBASEDONINDIVIDUALVERSUSSOCIALDISTANCESIt is common for individual distancesamong actors to be forced to be symmetric and then tobe subjected o a nonmetricclusteringalgorithm-often a smallest space analysis (see reviewby Bailey). Given this frequentpracticeby sociologists, an importantquestion in referencetothe location of the M structurallynonequivalentpositionsin Section 3 concerns the differentinferences which would result from a nonmetric cluster analysis of symmetrized ndividualdistancesinsteadof the social distances used to compareactors as networkpositions.A comparisonof the smallest space analyses in Figures 1, 2 and 3 with Figures 3, 4and5 in Laumannand Pappi (a) will illustrate he potentiallydifferentconclusions which canbe gleaned from analyses based on individual versus social distancesbetween actors. As ageneralrule of interpretation,an analysis based on individual distances (as exemplified byLaumannandPappi's careful analysis) will conclude that two actors occupy similarpositionsin a network f they interact frequentlyor if they have homophily in terms of some profile ofcharacteristics-i.e., if they have smallindividual distance between them. An analysis basedon social distances (as illustratedhere in Figures 1, 2 and 3) will conclude that two actorsoccupy similar positions if they have the above characteristicof small individual distancebetween themselves as well as having similar individual distances to and from the otheractors in the network-i.e., if they have small social distance. Analyses of individualdistancesemphasize linkages between pairsof actors. Analyses of social distancesemphasizesimilaritiesbetween patternsof relationshipswith the N actors in a system.There are two general characteristicsof a network analysis in terms of which ananalysisof individual distances can be comparedwith one based on social distances:(1) theexpressedstructurally onequivalentpositionsin a network,and (2) the distributionof actorsin referenceto those nonequivalentpositions. Concerningthe firstcharacteristic,notice thatFigures 1, 2 and 3, through the use of social distances, emphasize the similaritybetweenactorsin structurallyequivalent positions and emphasize the differences between actors innonequivalentpositions (under the weak criterion).The structurallynonequivalentpositionsin a network are thereby made more obvious and easier to detect in an analysis. Thisemphasison structures most vividly demonstratedn Figure3 wherethepreviousscatterofactorsthrough he smallestspace analysis of individual distances (Figure5 in LaumannandPappi) is here reduced to three nonequivalent network positions. These structurallynon-equivalentpositionsprovidea set of referencepointsfromwhicheach actor in a networkcanbe located and discussedin contrast o the idea of comparingactorsin terms of theirrelativedistancefrom the center of a smallest space. Concerning the second characteristicof anetworkanalysis given above, actors aredistributedn space with referenceto nonequivalentpositionsaccordingto the principleof interpretingan analysisbasedon individualdistancesversus one based on social distances whichwas emphasized above. Forexample, the isolatesin the economic exchange network are occupying a joint position in the networkinsteadofbeing scatteredaroundthe periphery of the network as they would be in an analysis ofindividual distances. Similarly, the six most influential actors in the community jointlyoccupy a position in the information-seekingnetwork (positionI, in Figure 3) because theyall havethe characteristic f receiving manycitationsfromotheractorsin the networkwhich



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120 / Social Forces / vol. 55:1, september1976they do notreciprocate.Their position in the network s placed on the peripheryof the socialspaceas represented n the smallestspace analysissince they occupy a position which is verydifferent from the positions of most actors in the network. In an analysis of individualdistances these actors would be nearthe center of the space since so many actors claim tointeractwith them (this characteristic s elaboratedby Laumannand Pappi, a, as a measureofthe integrativecentralityof the actor). In an analysis of social distances, on the other hand, anactor is close to the center of the social space when his position is equally dissimilar to thepositions of the other actorsin the network. The more that the actorsjointly occupying anetwork position receive a disproportionate umberof sociometric choices-more or less-then the moreperipheral he position will be in the social space as represented n a smallestspace analysis.REFERENCESAllen, M. P. 1974. The Structureof Interorganizational lite Cooptation: nterlockingCor-porateDirectorates. American Sociological Review 39:393-406.Alwin, D. F., and R. C. Tessler. 1974. CausalModels, Unobserved Variablesand Experi-mental Data. AmericanJournal of Sociology 80:58-86.Bailey, K. D. 1974. Cluster Analysis. In David R. Heise (ed.) Sociological Methodology1975. San Francisco:Jossey-Bass.Barton, A. H., B. Denitch, and C. Kadushin. 1973. Opinion Making Elites in Yugoslavia.New York:Praeger.Boissevain, J. 1974. Friends of Friends: Networks,Manipulators and Coalitions. Oxford:Blackwell.Bonacich, P. 1972. Factoring and WeightingApproaches to Status Scores and Clique

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