centrality in social networks: ii. experimental results*moreno.ss.uci.edu/29.pdfaccording to freeman...

Social Networks, 2 (1979/80) 119-141

@&evier Sequoia S.A., Lausanne - Printed in the Netherlands 119

Centrality in Social Networks: II. Experimental Results*

Lin ton C. Freeman University of California, Irvine”*

Dougias Roeder Stanford University ***

Robert R. Mulholland University of Culifornia, Irvine**

Three competing hypotheses about structural centrality are explored by means of u rep&&ion of the curly MIT experiments on communication structure and group ~rob~en~-soil~ing. It is showrl that ~~t~lo~~g~l two of the three kinds of measures of centrality have a demonstrable effect on individual responses and group processes, the classic measure of centrality based on dist~nee is u~~re~~ted to any experime~lt~~ variuble. A sl~ggestio~l is ?nade that the positive results provided by distunce-based centrality in curlier experiments is un artifact of the particulur structures chosen for experimen tation.

Introduction

In a recent report, Freeman ( 1979) uncovered three intuitive conceptions that have been used to try to capture the idea of structural ten trality in social networks. He refined these intuitions and specified a family of measures for each. One set of measures was based on the degree of a point and seemed to be an index of that position’s potential for activity in the network. Another was based on the extent to which a point fell between others on the shortest paths connecting them. It was taken to be an index of potential for control of ~ommunicatio1~. And the third was based on the closeness of a point to all others. This was viewed as a measure of either independence from control or of efficiency. Three measures were developed for each of these structural properties. Two of each set of measures referred

*The authors wish to express their sincere gratitude to Everett Rogers and Jack Ilunter for their careful reading and helpful criticisms of an earlier draft of this manuscript. **School of Social Sciences, University of California, Irvine, CA 92717, [J.S.A.

***Department of Sociology, Stanford University, Stanford, CA 94305, U.S.A.

120 I,. C. Freeman. 11. Roeder ad R. R. hlulholland

to attributes of positions or points as they were embeclclccl in network structures: one was based on absolute counts ofdegrec, betweenncss or closeness and the other was made relative by eliminating the efl’ect of network size. The third measure in each case was an index of centralization of the entire structure. It was designed to reflect the degree to which a network was dominated by a single point, again in terms of deg-ee. betwccnness or closeness respectively.

According to Freeman (1979) “these kinds of centrality imply three competing ‘theories’ of IIOW centrality might affect goup processes”. Thus. what is needecl now is an effort to sort out the effects of these several kinds of centrality. The most natural way to accomplish this is to return to the classic experiment that was designed in the late 1940s at the Massachusetts Institute of Tectlnology. It was dcvelopctl specifically to study the effects of structural centrality on human communication.

The experimental study of ten trality

The MIT experiment as reported by Bavclas (1950), Smith (1950) and Leavitt (195 1) is both simple and elegant. According to McWhinney ( 1964) s~ich experiments are “simple enough to permit observation of group processes and effective use of analytic tools, but not so simple that they remove the essence of group interaction”.

A group of subjects is seated around a table that is partitioned by large opaque walls that isolate subjects from any visual or auditory contact with one another. Each subjject, however, is provided with one or more slots through which he can pass and receive messages to and from designated others. The independent variable, then, is the pattern of permitted communication ~ the group structure.

Four structural forms were studied. They are shown in Figure 1. Each position or point was identified by a color, and each subject was provided with a stack of blank message cards in his own designated color.

Figure 1 Structural forms used irl the original MT experimen t.

.=; ;.

Chain

Y

“Y”

+ 0

Star or Wheel Circle

Centrality in social networks 121

The experiment itself consisted of a series of 15 problem-solving trials. On each trial each subject took up a card showing five symbols from a set of six (0, *, +, u, a, 0). The subjects had only one symbol in common on a given trial and their job was to discover that common symbol by passing messages to one another. When a subject thought he had the correct answer he so indicated by throwing a switch associated with that symbol. When all subjects had thrown switches the trial was ended.

When the 15 trials were over each subject was asked to fill out a questionnaire containing a number of questions, three of which have turned out to produce interesting results:

1. Did your group have a leader? Yes _~~_ No ~~.

2. If so, who? (Identify by color) _ _ ~~~ ~~___~

3. How did you like your job in the group? (place a vertical slash along the line below)

Disliked Liked It It I I _~.. ~~ ~~___

The information provided by the questionnaire along with data on the efficiency of the group in solving the problems constituted the dependent variables in the experiment. Thus, the MIT experiment was designed to study the effects of the structure of communication ~~ or, specifically, the centrality of points in the communication structure - on problem solving, perception of leadership and personal satisfaction.

Results showed that all three dependent variables were related to the structure of communication. Personal satisfaction and nomination for leadership, it turned out, were related to point centrality. Efficiency in problem solving, group satisfaction and the tendency to see leadership as operating were related to overall centralization of the network,

Results were impressive and they generated a good deal of derivative work’ As Rogers and Agarwala-Rogers ( 1976, pp. 119--- 123) have observed,

‘Experimental follow-up work was done by Heise and Miller (1951), Guetzkow (1951, 1954, 1960), Christie, Lute and Macy (1952, 1956), Hirota (1953), DeSoto (1953), Rogge (1953), Macy, Christie and Lute (1953), Lute, Macy, Christie and Hay (1953), Shaw (1953, 1954a, 1954b, 1954c, 1955a, 1955b, 1956, 1958, 1959), Shelley (1953), Karaneff(1954),Walker (1954),Christie (1954a, 1954b), Gilchrist, Shaw and Walker (1954), Schein, White and Hill (1955), Goldberg (1955), Guetzkow and Simon (1955), Mulder (1955, 1959a, 1959b, 1960a, 1960b), Berkowitz (1956), l-lament (1956, 1958a, 1958b, 1961), Lanzetta and Roby (1956a, 1956b, 1957), Roby and Lanzetta (1956a, 1956b, 1957a, 1957b), Shaw and Rothschild (1956), Trow (1957). Guetzkow and Dill (1957). Shaw. Rothschild and Strickland (1957). Shelley and Gilchrist (1958), Cohen, Bennis and Wolkon i1959a; 1959b, 1960, 1961, 1962), Faucheux and Moscovici (1960). Mohanna and ArcVle (1960). Cohen and Bennis (1960a, 1960b, 1961, 1962), Cohen, Bennis and Briggs (1960), Cohen (1961, 1462, 1964a, 1964b, 1964c, 1967), Sure, Rogers, Larson and Tassone (1962), Carzo (19631, Leavitt and Knight (1963), Lawson (1964a, 1964b, 1965), Morrisette, Pearson and Switzer (1965), Morrisette, Switzer and Cranncl (1965), Watson and Bromberg (1965), Faucheux and Mackenzie (1966), Morrisette (1966), Burgess (1968a, 1968b, 1968c, 1968d, 1969), Cohen and I.‘oerst (1968), Harshberger (1971), Cohen, Rosmer and Foerst (1973), Snadowski (1974), Liddell and Slocum (1976).

122 I,. C. Freenm, Il. Roeder am1 R. R. ~IfdholLmd

however, interest peaked in the 1950s and 1960s and has declined recently. The reasons for this decline are not entirely clear, but we can make some guesses about at least some of the factors involved.

Since the early fifties there has been a growing divergence between theoretical and experimental work in this area. The early experimental studies were concerned with the effects of structural centrality on communication in small groups. Exactly what centrality was, as Freeman (1979) showed. was not entirely clear. But at least these investigators were concerned both with specifying a potentially relevant structural attribute of networks and with determining its cffccts.

After these first studies, however, subsequent investigators tended to LX concerned either with conceptual problems of centrality or with the consequences of structure on prohlcm solving, not both. Some investigators, like Beauchamp (1965) and Sabidussi ( 1966) were concerned very little with experimental results. They rcturnetl to graph tlleory and redefined centrality in mathematically sophisticated (but empirically inappropriate) ways. Others, like I-leise and Miller ( 195 1) and Lawson ( 1965) dropped the notion of centrality entirely and concentrated on studyin g the unspecified at tribu tcs of structure on problem solving. They looked simply for differences in perfor- niancc aniong tile standard named structural forms, like the ‘wheel or the ‘circle’. Tlicir reports reflect a sort of blind empiricism. They show that the wheel is. say, faster than tlic circle in solving problems of a given class, but make no attempt to uncover what it is about these forms that might lead to this result.

A second difficulty with the later cxpcrimcntal work is in part at least - a consequence of this same blind concern with standard structures. In the

early MIT studies Bavelas and Leavitt made some attempt to sort out positional effects from overall structural effects. Positional effects could be seen when the value taken by the dependence variable could IX predicted from the centrality ofa point regardless of the structure in which it was eniL~etldccl. Overall structural effects. on the other Iiand. were observable when the value of the dependent variable could be predicted from the centralization of the structure itself regardless of individual point centralitios within the structure. By concentrating entirely on structural differences, the later studies lost sight of the sort of subtle complexity inipliccl by a concern with both positional and overall structure effects and their possible interaction.

These problems led to an emerging consensus that this kind of experiment had hit an intellectual dead end. Burgess (1968b) summarized this view when he concluded that the results were “contradictory as well as inconclusive”. He reasoned, and demonstrated empirically, that if subjects were run long enough (900+ trials). and rewarded for speed and efficiency. all previously dcnionstratccl differences in performance would disappear.

Burgess’s conclusion is probably correct. hut it has little to do with the importance of tile Bavclas-type experiment. Obviously learning takes place in the experiment. and cclually obviously ii‘ pushed hard cnou:gh and long enough suL>jects will find tlic organization structlire that provides an opti-


mum solution. Nevertheless, in seeking this solution, subjects will use strate- gies that reflect the influence of structural factors. As Guetkow and Simon (1955) and Guetzkow and Dill (1957) have suggested this is probably due to the fact that participants must solve two problems: (1) that of developing an organizational scheme suitable for finding the common symbol within the constraints of their particular network form and (2) that of actually finding the common symbol. Unless subjects are ‘beaten to death’ with a seemingly endless string of experimental trials they are likely to behave in a way that parallels their real-life communications. They will seek and find an organizational form that works, and then play with variations on it both to main- tain interest and to seek a ‘better’ form.

All this, of course, must be accomplished within the structural constraints of the communication network. And the major structural constraints of interest in this context are those having to do with centrality and centralization. Any or all combinations of three kinds of centrality might be appro- priate to a given application, either at a point level or at the level of the analysis of overall structural properties.

Existing data, however, provide no basis for sorting out the effects of various kinds of centrality on communication processes in small groups. The particular structural forms used in the MIT experiments and by subsequent investigators are all ranked in the same order by all three measures of centralization (see Table 1). Thus, it is impossible to make a decision about the kind of centralization involved in affecting results on the basis of existing data.

Table 1. Centralization index scores and ranks for the structures studied by Leavitt.

Form

Star or Wheel Y Chain Circle

Control Independence Activity

CB Rank cc Rank CC Rank

1 1 1 I 1 1 0.71 2 0.63 2 0.58 2 0.41 3 0.43 3 0.17 3 0 4 0 4 0 4

It is, however, possible to choose other structural forms that are ranked in differing orders by the three measures. Such choice permits a critical test among the three centrality concepts. That is the purpose of the empirical test used here. This test was designed in an attempt to determine the degree to which the results of an MIT-type experiment could be associated with centrality in terms of control, independence or activity.

A critical test

The experiment reported here was designed to determine which, if any, of the three kinds of structural centrality specified by Freeman (1979) was

124 L. C. Freeman, D. Roeder and R. R. Mulholland

relevant to small group problem solving. Except for the use of new structural forms and some female subjects, this experiment is an exact replication 01 the one reported by Leavitt (195 1).

Like Leavitt, we used four structural forms, each containing five positions. And, as in the Leavitt experiment, each form was replicated with five sets of subjects. The forms used here are shown in Figure 2. The overall centralization scores and their rankings for these structures are shown in Table 2. The point centrality scores for these structures are shown in Table 3. These rankings and scores are used to forecast possible outcomes, by form and by position, of the experimental trials.

Figure 2. Structural forms used in the current experiment (positions are identijied with colors.. R-red, Y-yellow, W--white, B-blue, and G-green).

Subjects were 100 volunteers from among the student body at Lehigh University. They ranged from freshmen to graduate students; 51 were male and 49 were female. None had previously participated in a group problem-solving experiment.

The first question addressed in analyzing our results involved evaluating the success of this experiment as a replication of Leavitt’s study. In choosing structural forms, we had purposely picked one, our Form A, that was identical with one of Leavitt’s forms, the chain. This choice, it was reasoned,

Table 2. Centralization index scores and ranks ftir the structural forms shown in Figure 2.

Control Independence Activity

C‘B Rank (‘c! klllk (‘D killk

0.41 ? 0.43 3 0.17 4

0.56 I 0.55 2 0.42 2

0.29 3 0.62 1 0.50 I 0.14 4 0.23 4 0.25 3


Table 3. Point centrality index scores for positions in the structural forms shown in Figure 2.

Point Form Position Centrality Index R Y W B G

CB A 4 3 3 0 0 13 4 0 3 0 0 C 2 0.5 0.5 0 0 D 1.5 1.5 0.5 0 0.5

CD A 2 2 2 1 1 B 3 2 2 2 1 C 4 3 3 2 2 D 3 3 2 2 2

C,’ A 6 1 I 10 10 B 5 I 6 I 9 C 4 5 5 6 6 D 5 5 6 6 6

would allow us to evaluate the degree to which our results were comparable with those produced by Leavitt.

Comparative results for six key variables are shown in Table 4. No one difference is especially large - particularly in view of the relatively small samples used. The interesting result, however, is the fact that our subjects performed ‘better’ with respect to every single variable. They used less time, sent fewer messages, made fewer errors, and in general, were more satisfied with the task.

Table 4. Comparison on key variables for Form A, the Chain, between Leavitt’s data and data from the current study.

Variable MIT ca. 1949 Lehigh 1916

1. Time (in seconds per trial) (trials 10-15) 2. Fastest correct trial (in seconds) 3. Number of messages per trial (trials 10-15) 4. Number of trials with at least one error per group 5. Satisfaction scores 6. Satisfaction scores by position

Center Intermediate End

77.1 63.4 53.2 48.4 14.8 13.2

1.8 1.6 60 65.2

78 92 76 82 34 30

It is difficult to explain the consistent pattern of these differences. It might simply be due to sampling variation. Or it might result from a change in the attitudes or skills of students in general over almost 30 years, to an institutional difference between MIT and Lehigh, or to our use of female subjects and (one would assume) a greater proportion of non- engineering students. There is no way to determine the bases for these differences from available data. Instead. we must be content that the differences

126 I,. C. Freeman. D. Roeder and R. R. Mulhollarzd

are rather small and that the results of our replication are reasonably close to those produced by the original experiment.

The main analyses of the data of the present experiment were made in terms of three classes of dependent variables: (1) leadership, (3) satisfaction and (3) efficiency. These will be examined, in turn, below.

Concern with leadership was embodied in two questions in the post- oxpcrimental questionnaire. These questions were used to determine whether or not subjects perceived their group as liavin g a leader and whether they could identify that leader. If 3 subject identified the occupant of a dcsig- nated position 3s the leader for his or Iier goup that was counted as an instance of perceived leadership and tabulated for the position chosen. These results were recorded for positions in each of the four structural forms. Results are shown in Table 5.

Table 5. Number of times each position was idrnt$ied as a leader fly structural jbrnz.

I~orm Posit ion

K cd Yellow CVhite IIIUC Green

A 3 1 1 I 3

3

1% 3 3

3 3

(’ 4 I 5 3 I

D 4 2 I

3

2

Red was the most central position in forms A, U and C and red and yellow shared dominance in form D. Table 5 shows that most participants who chose a leader picked the most central point. Moreover, the overall number of leadership choices made as well as the number of non-central choices varied by structural form, but only slightly. The analysis of variance of average number of leadership choices per trial by structural form is shown in Table 6. The differences, it seems, arc far from significant.

The means of leader identification are, however, in exactly the order predicted by tile control-based mcasurc of centrali:<ation, C’, Thus, though the


Table 6. Analysis of variance of average number of leadership nominations per trial and position by structural forms.

Form xx N XX2 (2X)*/N i

A 17 25 5.5 11.56 0.68 B 18 25 62 12.96 0.72 c 16 25 56 10.24 0.64 D 14 25 36 7.84 0.56

65 100 209 42.60

65*/100 = 42.25

TSS = 209 ~ 42.25 = 166.75 BSS = 42.6 - 42.25 = 0.35 WSS = 166.75 ~ 0.35 = 166.4

SOUICC ss df MS F

Between Means 0.35 3 0.117 0.067 Within Groups 166.4 96 1.731

_____ -.~

Table 7. Analysis of variance of number of leadership nominations per trial by betweenness-based point centrality score.

c',(Pk) cx N XX2 (~X)*/N i

4 32 10 114 102.4 3.2 3 1 15 1 0.067 0.067 2 13 5 51 33.8 2.6 1.5 11 10 31 12.1 1.1 0.5 5 20 9 1.25 0.25 0 3 40 3 0.225 0.075

65 100 209 149.842

65*/100 = 42.25 TSS = 209 ~ 42.25 = 166.75 BSS= 149.842 ~ 42.25 = 107.592 WSS = 166.75 ~ 107.592 = 59.159

Between means 107.592 5 21.52 34.21 Within groups 59.159 94 0.629

**Significant at 01= 0.01.

differences are small, their order suggests that they may be associated with betweenness.

This conclusion is supported, in part, by the analyses of variance of leadership choices by point centrality scores shown in Tables 7, 8 and 9. Here, means are calculated for individual point centrality scores regardless of the form in which they are embedded. All three point centrality measures yield significant Fs, but the residual variance produced by the partition based on the betweenness measure is less than half of that shown by the

128 I,. C Freeman, D. Koeder and R. R. Mztiholland

Table 8. Analysis of variance of number c?f’leadership nominations per trial by degree- based point centrality score.

4 13 5 51 33.8 2.6 3 30 2s 97 36.0 1.2 2 21 55 60 8.018 0.38 1 1 10 I 0.067 0. I

65 1 00 209 77.885

6S2/l 00 = 42.25 TSS = 209 42.25 = 166.75 BSS = 77.885 ~ 42.25 = 35.635 WA’S = 166.75 35.635 = 131.1 I5

SOUK~ ss d!~ MS f,‘

13ctween means 35.635 3 11.88 8.70** Within groups 131.115 96 1.366

“*Si-nilicant at a = 0.01 c

Table 9. Analysis of variance of number of leadership nominations per trial by close-

Iless-based point centrality score.

+(/‘k)-’ xx N 2x2

4 13 5 51 s 30 25 97 6 19 35 5x 7 2 20 2 Y 0 5 0

10 I 10 1

65 100 209

65’/100 = 42.25 TSS = 209 42.25 = 166.75 BSS = 77.885 ~ 42.25 = 38.164 WSS= 166.75 38.164 = 128.58

(cX)‘/N i

33.8 2.6 36.0 I .I IO.314 0.54

0.2 0.1 0 0 0.1 0.1

80.414

SULUCC SS dl

Iktween means 3X.164 S Within groups 128.58 94

“*Significant 31 a y 0.01.

AIS I<’

7.6328 5.58*” 1.368

others. Betweenness, then, seems to be the point centrality measure of choice when it comes to understanding leadership nominations.

There is a problem, however, in considering the relationship between point centralities based on betweenness and nominations for leadership. Their relationship is not strictly monotone; points with a betweenness index of 3 arc actually lowest in receiving leadership choices. inspection of the

means suggests that the really important partition is neither by forms nor by point centralitics but by the most central point (in terms of betweenncss)


in each form versus all other points. The analysis of variance of leadership nominations by this partition is shown in Table 10. Clearly the results are significant.

Table 10. Analysis of variance of number of leadership nominations per trial by central position versus all other ___- Partition CX N CX2 (xX)2/N x

Betweenness Center 56 25 196 125.44 2.24 Others 9 15 13 1.08 0.12

65 100 209 126.52

652/100 = 42.25 TSS = 209 -- 42.25 = 166.75 BSS = 126.52 - 42.25 = 84.21 WSS = 166.75 - 84.27 = 82.48

Source ss df MS b

Between means Within groups

84.27 1 84.27 100.08** 82.48 98 0.842

**Significant at 01= 0.01.

Betweenness, then, seems to be the key to understanding choice as leader. Since it is based on potential for control of communication, this outcome makes good intuitive sense; it is reassuring to find that perceived leadership is related to what we have called ‘control potential’.

Satisfaction was recorded by having subjects place a check mark along a line. It was scaled by measuring the distance from the zero point to the check and converting to a base of 10. Satisfaction ratings are shown in Table 1 1.

Again, analysis of variance of satisfaction scores by structural form shows that differences in structural forms are not significant (F = 0.28 with df = 3, 96). And again, as Table 12 shows, F-tests of point centralities are all significant with the betweenness-based measure yielding the smallest residuals.

In the case of satisfaction, examination of the means suggests that differences are not simply a matter of whether points are dominant in betweenness or not. Instead the satisfaction of the occupant of a point seems to depend on a combination of betweenness and degree. Participants whose degree is less than two are connected to only one other person; they report feeling powerless and more or less isolated; they are dissatisfied. Those who are connected to two or more others are more likely to be satisfied. If, however, they are connected to at least two others and also have a necessary role in passing messages (betweenness greater than or equal to 1) they seem quite satisfied indeed.

This suggests that ~ regardless of form -- there are three classes of subjects with respect to satisfaction. Results using this partition are shown in Table 13. They show a reduction in residual variance and suggest that this

130 L. C. Freenlan, 1). Roeder and R. R. iZlulho16and

Table 1 I . Satisfktion scores by structural fbrnr and position

A IO IO 8 IO 9 8 x 8 9

10 9 9

x 9 x

H 9 Y 8

IO 8 7 I 0 5 I 7 9 IO

IO 3 5

<’ 5 7 I 0 I 0 x 3

8 I X

9 6 I

9 IO 7

I) I 0 Y 3

9 I 2 0 I 0 2 x 9 5

I 0 I 0 6

3 2 5

3 0

4 2 0 9 8

Table 12. -4 naZyses of variarzce oj’satisfaction by three point-centrality measures

c’B(/‘k)

4

3 , ;.5

0.5 0

22 :V ?;x* (LY)*/N

92 I 0 858 X46.4 I24 I5 1048 1025.1

41 82 IO 5 35 756 I 612.4 336.2

114 20 194 649.X 152 40 898 571.6

605 I 00 4705 4107.5

605*/l 00 = 3660.25

x

TSS = 4705 3660.25 = 1044.75 RSS = 4107.5 ~- 3660.25 = 441.25

II’SS = 1044.75 447.25 = 597.5

2 .y xu2 ( xx)2/N

41 5 351 336.2 I95 25 1707 1521.0 324 55 2404 1908.65

45 IS 243 135.0

605 IO0 4705 3900.85

RSS = 3900.85 3660.25 = 240.6 It’ss = 1044.75 240.6 = 804. I 5

9.2

X.27 x.2 X.2 5.7

3.x

A-

8.2

7.x 5 .Y 3.0

Table 12. (Continued)

Centrality in social networks 13 1

Cc(pk)-’ XX N XX2 (xf12 IN x 4 41 5 5 195 25 6 188 35 I 136 20 9 15 5

10 30 10

605 100

BSS = 3926.83 WSS = 1044.75

351 336.2 8.2 1717 1521.0 7.8 1322 1009.83 5.31 1082 924.8 6.8

95 45.0 3.0 148 90.0 3.0

4705 3926.83

3660.25 = 266.58 266.58 = 778.17

Source ss df MS F

Means on CB~&) 477.25 5 89.45 14.06** Within Groups 597.5 94 6.36 Means on CD&) 240.6 3 80.2 9.57** Within Groups 804.15 96 8.38 Means on C&k) 266.58 5 53.32 6.44** Within Groups 778.17 94 8.28

**Significant at (Y= 0.01.

Table 13. Analysis of variance of satisfaction by a three class partition of positions.

Partition

C&$k) > 2, C&k) > 1 339 40 3013 2873.0 8.47 C&k) a 2, C&k) < 1 221 45 1449 1085.3 4.9 1 cD@k) < 2 45 15 243 135.0 3.0

100 4705 4093.3

605’/100 = 3660.25 TSS = 4705 ~ 3660.25 = 1044.75 BSS = 4093.3 ~ 3660.25 = 433.05 WSS= 1044.75 ~~ 433.05 = 611.7

Source ss df‘ MS b

Between subgroups 433.05 2 216.525 34.31** Error 611.7 97 6.31

**Significant at (Y = 0.0 1.

three-way partition may indeed be a relevant determinant of satisfaction for participants. Participants require some connection with more than one other to be satisfied at all, but real satisfaction requires some betweenness.

The problem of efficiency is a bit more complicated. Traditionally, there are three measures of efficiency: (1) the number of errors, (2) speed per trial, and (3) the number of messages sent. These will be examined in turn.

An error was tabulated whenever a subject had selected the wrong symbol when a trial had ended. Data on errors by position, structural form and experimental group are shown in Table 14.

132 I,. C. Freerna~l, II. Roeder and R. R. i2lulkdland

Table 14. Errors in 15 trials by position ad stntchlral jbrtn.

K

A 0 4 I 0 I

H (I 0 0 0 0

(‘ 1 0 0 0 1

I> I 0 Cl 2 0

0 4 I 0 1

0 0 I 0 0

0 0 I 0 0

0 0 0 2 0

w I3 (;

0 0 0 4 4 4 I I I 0 0 0

?. I 3

0 0 0 0 0 0 0 1 0 0 0 0 1 0 I

0 0 0 0 0 0 I I 2

0 0 0 2 (1 I

0 I 0 0 0 0

0 (1 0

I I 0 I 0 7

It is difficult to draw conclusions from the data in Table 14. It is obvious that form A, the chain, generates far more errors than the others, but beyond that, no pattern emerges. For forms, I+’ = 7.48 with (/I’= 3 and 96 which is significant at OL = 0.01, this seems almost entirely the result of the high error rate of form A.

It might be noted in this context that most of the errors produced by form A arc’ of a different kind than those generated in other forms. Form A’s errors are ~~ 30 of 33 cases ~ shared by all subjects for a given trial. All subjects agreed on the wrong answer. Other errors are individual or shared at most by one or two people. As a matter of fact, when consensual errors are dropped out, I.‘ is 110 longer significant.

All this suggests that this experiment generates two kinds of errors: ( 1) seemingly random individual mistakes and (2) systematic and COIISCIISLI~

incorrect responses. The second type of error is found only among participants using form A. The most likely explanation of this phenomenon is that in form A there are no cycles and no redundant paths connecting pairs of points. There is, therefore, less opportunity for checking answers.

If a mistake is made everyone is stuck with it. This notion is supported by the fact that, as we shall report below, form A participants used far fewer messages in solving problems than did those from any other structure. It would seem, then, that systematic errors result from a structure to the degree that it inhibits the checking of answers.

Time data were collected for each trial. The range of variation in solution times C‘VL’II replications of a single participating group is far greater than

Centrality in social nehvorks 133

that for any other dependent variable in this experiment. A number of partitions were tried, but the error variance for each was so large and mean differences were so small that any conclusion from these data was impossible. Interpretation of time data awaits further study with considerably larger samples and, hopefully, detailed records of times involved in the sub- processes that make up the solution.

The most natural way to check efficiency in terms of message flow is to total the number of messages sent. In what is essentially a learning experiment, however, early trials involve huge numbers of messages as compared with later trials. Later trials are nearly stable and give a better indication of the more-or-less steady state of communication. For purposes of the present analysis, then, message data were recorded only for trials 10 to 15 where communication frequencies had settled down to a fairly stable pattern. In addition, the small proportion of messages that were irrelevant to the experiment - “What are you doing this weekend?” - were also eliminated from the analysis.

Data on overall communication by position and form are shown in Table 15. It is clear that there are massive differences both between structural forms and between positions in forms.

Table 15. Number of messages sent in trials lo-15 by position and structural form

Form Position

R Y W B G

A 33 26 25 8 6 12 12 12 6 6 13 12 12 11 6 13 12 15 6 7 22 19 20 6 10

B 55 41 22 30 I 27 26 29 13 12 18 18 12 15 6 32 32 17 40 6 26 18 12 28 6

C 29 56 47 20 29 29 27 20 12 30 33 28 29 25 26 25 20 17 11 18 41 40 35 23 34

D 36 44 25 15 21 37 28 18 22 23 31 17 19 14 24 40 44 29 30 33 34 43 33 36 28

The results of the F-test for structural forms are shown in Table 16. These results do not order the structural forms according to any of the measures of centrality. Instead the forms are ordered in terms of their density. This suggests that communication is simply a function of the

134 L. C. Freeman, L). Roeder and R. R. Mulholkznd.

Table 16. Means, variances and analysis of variance of messages sent in trials 1 O-1 structural form

l~‘Orlll 7 x s*

A 13.2 5 0.24 Px 21.9 147.03 (‘ 2x.14 103.25 1) 29.22 76.32

Source ss d/’ !W 1..

Ilcan? on t‘orms 4055.36 3 1351.79 13.77”” Within tirrnls 9421.2 96 98.14

bv

Table 17. Means, variances and analysis of variance of messages sent in trials lo-15 bv degree-based poin t centralities

C’&‘k) a s2

4 31.40 29.44 3 33.24 117.78 2 22.09 65.35 1 7.27 3.93

Source ss dl MS I.‘

Means of‘ C’D(&) 6731.76 3 2243.92 31.93** N’ithin groups 6744.8 96 70.26

**Significant at a = 0.01.

number of different others with whom each participant leas the opportunity to communicate.

This interpretation is supported by the positional analysis shown in Table 17. Except for the reversal of means at degrees 4 and 3, these results suggest that communication activity is strongly a function of degree for points. Moreover, this reversal seems to be the result of one very confused trial where the information was finally organized by a person in a non-central position.

Overall communication seems to depend simply on the number of tiif- ferent people with which each subject has the opportunity to communicate. It is possible, however, to break down various kinds of communication to see if we can learn any more about how this process works. Message data were, thercforc, broken down into four categories:

(1 ) organizational suggestions. (2) requests for information, (3) information (but not answers). and (4) answers.

Centrality in social networks 13.5

For the most part, organizational suggestions had dropped out by trial 10. But the few that were made occurred more frequently in forms with more edges. The same is true for answers and for messages containing non-answer information. Only requests for information depart slightly from this order. In that category, form C (with fewer edges) involved slightly more messages than form D (with more). This probably simply reflects the unresolved competition for the job of pooling information between the two equally central points in form C. But overall, each of the message categories displays the same relationship to structural form as that shown by the total number of messages.

When types of messages are analysed in terms of positions within forms, the presence of communication roles is suggested. The data are shown in Table 18.

Table 18. Means of number of messages sent by message type and form for trials 10 15.

Form Structural role Answers Information Information requests

Organizational suggestions

Center Midpoint Endpoint

Center Midpoint Endpoint

Center Midpoint

Center Midpoint

12.4** 6.2** 0**

14.8** 5.61** 0**

15.2** 4.95**

11.4** 4.87**

5.8 10.9

7.2

14.2 17.0

7.2

15.4 19.25

21.0 18.27

0.4 0 0 0 0 0

2.6 0 0.8 0.07 0.2 0

3.2 0 2.6 0.05

2.8 0.1 1.73 0.27

**E‘s on these differences are significant at Q= 0.01

Messages containing answers clearly specify positional roles. All differences are significant. The most central point always passes more answers than any other point. Endpoints - points that are connected to only one other - do not pass messages. And others pass an intermediate number.

It should be noted that centers here are specified in terms of betweenness. If degree is used, the systematic differences exhibited in form A disappear since the center and the two midpoints all have degrees of two.

Informational messages do not generate significant differences, but their means are, for the most part, in a predictable order. Center points pass less information than mid points (except for form D where confusion seems consistently to reign). Centers, it seems, are collectors, not passers, of information. And, of course, endpoints pass minimal information along the one channel available to them.

Differences are small and non-significant for the other two message types. But again their order is reasonable. Centers request information in

136 L. C Freeman, D. Roeder and R. R. Mulholland

order to solve the problem. And midpoints make the most organizational suggestions presumably because of their ambiguous positions.

All this suggests that people find their place in this experiment according to their structural position. Clearly, there are effects of forms themselves, but there are also internal positional effects that govern communication. This result is, again, intuitively plausible and helps to clarify the results of the MIT experiment. It completes our analysis of the experimental data and helps to fill in the picture of the relationship between communication structure and group performance.

Overall, the analysis of experimental results shows that centrality is an important but not the only ~ structural factor influencing leadership, satisfaction and efficiency in small groups. Another structural factor ~~ the overall density of edges in the structural form also turned out to be relevant.

With respect to the three concepts of structural centrality, the experimental results were interesting. Both the control based measure of betweenness and the activity based measure of degree turned out to be important in understanding group performance. But at no point was the independence index (traditionally most commonly ~lsed) based on closeness even vaguely related to experimental results. It would seem that the closeness-based measure held up in the earlier experiments only because in the particular forms studied it yielded the same predictions of order as the other measures: its apparent utility was simply an artifact of the forms chosen for study. Perhaps a good deal of the historical confusion surrounding the results of this experiment have been the result of using an inappropriate index of the main structural dimension.

Summary and conclusions

This essay has been concerned with structural centrality. We have raised and attempted at least preliminary solutions to the problem of, not what centrality is, but what centrality does.

The three measures of overall network centrality introduced by Freeman ( 1979) agree on assignment of extremes. They all assign the star or wheel the maximum centrality score and the circle and the complete graph the minimum score. Between these extremes, however, agreement breaks down; they differ in their relative ranking of intermediate fonns.

These differences, it turns out, allow a re-examination of the classic MIT experiment on small-group structure and communication. The standard structural forms, wheel, Y, chain and circle, could not be used since they were all ranked in the same order by all three measures. But alternatives were specified and a replication of the earliest experiment was conducted using these new structural forms.

Centrality did emerge as an important structural variable, but not the traditional kind of centrality based on closeness. Instead, the experimentally


important kinds of centrality were those based on potentials for activity and for control.

What, then, does this all come down to? Where does the current report fit into an ongoing research process?

Hopefully, this work has the potential for breathing new life into the classic Bavelas experiment. Recent reviewers have suggested this experiment is an intellectual dead end. At this point, however, it would seem that many of the contradictory results generated by earlier investigators were the result of fuzzy structural concepts along with a tendency to design and analyze experiments in such a way that they succeed only in masking structural effects. Here we have demonstrated that structural effects are present - and present in intuitively plausible ways.

However, like research in any ongoing field, the current work raises many questions for further study. Perhaps the most important single question involves the stability of the reported results. How well would they stand up to replication? Before we can realistically begin to vary the kinds and diffi- culties of problems to be solved, or other fundamental components of the design, we must develop some reliable estimates of experimental parameters.

A related question involves the portability of the general findings of this study to other applications. How far can the observed relationship be generalized? In this context we need to explore variations on the experimental design as well as the possibilities of studying the implications of the various kinds of centrality in other settings including naturally occurring human social networks.

The problem of how to predict trial times was not solved in the current experiment. We were left with the impression that the variance in times was too great for the current exceedingly small samples. Further study, using large samples, is needed to evaluate this impression.

We were able to observe - at least indirectly ~- the emergence of the role of ‘problem solver’ who pooled information in the current experiment. Although it was clear that information-pooling became the task of the most central position, it was not possible to determine what kind of centrality was operating in role assignment. We are left with a question as to whether information-pooling roles are assigned to positions that are central in terms of control or activity. This is a problem for subsequent study.

And finally, along with replications, extensions and parameter estimates based upon the concepts and experiments reported here, the next theoretical problem must be specified and solved. As results become more reliable and detailed it will be possible to build a process model of this kind of communication. Only then will we be able really to untangle the impact of structural centrality on communication networks.

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