An Experimental Study of the Small World Problem Author(S): Jeffrey Travers and Stanley Milgram Source: Sociometry, Vol

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An Experimental Study of the Small World Problem Author(s): Jeffrey Travers and Stanley Milgram Source: Sociometry, Vol. 32, No. 4 (Dec., 1969), pp. 425-443 Published by: American Sociological Association Stable URL: http://www.jstor.org/stable/2786545 Accessed: 23/09/2010 13:05

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http://www.jstor.org An ExperimentalStudy of the Small World Problem* JEFFREY TRAVERS Harvard University

AND STANLEY MILGRAM The City Universityof New York

Arbitrarilyselected individuals (N-296) in Nebraskaand Boston are asked to generateacquaintance chains to a targetperson in Massachusetts,employ- ing "the small world method" (Milgram, 1967). Sixty-fourchains reach the targetperson. Within this group the mean numberof intermediariesbe- tweenstarters and targetsis 5.2. Boston startingchains reach the target personwith fewer intermediaries than those starting in Nebraska; subpopula- tionsin the Nebraskagroup do not differamong themselves. The funneling of chainsthrough sociometric "stars" is noted,with 48 per centof the chains passingthrough three persons before reaching the target.Applications of the methodto studiesof large scale social structureare discussed.

The simplestway of formulatingthe small worldproblem is "what is the probabilitythat any twopeople, selected arbitrarily from a largepopulation, such as thatof the UnitedStates, will knoweach other?"A moreinteresting formulation,however, takes accountof the fact that,while persons a and z may not know each other directly,they may share one or more mutual acquaintances;that is, theremay exist a set of individuals,B, (consisting of individualsbi, b2 . . . bn) who know both a and z and thus link them to one another.More generally,a and z may be connectednot by any single commonacquaintance, but by a seriesof such intermediaries,a-b-c- . . . -y-z; i.e., a knowsb (and no one else in the chain); b knowsa and in addition knowsc, c in turnknows d, etc. To elaboratethe problemsomewhat further, let us representthe popula-

* The study was carried out while both authors were at Harvard University,and was financedby grants from the Milton Fund and from the Harvard Laboratory of Social Relations.Mr. JosephGerver provided invaluable assistancein summarizingand criticizing the mathematicalwork discussedin thispaper. 425 426 SOCIOMETRY tion of the United States by a partiallyconnected set of points.Let each point representa person,and let a line connectingtwo points signifythat the two individualsknow each other. (Knowingis hereassumed to be symmetric:if a knowsb then b knowsa. Substantively,"knowing" is used to denotea mutualrelationship; other senses of the verb,e.g. knowingabout a famousperson, are excluded.) The structuretakes the formof a cluster of roughly200 millionpoints with a complexweb of connectionsamong them.The acquaintancechains describedabove appear as pathwaysalong connectedline segments.Unless some portionof the populationis totally isolated fromthe rest,such that no one in that subgroupknows anyone outsideit, theremust be at least one chain connectingany twopeople in the population.In generalthere will be manysuch pathways,of variouslengths, betweenany two individuals. In view of such a structure,one way of refiningour statementof the smallworld problem is thefollowing: given two individuals selected randomly fromthe population,what is the probabilitythat the minimumnumber of intermediariesrequired to link them is 0, 1, 2, . . . k? (Alternatively,one mightask not about the minimumchains betweenpairs of people, but mean chain lengths,median chain lengths,etc.) Perhaps the most directway of attackingthe small worldproblem is to tracea numberof real acquaintancechains in a largepopulation. This is the techniqueof the study reportedin this paper. The phrase "small world" suggeststhat social networksare in some sense tightlywoven, full of unex- pected strandslinking individuals seemingly far removedfrom one another in physicalor social space. The principalquestion of the presentinvestiga- tion was whethersuch interconnectednesscould be demonstratedexperi- mentally. The only example of mathematicaltreatment dealing directlywith the small world problemis the model providedby Ithiel Pool and Manfred Kochen (unpublishedmanuscript). Pool and Kochen assume a population of N individuals,each of whom knows,on the average, n othersin the population.They attemptto calculatePk, the probabilitythat two persons chosenrandomly from the group can be linkedby a chainof k intermediaries. Theirbasic modeltakes the formof a "tree" or geometricprogression. Using an estimateof averageacquaintance volume provided by Gurevitch(1961), theydeduce that two intermediarieswill be requiredto link typicalpairs of individualsin a populationof 200 million.Their model does not take account of social structure.Instead of allowing acquaintance nets to define the boundariesof functioningsocial groups,Pool and Kochen must,for the pur- poses of theirmodel, conceive of societyas beingpartitioned into a number of hypotheticalgroups, each withidentical populations. They are thenable STUDY OF THE SMALL WORLD PROBLEM 427 to devisea way to predictchain lengths within and betweensuch hypothesized groups. In an empiricalstudy related to the small worldproblem Rapoport and Horvath (1961) examinedsociometric nets in a junior high school of 861 students.The authors asked studentsto name in order their eight best friendswithin the school.They thentraced the acquaintancechains created by the students'choices. Rapoportwas interestedin connectivity,i.e. the fractionof the total populationthat would be contactedby tracingfriend- ship choices from an arbitrarystarting population of nine individuals. Rapoportand his associates (Rapoport and Horvath, 1961; Foster et al., 1963; Rapoport, 1953; 1963) have developed a mathematicalmodel to describethis tracingprocedure. The model takes as a point of departure randomnets constructedin the followingmanner: a small numberof points is chosenfrom a largerpopulation and a fixednumber of "axones" is ex- tendedfrom each of thesepoints to a set of targetpoints chosen at random fromthe population. The same fixednumber of axonesis thenextended from each of the targetpoints to a set of second generationtarget points, and the process is repeatedindefinitely. A targetpoint is said to be of the tth removeif it is of the tth generationand no lowergeneration. Rapoport then suggestsa formulafor calculatingthe fraction,Pt, of the population pointswhich are targetsof the tth remove.He is also able to extendthe formulato nonrandomnets, such as those created in the Rapoport and Horvath empiricalstudy, by introducinga numberof "biases" into the randomnet model.Rapoport shows that two parameters, obtainable from the data, are sufficientto producea close fitbetween the predictions of the model and the empiricaloutcome of the traceprocedure.1 Rapoport'smodel was designedto describea trace procedurequite dif- ferentfrom the one employedin the presentstudy; however,it has some relationto the small worldproblem. If we set the numberof axones traced froma given individualequal to the total numberof acquaintancesof an average person, the Rapoport model predicts the total fractionof the populationpotentially traceable at each remove from the start, serving preciselythe aims of the model of Pool and Kochen. (It should,however, be noted that Rapoport'smodel deals with asymmetricnets, and it would be difficultto modifythe model to deal withgeneral symmetric nets, which characterizethe small world phenomenon.) Despite the goodnessof fitbetween Rapoport's model and the data from

1 Thereis additionalempirical evidence (Fararo and Sunshine,1964) and theoretical support(Abelson, 1967) forthe assumption that two parametersare sufficientto describe the Rapoporttracing procedure, i.e. thatmore complex biases have minimaleffects on connectivityin friendshipnets. 428 SOCIOMETRY two large sociograms,there are unsolved problems in the model, as Rapoport himselfand others (Fararo and Sunshine,1964) have pointed out. The Pool-Kochenmodel involvesassumptions difficult for an empiri- cally orientedsocial scientistto accept,such as the assumptionthat society may be partitionedinto a set of groupsalike in size and in internaland externalconnectedness. In the absence of empiricaldata, it is difficultto knowwhich simplifying assumptions are likelyto be fruitful.On the other hand, with regardto the empiricalstudy of Rapoport and Horvath,the fact that the total populationemployed was small,well-defined, and homo- geneousleaves open manyquestions about the natureof acquaintancenets in thelarger society.2 An empiricalstudy of Americansociety as a wholemay well uncoverphenomena of interestboth in theirown rightand as con- straintson the natureof any correctmathematical model of the structure of large-scaleacquaintanceship nets.

PROCEDURE This paper followsthe procedurefor tracingacquaintance chains devised and firsttested by Milgram (1967). The presentpaper introducesan ex- perimentalvariation in this procedure,by varying"starting populations"; it also constitutesa firsttechnical report on the small world method. The proceduremay be summarizedas follows:an arbitrary"target person" and a groupof "startingpersons" were selected,and an attemptwas made to generatean acquaintancechain fromeach starterto the target.Each starterwas providedwith a documentand asked to beginmoving it by mail towardthe targetThe documentdescribed the study,named the target,and asked the recipientto become a participantby sendingthe documenton. It was stipulatedthat the documentcould be sent only to a first-name acquaintanceof the sender.The senderwas urged to choose the recipient in such a way as to advancethe progress of the documenttoward the target; several itemsof informationabout the targetwere providedto guide each new senderin his choice of recipient.Thus, each documentmade its way along an acquaintancechain of indefinitelength, a chain whichwould end only when it reached the targetor when someonealong the way declined to participate.Certain basic information,such as age, sex and occupation, was collectedfor each participant. 2 In additionto thePool-Kochen and Rapoportwork, there are numerousother studies of socialnetwork phenomena tangentially related to the small-worldproblem. Two well- knownexamples are Bailey'sThe MathematicalTheory of Epidemicsand Coleman,Katz and Menzel'sMedical Innovation. Bailey's work deals withdiffusion from a structured source,rather than with convergence on a targetfrom a set of scatteredsources, as in the presentstudy. The Coleman,Katz and Menzelstudy deals withan importantsub- stantivecorrelate of acquaintancenets, namely information diffusion. STUDY OF THE SMALL WORLD PROBLEM 429

We wereinterested in discoveringsome of the internalstructural features of chains and in making comparisonsacross chains as well. Among the questionswe hoped to answerwere the following:How manyof the starters -if any-would be able to establishcontact with the target througha chain of acquaintances?How many intermediarieswould be requiredto link the ends of the chains? What formwould the distributionof chain lengthstake? What degreeof homogeneityin age, sex, occupation,and other characteristicsof participantswould be observedwithin chains? How would completechains differ from incomplete on theseand otherdimensions? An additional comparisonwas set up by using three distinctstarting subpopulations.The targetperson was a Boston stockbroker;two of the startingpopulations were geographicallyremoved from him, selected from the state of Nebraska. A thirdpopulation was selected fromthe Boston area. One of the Nebraska groupsconsisted of bluechipstockholders, while the secondNebraska group and the Boston groupwere "randomly"selected and had no special access to the investmentbusiness. By comparisonsacross thesegroups we hoped to assess the relativeeffects of geographicaldistance and of contactwith the target'soccupational group. Moreoverwe hoped to establisha strategyfor futureexperimental extensions of the procedure, in whichthe sociological characteristics of the startingand targetpopulations wouldbe systematicallyvaried in orderto exposefeatures of social structure. The primaryresearch questions, then, involved a test of the feasibility and fruitfulnessof the methodas well as an attemptto discoversome ele- mentaryfeatures of real social nets. Several experimentalextensions of the procedureare alreadyunderway. A moredetailed description of the current methodis givenin the followingsections. PARTIcIPANTS.Starting Population. The startingpopulation for the study was comprisedof 296 volunteers.Of these,196 were residentsof the state of Nebraska,solicited by mail. Withinthis group,100 were systematically chosen owners of blue-chipstocks; these will be designated"Nebraska stockholders"throughout this paper. The restwere chosenfrom the population at large; thesewill be termedthe "Nebraska random"group. In addition to the two Nebraska groups,100 volunteerswere solicitedthrough an advertisementin a Boston newspaper(the "Boston random"group). Each memberof the startingpopulation became the firstlink in a chain of acquaintancesdirected at the targetperson. Intermediaries.The remainingparticipants in the study,who numbered 453 in all, werein effectsolicited by otherparticipants; they were acquain- tancesselected by previousparticipants as people likelyto extendthe chain towardthe target.Participation was voluntary.Participants were not paid, nor was moneyor otherreward offered as incentivefor completion of chains. 430 SOCIOMETRY

THE DOCUMENT.The 296 initialvolunteers were sent a documentwhich was the-principal tool of the investigationsThe documentcontained: a. a descriptionof the study,a requestthat the recipientbecome a participant,and a set of rules forparticipation; b. thename of the targetperson and selectedinformation concerning him; c. a roster,to whicheach participantwas asked to affixhis name; d. a stack of fifteenbusiness reply cards asking informationabout each participant.

Rules for Participation.The documentcontained the followingspecific instructionsto participants: a. Add yourname to the rosterso that the nextperson who receivesthis folderwill knowwhom it came from. b. Detach one postcardfrom the bottomof this folder.Fill it out and returnit to Harvard University.No stamp is needed. The postcard is very important.It allows us to keep track of the progressof the folderas it movestoward the targetperson. c. If you know the targetperson on a personalbasis, mail this folder directlyto him (her). Do this only if you have previouslymet the targetperson and know each otheron a firstname basis. d. If you do not know the targetperson on a personal basis, do not try to contact him directly.Instead, mail this folderto a personal acquaintancewho is morelikely than you to know the targetperson. You may send the bookleton to a friend,relative, or acquaintance,but it mustbe someoneyou knowpersonally.

TargetPerson. The targetperson was a stockholderwho lives in Sharon, Massachusetts,a suburb of Boston, and who worksin Boston proper.In additionto his name,address, occupation and place of employment,participants were told his college and year of graduation,his militaryservice dates, and his wife'smaiden name and hometown.One questionunder in- vestigationwas the typeof informationwhich people would use in reaching the target. Roster.The primaryfunction of the rosterwas to prevent"looping," i.e., to preventpeople fromsending the documentto someonewho had already receivedit and sent it on. An additionalfunction of the rosterwas to moti- vate people to continuethe chains.It was hoped that a list of priorpartic- ipants,including a personalacquaintance who had sent the documentto A photographicreproduction of this experimentaldocument appears in Milgram, 1969: 110-11. STUDY OF THE SMALL WORLD PROBLEM 431 the recipient,would createwillingness on the part of thosewho receivedthe documentto send it on. TracerCards. Each participantwas asked to returnto us a businessreply card givingcertain information about himselfand about the personto whom he sent the document.The name, address,age sex and occupationof the senderand sender'sspouse were requested,as were the name, address,sex and age of the recipient.In addition,the natureof the relationshipbetween senderand recipient-whetherthey were friends,relatives, business associates, etc.-was asked. Finally, participantswere asked why they had selectedthe particular recipient of the folder. The businessreply cards enabledus to keep runningtrack of the progress of each chain. Moreover,they assured us of gettinginformation even from chainswhich were not completed,allowing us to make comparisonsbetween completeand incompletechains.

RESULTS COMPLETIONS.217 of the 296 startingpersons actually sent the document on to friends.Any one of the documentscould reach the targetperson only if the followingconditions were met: 1) recipientswere sufficiently motivated to send the documenton to the nextlink in the chain; 2) participantswere able to adopt some strategyfor moving the documentscloser to the target (this conditionfurther required that the given informationallow them to select the next recipientin a mannerthat increasedthe probabilityof con- tactingthe target); 3) relativelyshort paths were in fact requiredto link startersand target(otherwise few chainswould remainactive long enough to reach completion).Given these contingencies,there was serious doubt in the mindof the investigatorswhether any of the documents,particularly thosestarting in an area remotefrom the targetperson, could move through interlockingacquaintance networks and convergeon him. The actual out- comewas that64 of the folders,or 29 per centof thosesent out by starting persons,eventually reached the target. DISTRIBUTION OF CHAIN LENGTHS. CompleteChains. Figure 1 showsthe frequency distribution of lengths of the completed chains. "Chain length" is heredefined as the numberof intermediariesrequired to link startersand target.The mean of the distributionis 5.2 links. It was unclearon firstinspection whether the apparentdrop in frequency at the medianlength of fivelinks was a statisticalaccident, or whetherthe distributionwas actually bimodal. Furtherinvestigation revealed that the summaryrelation graphed in Figure 1 concealed two underlyingdistribu- tions: whenthe completedchains were dividedinto thosewhich approached the target throughhis hometownand those which approached him via 432 SOCIOMETRY

20 -

15 - Z~~~

U : A \ Nx64 0 10 a: w

0 0 I 2 3 4 5 6 7 8 9 10 I 1 12 NUMBEROF INTERMEDIARIES FIGURE 1

Lengthsof CompletedChains

Boston business contracts,two distinguishabledistributions emerged. The mean of the Sharondistribution is 6.1 links,and thatof the Bostondistribu- tion is 4.6. The differenceis significantat a level betterthan .0005, as assessed by the distribution-freeMann-Whitney U test. (Note that more powerfulstatistical tests of the significanceof differencesbetween means cannotbe appliedto thesedata, sincethose tests assume normality of under- lying distributions.The shape of the true or theoreticaldistribution of lengthsof acquaintancechains is preciselywhat we do not know.) Qualitatively,what seems to occur is this. Chains which convergeon the targetprincipally by using geographicinformation reach his hometown or the surroundingareas readily,but once thereoften circulate before en- teringthe target'scircle of acquaintances.There is no available information to narrowthe fieldof potentialcontacts which an individualmight have withinthe town.Such additionalinformation as a list of local organizations -STUDY OF THE SMALL WORLD PROBLEM 433 of which the targetis a membermight have provideda natural funnel, facilitatingthe progressof the documentfrom town to targetperson. By contrast,those chains which approach the target throughoccupational channelscan take advantageof just such a funnel,zeroing in on him first throughthe brokeragebusiness, then throughhis firm. IncompleteChains. Chains terminateeither through completion or dropout: each dropoutresults in an incompletechain. Figure 2 showsthe number of chainswhich dropped out at each "remove"from the startingpopulation. The "Oth remove"represents the startingpopulation itself: the "firstremove" designatesthe set of people who receivedthe documentdirectly from membersof the startingpopulation. The "second remove" received the documentfrom the startersvia one intermediary,the third throughtwo intermediaries,etc. The lengthof an incompletechain may be definedas the numberof removesfrom the startat whichdropout occurs, or, equiv- alently,as the numberof transmissionsof the folderwhich precede dropout. By this definition,Figure 2 representsa frequencydistribution of the lengthsof incompletechains. The mean of the distributionis 2.6 links. The proportionof chains which drop out at each removedeclines as

Z 60X60 _

U I \ F \ Nx232 _J\ o 40

U. 0

CUZ\

REMOVE FROM START AT WHICH TERMINATIONOCCURRED FIGURE 2 Lengthsof IncompleteChains 434 SOCIOMETRY chainsgrow in length,if thatproportion is based on all chainsactive at each remove(those destinedfor completionas well as incompletion).About 27 per cent of the 296 folderssent to the startingpopulation are not sent on. Similarly,27 per cent of the 217 chains actually initiatedby the starters die at the firstremove. The percentageof dropoutsthen appears to fall. It also beginsto fluctuate,as the totalnumber of chainsin circulationgrows small,and an increasingproportion of completionsfurther complicates the picture. It was argued earlierthat, in theory,any two people can be linked by at least one acquaintancechain of finitelength, barring the existenceof totallyisolated cliques withinthe populationunder study. Yet, incomplete chainsare foundin our empiricaltracing procedure because a certainpropor- tion of thosewho receivethe documentdo not send it on. It is likelythat thisoccurs for one of two major reasons: 1) individualsare not motivated to participatein the study; 2) they do not know to whom to send the documentin orderto advance it towardthe target. For purposesof gaugingthe significanceof our numericalresults, it would be useful to know whetherthe dropoutsare random or systematic,i.e., whetheror not theyare relatedto a chain's prognosisfor rapid completion. It seemspossible, for example, that dropoutsare preciselythose people who are least likelyto be able to advance the documenttoward the target.If so, the distributionof actual lengthsof completedchains would understate the truesocial distancebetween starters and targetby an unknownamount. (Even if dropoutsare random,the observeddistribution understates the true distribution,but by a potentiallycalculable amount.) We can offer some evidence,however, that this effectis not powerful. First,it shouldbe clear that,though people may drop out because they see littlepossibility that any of theiracquaintances can advance the folder towardthe target,their subjective estimates are irrelevantto the question just raised. Such subjectiveestimates may account for individualdecisions not to participate;they do not tell us whetherchains that die in factwould have been longerthan others had theygone to completion.People have poor intuitionsconcerning the lengthsof acquaintancechains. Moreover,people can rarelysee beyondtheir own acquaintances;it is hard to guess the circles in whichfriends of friends-notto mentionpeople even moreremotely con- nectedto oneself-may move. More directevidence that dropoutsmay be treatedas "random"can be gleanedfrom the tracercards. It will be recalledthat each participantwas asked for informationnot only about himselfbut also about the person to whom he sent the document.Thus some data were available even for dropouts,namely age, sex, the nature of theirrelationship to the people TABLE 1 Activityof Chainsat Each Remove on All Chains Incomplete Chains Only

Chains Reaching Completions Dropouts Per cent Chains Reaching Dropouts at Per cent Remove this Remove at this Remove at this Remove Dropouts Remove this Remove this Remove Dropouts ? 0 296 0 79 27 0 232 79 34 m 1 217 0 59 27 1 153 59 39 j 2 158 2 34 22 2 94 34 36 v 3 122 3 24 20 3 60 24 40 4 95 8 11 12 4 36 11 31 5 76 14 12 16 5 25 12 48 6 50 8 4 8 6 13 4 31 S 0 7 38 16 5 13 7 9 5 55 8 17 6 1 6 8 4 1 25 t- 9 10 2 2 20 9 3 2 67 10 6 2 0 0 10 1 0 0 11 4 3 0 0 11 1 0 0 0 12 1 0 0 0 12 1 0 0 r 13 1 0 0 0 13 1 0 0 14 1 0 1 100 14 1 1 100

. _ . . .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' 436 SOCIOMETRY precedingthem in the chain,and the reason the dropouthad been selected to receivethe document.These fourvariables were tabulatedfor dropouts versusnon-dropouts. None of the resultingcontingency tables achieved the .05 level of statisticalsignificance by chi-squaretest; we are thereforeled to accept the null hypothesisof no differencebetween the two groups,at least on this limitedset of variables.Of course,a definitiveanswer to the questionof whetherdropouts are reallyrandom must wait until the deter- minantsof chain lengthare understood,or until a way is foundto force all chains to completion.4 SUBPOPULATIONCOMPARISONS. A possible paradigmfor futureresearch using the tracingprocedure described here involvessystematic variation of the relationshipbetween the startingand targetpopulations. One such study, usingNegro and White startingand targetgroups, has already been com- pletedby Korteand Milgram(in press). In thepresent study, which involved only a singletarget person, three starting populations were used (Nebraska random,Nebraska stockholders,and Boston random.) The relevantexperi- mentalquestions were whether the proportionof completedchains or mean chain lengthswould vary as a functionof startingpopulation. Chain Length.Letters from the Nebraska subpopulationshad to cover a geographicdistance of about 1300 milesin orderto reachthe target,whereas lettersoriginating in the Boston group almost all startedwithin 25 miles of his home and/orplace of work. Since social proximitydepends in part on geographicproximity, one mightreadily predict that completechains originatingin the Boston area would be shorterthan those originatingin Nebraska.This presumptionwas confirmedby the data. As Table 2 shows, chains originatingwith the Boston randomgroup showed a mean length of 4.4 intermediariesbetween starters and target,as opposed to a mean lengthof 5.7 intermediariesfor the Nebraska randomgroup. (p?.001 by

4 ProfessorHarrison White of HarvardUniversity has developeda techniquefor adjust- ing raw chainlength data to take accountof the dropoutproblem. His methodassumes that dropoutsare "random,"in the followingsense. An intermediarywho knowsthe targetsends him the folder,completing the chain,with probability1. Otherwise,an intermediarythrows away the folderwith fixedprobability I-a, or sendsit on with probabilitya. If senton, there is a probabilityQi (whichdepends on numberof removes fromthe origin) that the next intermediary knows the target. The data is consistentwith a value fora of approximately0.75, independent of removefrom the origin,and hence witha "random"dropout rate of 25 per cent.The limiteddata furthersuggest that Q, growsin a "staircase"pattern from zero (at zeroremoves from the starting population) to approximatelyone-third at six removes,remaining constant thereafter. Based on these values,the hypotheticalcurve of completionswith no dropoutsresembles the observed curveshifted upward; the medianlength of completedchains rises from 5 to 7, but no substantialalteration is requiredin conclusionsdrawn from the raw data. TABLE 2 0 Lengths of Completed Chains Frequency Distribution MI Number of Intermediaries Means Population 0 1 2 3 4 5 6 7 8 9 10 11 Total StartingPopulation Mean Chain Length ? NebraskaRandom 0 0 0 1 4 3 6 2 0 1 1 0 18 NebraskaRandom 5.7 NebraskaStock 0 0 0 3 6 4 6 2 1 1 1 0 24 NebraskaStockholders 5.4 Boston Random 0 2 3 4 4 1 4 2 1 0 1 0 22 All Nebraska 5.5 All 0 2 3 8 14 8 16 6 2 2 3 0 64 BostonRandom 4.4 All 5.2

0A 438 SOCIOMETRY a one-tailedMann-Whitney U test.) Chain lengththus provedsensitive to one demographicvariable-place of residenceof startersand target. The Nebraska stockholdergroup was presumedto have easy access to contactsin the brokeragebusiness. Because the targetperson was a stockbroker,chains originatingin this groupwere expectedto reach the target moreefficiently than chains fromthe Nebraska randomgroup. The chain- lengthmeans forthe two groups,5.7 intermediariesfor the randomsample and 5.4 for the stockholders,differed in the expected direction,but the differencewas not statisticallysignificant by the Mann-Whitneytest. The stockholdersused the brokeragebusiness as a communicationchannel more oftenthan did the randomgroup; 60.7 per cent of all the participantsin chainsoriginating with the stockholdergroup reported occupations connected with finance,while 31.8 per cent of participantsin chains originatingin the Nebraska randomgroup were so classified. Proportionof Completions.As indicatedin Table 3, the proportionsof chains completedfor the Nebraska random,Nebraska stockholder,and Boston subpopulationswere 24 per cent, 31 per cent and 35 per cent, respectively.Although the differencesare not statisticallysignificant, there is a weak tendencyfor highercompletion rates to occur in groupswhere mean length of completedchains is shorter.This result deserves brief discussion. Let us assumethat the dropoutrate is constantat each removefrom the start. If, for example,the dropoutrate were 25 per cent then any chain wouldhave a 75 per centprobability of reachingone link,(.75)2 of reaching twolinks, etc. Thus, thelonger a chainneeded to be in orderto reachcomple- tion,the less likelythat the chain would survivelong enoughto run its full course.In thiscase, however,chain-length differences among the threegroups were not sufficientlylarge to produce significantdifferences in completion rate. Moreover,if the dropoutrate declinesas chainsgrow long, such a de- creasewould off-set the effect just discussedand weakenthe observed inverse relationbetween chain lengthand proportionof completions.

TABLE 3 Proportionof Completionsfor ThreeStarting Populations StartingPopulation NebraskaRandom NebraskaStock. Boston Total Complete 18 (24%) 24 (31%) 22 (35%) 64 (29%) Incomplete 58 (76%) 54 (69%) 41 (65%) 153 (71%) 76 (100%o) 78 (100%) 63 (100o) 217 (100%) X3=2.17,df.=2, P>.3, N.S. STUDY OF THE SMALL WORLD PROBLEM 439

COMMONCHANNELS. As chainsconverge on the target,common channels appear-that is, some intermediariesappear in morethan one chain. Figure 3 showsthe patternof convergence.The 64 letterswhich reached the target were sent by a total of 26 people. Sixteen,fully 25 per cent, reached the targetthrough a singleneighbor. Another 10 made contactthrough a single businessassociate, and 5 througha second businessassociate. These three "penultimatelinks" togetheraccounted for 48 per cent of the total completions.Among the three,an interestingdivision of labor appears. Mr. G,

64 INDIVIDUALS 55 INDIVIDUALS 5 CHAINS

CHAINS 26 2 CHAINS3 INDIVIDUALS 16 CHAINS

FIGURE 3 CommonPaths Appearas ChainsConverge on the Target 440 SOCIOMETRY who accountedfor 16 completions,is a clothingmerchant in the target's hometownof Sharon; Mr. G funnelledtoward the targetthose chains which wereadvancing on the basis of the target'splace of residence.Twenty-four chains reachedthe targetfrom his hometown;Mr. G accountedfor 2/3 of thosecompletions. All the letterswhich reached Mr. G came fromresidents of Sharon. By contrast,Mr. D and Mr. P, who accountedfor 10 and 5 completions,respectively, were contactedby people scatteredaround the Boston area, and in several cases, by people livingin othercities entirely. On the otherhand, whereas Mr. G receivedthe folderfrom Sharon residents in a wide varietyof occupations,D and P receivedit almost always from stockbrokers.A scatteringof names appear two or three times on the list of penultimatelinks; seventeennames appear once each. Convergenceappeared even beforethe penultimatelink. Going one step furtherback, to people two removesfrom the target,we findthat the 64 chains passed through55 individuals.One man, Mr. B, appeared 5 times, and on all occasionssent the documentto Mr. G. Otherindividuals appeared twoor threetimes each. ADDITIONALCHARACTERISTICS OF CHAINS. Eighty-sixper cent of the par- ticipantssent the folderto personsthey described as friendsand acquaintances; 14 per cent sent it to relatives.The same percentageshad been observedin an earlierpilot study. Data on patternsof age, sex and occupationsupport the plausible hy- pothesisthat participants select recipients from a pool of individualssimilar to themselves.The data on age supportthe hypothesisunequivocally; the data on sex and occupationare complicatedby the characteristicsof the targetand the special requirementof establishingcontact with him. Age was bracketedinto ten-yearcategories and the ages of those who sent the documenttabled against the ages of those to whom they sent it. On inspectionthe table showed a strongtendency to clusteraround the diagonal,and a chi-squaretest showedthe associationto be significantat betterthan the .001 level. Similarly,the sex of each sender was tabled against the sex of the correspondingrecipient. Men were ten timesmore likely to send the document to other men than to women,while women were equally likely to send the folderto malesas to females(p<.001). These resultswere affected by the fact that the targetwas male. In an earlier pilot study using a femaletarget, both men and womenwere threetimes as likelyto send the documentto membersof the same sex as to membersof the oppositesex. Thus thereappear to be threetendencies governing the sex of the recipient: (1) thereis a tendencyto send the documentto someoneof one's own sex, but (2) womenare morelikely to cross sex lines than men,and (3) there STUDY OF THE SMALL WORLD PROBLEM 441 is a tendencyto send the documentto someoneof the same sex as the target person. The occupations reportedby participantswere rated on two com- ponents-one of social status and one of "industry"affiliation, that is, the subsectorof the economywith which the individualwould be likely to deal. The coding systemwas ad hoc, designedto fit the occupational titles suppliedby participants.Tabling the status and "industry"ratings for all sendersof the documentagainst those of respectiverecipients, we observed a strong tendencyfor people to select recipientssimilar to themselveson both measures (p<.001 for both tables). However, the strengthof the relationshipfor industryseemed to be largely due to a tendencyfor the folderto stay withinthe financefield once it arrived there,obviously because the targetwas affiliatedwith that field. More- over, the participantsin the study were a heavily middle-classsample, and the targetwas himselfa memberof that class. Thus there was no need for the documentto leave middle-classcircles in progressingfrom startersto target. When separatecontingency tables were constructedfor completeand incompletechains, the above resultswere obtainedfor both tables. Similarly, whenseparate tables were constructed for chains originating in the 3 starting populations,the findingsheld up in all 3 tables. Thus, controllingfor com- pletionof chains or for startingpopulation did not affectthe findingof demographichomogeneity within chains.

CONCLUSIONS

The contributionof the study lies in the use of acquaintancechains to extendan individual'scontacts to a geographicallyand sociallyremote target, and in the sheersize of the populationfrom which members of the chains were drawn.The study demonstratedthe feasibilityof the "small world" technique,and took a step towarddemonstrating, defining and measuring inter-connectednessin a large society. The theoreticalmachinery needed to deal with social networksis still in its infancy.The empiricaltechnique of this researchhas two major con- tributionsto make to the developmentof that theory.First, it sets an upper bound on the minimumnumber of intermediariesrequired to link widelyseparated Americans. Since subjectscannot always foreseethe most efficientpath to a target,our traceprocedure must inevitably produce chains longerthan those generatedby an accurate theoreticalmodel which takes full accountof all paths emanatingfrom an individual.The mean number of intermediariesobserved in this study was somewhat greaterthan five; 442 SOCIOMETRY additionalresearch (by Korte and Milgram) indicatesthat this value is quite stable,even when racial crossoveris introduced.Both the magnitude and stabilityof the parameterneed to be accountedfor. Second, the study has uncoveredseveral phenomenawhich futuremodels should explain. In particular,the convergenceof communicationchains throughcommon individuals is an importantfeature of small world nets, and it should be accountedfor theoretically. There are many additionallines of empiricalresearch that may be ex- amined with the small world method.As suggestedearlier, one general paradigmfor researchis to vary the characteristicsof the startingperson and the target.Further, one mightsystematically vary the informationpro- vided about the targetin orderto determine,on the psychologicalside, what strategiespeople employin reachinga distanttarget, and on the sociological side, what specificvariables are critical for establishingcontact between people of given characteristics.

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