American Association for Public Opinion Research
Formulas for Spreading Opinions
Author(s): Stuart Carter Dodd
Source: The Public Opinion Quarterly, Vol. 22, No. 4 (Winter, 1958-1959), pp. 537-554
Published by: Oxford University Press on behalf of the American Association for Public
Opinion Research
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Formulas for Spreading Opinions
BY STUART CARTER DODD
The diffusion of messages by leaflets dropped from artillery shells or airplanes over
military or civilian populations is a technique of propaganda that was widely used in
Korea and in World War II. Project Revere was established to test experimentally the
distribution and effects of airborne messages and their diffusion as they passed by word
of mouth from the recipient of the leaflets to other members of the population. This
is a summary report of the project.
Stuart Carter Dodd is Professor of Sociology and Director of the Washington Public
Opinion Laboratory at the University of Washington.
T HIS PUBLIC ACCOUNTING on a major research with taxpayers' funds will
take the form of a self-interview by the director of Project Revere, which is a
study of message diffusion.' An initial report in the Public Opinion Quarterly (12)2 six years ago told what the project was expected to do. Now we
report to what extent expectations were fulfilled-and some unexpected results also.
PRACTICAL PROBLEMS FOR LEAFLET OPERATIONS
My first question is: Why did the Air Force contract for research on
leaflet operations? They told us in effect: We need to learn principles for
predicting and producing message diffusion from airborne leaflets. We have
dropped over 2 billion leaflets-with very little firm knowledge of their
effect. We must be ready to drop billions more-perhaps in very different
situations. Some drops go agley-for any of eighty physical and social reasons
we can list. In one early drop on France in World War II, air currents finally
deposited leaflets on a director of leaflet operations while he was playing golf
in England. Some seem highly effective if the conditions are just right. The
"Safe Conduct" passes dropped in North Africa when German resistance
was crumbling induced large numbers of Nazi soldiers to surrender. But
IThis research was supported in part by the United States Air Force under Contract AF
33(038)-27522, monitored by the Human Resources Research Institute (now, Officer Education
Research Laboratory, Air Force Personnel and Training Research Center), Air Research and Development Command, Maxwell Air Force Base, Alabama. Permission is granted for reproduction,
translation, publication, and distribution in part and in whole by or for the United States Government.
2 Numbers in parentheses refer to the 57 bibliographic items at the end of this paper. Else-
where reports have been made to sociologists, to statisticians, to semanticists, to scientists at large,
and of course to the Air Force monitors. The latter have received some twenty-seven technical
reports on various aspects of Project Revere. The professional public has so far received some
forty-nine articles in the journals (see Bibliography below). Both may expect to receive the summary volume, Revere Studies on Interaction, when the 700-page manuscript is published in 1959.
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538 PUBLIC OPINION QUARTERLY
exactly what are these conditions? Can they be reduced to rules which we
can use in new situations and future theaters of war? Can the conditions be
stated in general, yet measurable, terms that are not limited to Korea or to
previous uses of leaflets in wartime? What rules, for example, can you find
for deciding how many leaflets to print or drop? What rules may be discovered for guiding the pilots as to where to drop most effectively? What
rules for the best timing? What rules may help us write the most potent
leaflets? In short, what principles may maximize the effect of this leaflet
weapon?
During the research we learned to restate many of the questions put by
the Air Force. We learned to phrase the questions in more experimentally
testable form. Thus we asked: Under specified and controllable conditions
B, C (as to number of leaflets dropped, locating, timing, type of motivation,
etc.), how far will a message spread? How fully or to what percentage of the
population will it spread? How fast will it spread? How faultlessly? Such
"4F" questions, as we called them, guided our program of tests as we tried
to isolate the effect of each basic factor in turn.
THEORETICAL PROBLEMS FOR SOCIAL SCIENTISTS
The next question is: Why did the Washington Public Opinion Laboratory, dedicated to basic research in a university social science department,
undertake this contract? What promise of developing scientific laws of
opinion and behavior was there in thus servicing psychological warfare?
The answer was that it promised a coveted opportunity to begin testing
our interactance formula, or hypothesis of demographic gravitation (7).
This general formula for any human interaction is part of our more comprehensive actance formula which specifies, or serves as a mathematical
model for, our theory of social action in general. This interactance formula
describes (and so expects to predict) any social action as a product of a few
necessary, standardizable, and measurable factors. Six broad factors or dimensions of human behavior were taken as (1) the acts themselves, (2) their
actors, and the (3) temporal, (4) spatial, (5) stimulational (including motivational), and (6) residual conditions. These factors were chosen as highly
measurable and standardizable throughout science. They were also chosen
as being necessary factors in any behavioral product, not alternative addends
in a sum.3 They furthermore deal directly with the Air Force's queries about
(1) what acts will spread messages best? (2) by whom? (3) when? (4)
where? (5) why? and (6) how? The six classes of variable are hypothesized
3 A test to determine whether variables should be combined as a sum or as a product is their
"vanishing result." A zero factor makes the result vanish, but a zero addend does not. Thus any
human behavior will vanish if there is no act, or no actor, or no time whatever to happen in,
or absolutely no space, or no stimulation of any sort, or no context of any kind.
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FORMULAS FOR SPREADING OPINIONS 539
to be categories which will analyze behavioral situations' in most predictive
ways-and this is the major function of science.
But we may probe deeper and ask: Just how did the Laboratory hope
to test the interactance formula by dropping leaflets from the sky upon
people?
Our answer to this 64-dollar question was that the testing would follow
the steps of the scientific method using the currently favored language of
mathematical models (7, 10). The effect on message diffusion for each isolated factor in turn would be studied with the aid of the following research
design:
1. Operationally define the variables.
2. Hypothesize the conditions (i.e., "assumptions," mathematically), re-
lating the variables (such as the probabilistic pre-condition of "equal
opportunities").
3. Deduce (from 1 and 2 exclusively and rigorously) the formula or
model proper which predicts what should be observed.
4. Specify the experiment or controlled observations for testing that
formula.
5. Specify the statistical indices for measuring the agreement between
model and observations.
Since the variables are here observed in polls, these steps specify a controlled
experiment in polling which combines field work and laboratory research.
Probing still deeper we may ask: Why was message diffusion from airborne leaflets selected for this modeling?
Part of the answer, of course, was that the leaflet contract provided an
ample budget for research. But this was only part of the answer, for other
projects were canceled to concentrate on Project Revere.5 Chiefly, we saw
unusual research opportunities, such as a controllable stimulus applicable in
graded amounts in any culture to persons and to whole communities alike;
a measurable response caused solely by that stimulus, isolatable from all context, and serving as a clean-cut criterion of any predictive hypotheses; a con4We use Dewey's term "transaction," meaning an action-in-total-context as a convenient
label for behavior-in-a-situation. Our whole system of dimensional analysis is a comprehensive
model for "transactions" of man. This dimensional system for the social sciences was the full
answer to the question as to what opportunity for basic research we saw in this leaflet project.
The leaflet operations promised experimentally controllable situations to test many of the author's ideas from the two volumes Dimensions of Society (New York, Macmillan, 1942) and
Systeumatic Social Science (Seattle, University Bookstore, 1947, offset).
5 This name for the project had more than an appropriate reference to Paul Revere's historic
spreading of a message in a national emergency. It had an aura of patriotic importance that
helped get the necessary cooperation of officials and citizens-whether in suspending ordinances
against dropping leaflets or in mailing back leaflets picked up-which could not be expected for
a mere scientific experiment.
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540 PUBLIC OPINION QUARTERLY
trollable context where to a high degree the other factors could be held
constant or shown to be irrelevant; and ample resources in funds, manpower,
authority, and public cooperativeness. Rarely does a poller get such an opportunity for basic research.
PRACTICAL FINDINGS FOR DIFFUSING MESSAGES
We come now to results and ask: What advice could be given to leaflet
operators to produce diffusion? Our findings, contributing a chapter to a
Manual of Psychological War, were labeled by letter A, B, or C according as
they were: tested by controlled experiments in the United States (A), judgments from less rigorous empirical data (B), or guesses of experts (C). These
ratings of the firmness of our findings are noted in parentheses at the end of
each of the following examples:
1. What were some findings about the spatial factors?
a. Diffusion, defined as the percentage of message knowers, waned with
the distance the message traveled (A). "Many hear nearby; few hear far
off." This was a case of the harmonic curve, or "inverse distance" model (13,
15, 16). The rate of waning depended on mode of travel-by foot, conveyance, telephone, mail, etc.-for each of which average rates seem determinable in a given culture (C).
b. Diffusion is greatest when leaflets are dropped near to most people,
i.e., in the densely populated areas, in pedestrian trafficways and intersections,
in a uniform scatter over residential areas, not concentrated in spots (B)
(6, 25). This and other obvious generalizations were more exactly measured
than hitherto. (We invented a "Flowdrop" device to feed leaflets out
smoothly from a plane and measured its superiority to packaged or bomb
delivery.)
2. What wiere some findings about the timi'ng factors?
a. Physical and social diffusion (which mean, respectively, picking up
leaflets from the ground or passing them on to other persons) have characteristic growth curves (see below) (A). Exponential and logistic growth
curves were found to be the best growth models. These curves predicting the
growth of knowers depend on how uniform and stable the conditions are
(A) (15, 16, 18, 23, 25, 27, 28, 49, 54).
b. Pre-dawn or noon-hour drops diffused alike in Salt Lake City, indicating, if confirmed elsewhere, that safety of aircrews, etc., can determine
timing (A).
c. Split delivery of n leaflets on 4, 3, 2, or 1 days suggested in one series
of tests that repetitions of the same leaflet increase diffusion (though with
diminishing returns) (A). But ambiguity in another testing calls for further
research here (35).
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FORMULAS FOR SPREADING OPINIONS 541
d. A "chain-tags" technique whereby each recipient of a small leaflet
package keeps one, mails in post-free a "tracer-stub," and passes the rest on,
proved a swift method of communicating to a population when all other
channels may be knocked out (A) (19). The average transmission time from
one recipient to the next in small communities in peacetime (1952) (under
the motivation of "Be a modern Paul Revere. Help spread the word.") was
just ten minutes. At this rate each leaflet pack, if split into thirds, could
reach 1,000 persons within two hours. (Virgil noted that rumor flew on
swift wings-which our experiments clocked.)
3. What were some findings about the populational factors?
a. Diffusion grew absolutely but decreased per capita in a somewhat
harmonic curve as community size went from 1,000 to over 300,000, i.e.,
"small towns seem to gossip more" (A) (25).
b. Diffusion was largest when leaflets were dropped on most people, i.e.,
on denser concentrations, centers of interaction, cities, etc. (B).
c. Diffusion can be greatly mnultiplied by interpersonal retelling if stimulated appropriately (A) (25).
d. Children (in the United States) collected leaflets enthusiastically (B)
(31). Coupled with children being less controllable by police, this fact is
useful for the leaflet operator.
4. What were some findings about the activity factors?
a. Diffusion is best measured by a "potency" index or activity rate defined as "first-time hearers per teller in a unit period" (A). This potency
rate is a behavioristic summary of the interests of the target population, the
meaning of the particular message, the current situation (e.g., war or peace
pressures, etc.), any social resistances operating (e.g., threat of police action,
hostile sympathies, etc.), and timing factors. It proved the central index for
measuring and predicting the spread of messages (25). It measures a social
force in being an acceleration of acting in a population and period.6 We be-
lieve that further research can increasingly make the potency itself predictable (C) (2).
b. Diffusion as measured by the all-or-none index of knowing or not
knowing the message proved most useful (A). Probability theory and
probabilistic models became applicable by this definition. Effects of other
factors then became measurable as degrees of departure from "most probable"
expectations.
c. "Compliance," defined by "doing-as-told" (as in passing on a message,
6 See Stuart C. Dodd, "A Theory for the Measurement of Some Social Forces," Scientific
Monthly, Vol. 43, July 1936; and "The Standard Error of a Social Force," Annals of Mathematical Statistics, Vol. 7, December 1936.
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542 PUBLIC OPINION QUARTERLY
mailing in a tracer leaflet, etc.) proved less predictable than diffusion (A).
Degree of compliance among the fifty-seven census tracts of Birmingham,
Alabama, was best predicted, for example, by years of schooling and per cent
of managers among the sixteen census variables explored (r = .55 and .60)
(47). These sixteen demographic variables gave a raw multiple correlation
of .80 with the number of leaflets mailed in (the index of compliance) in
these populations of census tracts. Three variables alone did almost as well
as the sixteen.
d. With individuals as the units instead of census tracts and with cleaner
controls (eight subclasses by the two sexes, two races, and physical vs. social
diffusion) the average of the eight correlations between per cent of mailbacks and the log of years of schooling (among persons over twenty-five
years of age) rose to .92 (N > 10,000).
5. What were some findings about the values or motivational factor?
a. Values or desiderata defined as "what pollees say they want" seemed
an important but complex predicter of the potency and the diffusion of messages (B).
b. The utility or worth of each want or value is measurable by polls
which ask the respondent to choose between wants (B). Standardizable
wants like money may be included in the set of wants. Thus we found by
a paired comparison experiment that a sample of ministers could readily
scale their diverse supreme and "absolute" values in a single relative ranking
(A). This tended to assure that the most diverse leaflet messages dealing
with different motivations could be scaled in one ranking of effectiveness and
so compared with each other. The principle here is simply: Any motley set
of items can be scaled along one dimension if people can express preferences
among all of them (2, 4, 9, 22, 48, 50, 52).
c. Relative motivating power, whether of a leaflet message or of any
other value, can be measured by give/get ratios or exchange ratios whereby
respondents say what they will give or do to get (or keep) a specified value
under specified conditions (B).
d. The relative motivating power which resulted in seventeen leaflet
versions of one theme being recognized by a target population on whom
they have been dropped could be estimated in advance by rankings which
yielded correlations around .7 in each of four different populations of estimators (A).
e. As long as these rankings were estimated by people of the same culture
as the target population, the former did not need to be representative samples
of the target population (A) (2). If confirmed more widely, this implies
that some motivational aspects of candidate leaflets can be gauged from
prisoners of war or other available representatives of the target population.
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FORMULAS FOR SPREADING OPINIONS 543
However, further experiments estimating compliance yielded zero correlations (A), so that more research is needed to map this motivational field
more clearly.
f. A comprehensive theory of motivation was developed specifying fifty
conditions which seem predictively measurable by polls in any culture (B)
(22). Further research on these may develop laws of motivation which we
believe could revolutionize human engineering. We consider this as yet
untested model for motivating men to be the most important result of all of
Project Revere.
6. What were some findings about the stimulational factors in diffusion?
a. Diffusion increased but with diminishing returns as the strength of
stimulation increased (A). The logarithmic Weber-Fechner law of individual sensory psychology seems to apply here to whole communities. In the
case of eight matched small towns, for each increment of 9 per cent of
message knowers (as the community response) it was necessary to double
the number of leaflets per capita dropped (as the stimulation) (A) (16, 53).
b. This empirical rule-that to add to the response requires multiplying
the stimulation-urgently needs research to chart its rationale or conditions
under which it holds and can be predicted with new messages in new cul-
tures (B). We suspect a probability explanation will be achieved soon. Its
potential social importance seems very great.
7. What were some findings about the residtal class of factors?
a. This large class of variables, mostly unknown, yielded on exploration
a few variables correlating highly and significantly with the diffusion. Thus
friendships correlated up to .5 with diffusion in one study (B). People tell
their friends more than they tell acquaintances and both of these more than
they tell strangers, of course (A). Emergencies may be expected to cut across
such normal networks (C).
b. Yet that very message at the same time spread randomly through that
population, as evidenced by the fact that its growth fit one of the logistic
curves which is the cumulation of random meetings and tellings (B). We
suspect here an important principle-that behavior may be individually
purposive and planned, yet randomlike in the aggregate. This can happen
in "granular" situations where the purposes of many individuals are uncorrelated with each other and function much as separate granules of sand (B).
c. A point of view that is not yet a factual finding emerged from Revere
studies dealing with the social meaning of randomness. In seeking to specify
randomness (or any other concept) by an operational definition we try to
describe its causes, its contents, and its consequences. The social causes of
randomlike behavior we see as many, small, uncorrelated influences, none of
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544 PUBLIC OPINION QUARTERLY
which dominate. Such a situation could occur in either a very homogeneous
population or an extremely heterogeneous one-so that a leaflet message
could spread randomly simply because many, small, diverse determiners were
active, with none dominating the others (C).
The logistic curve arises whenever entities pair off randomly and steadily
(A). The fact that in our tests messages spread from person to person predominantly according to one of the family of logistic curves effectively indicates random situations.
One social consequence of a random distribution is that every person
has a democratically and mathematically equal opportunity (A). If one
wishes to spread a message in such a democratic way, we have learned how
to do it. The recipe is to set up a logistic distribution to order.
One way an experimenter can control or eliminate the effect of all variables other than the one at issue is by randomizing that one variable (A4).
He thereby guarantees it will not be correlated with his sampling, for example. Pollers know well how to randomize with a book of random numbers.
And now we are just learning how to prove randomness exists when it occurs
of itself in society. Fitting a logistic curve is one way to test for randomness
wherever the spreading of a single attribute is at issue (A).
The twenty-five items above skim off some findings concerning content
in our review of the factors of diffusion analyzed in our interactance formula.
But the large program of testing in Project Revere also came up with findings
as to methods.
SOME METHODOLOGICAL FINDINGS
What were some findings about techniques? What was learned about
leaflet techniques for spreading opinions and polling techniques for evaluating that spreading? Again only a sampling of items can be served up
here (25).
1. Preparing leaflets. Extensive studies of the literature on leaflet format
were made to find the most effective size, shape, colors, print, illustration,
etc., for specified conditions of weather, terrain, culture, police control, and
other situations. Our standard Revere leaflet was one outcome of these
studies. An unexpected by-product of one field test was a warning concerning
yellow leaflets (A). Yellow has a strong tropistic attraction for aphids, a
harmless garden variety of plant lice. They swarmed over our yellow leaflets,
blackening out the text, while not one aphid was found on our blue leaflets
beside them on the grass. In ignorant or hostile lands a UN leaflet crawling
with harmless plant lice could well be photographed and propagandized as
germ infested.
2. Reliability. The degree of agreement of re-observations was tested in
many ways: (a) Two persons: independent counts of leaflets on the ground
from inspecting air photos correlated at r = .99 (A). (b) Diffusion as ob-
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FORMULAS FOR SPREADING OPINIONS 545
served through face-to-face interviews (with 50 per c
correlated with diffusion measured by mail-ins of lea
matched towns (A). (Consequently, for studying the reaction of whole
communities the cheaper mail-back technique was thereafter substituted for
polling.) (c) The most promising of our sixteen series of "pretests" were
developed and repeated in twelve "tests" to re-observe the findings with fuller
controls. (A third year of "retests" of the chief findings in new situations and
other cultures was planned in the original prospectus but was cut for lack of
funds.) (d) Since the samples of respondents we studied (in the course of
dropping three quarters of a million leaflets on as many American citizens in
thirty communities) varied from 20 to around 20,000, a minimum standard
of statistical significance at the 5 per cent level was uniformly used (B).
(e) Eight techniques for observing a population's response to our leaflet
stimulation were tried, namely ground observers as leaflets fell, a center
for come-ins, telephone call-ins, leaflet mail-ins, group interviews, and polls
by newspaper, by phone, and by face-to-face interviewers. Five failed completely for various reasons, while group interviews, mail-ins, and face-to-face
interviews succeeded, though group interviews were limited to captive audiences, eg., in a classroom.
3. Validity. The degree of agreement of our observations with a criterion
was also tested in several ways: (a) Face-to-face interviews were five times
as valid as telephone interviews, by the criterion of actual compliance behavior of mailing in a leaflet as claimed in the interviews (29). (b) The
agreement of each model with the criterion of observed facts-how things
work-is a form of validity. A validity correlation above .9 measuring descriptive closeness of fit of model to data was set as a standard throughout the
project (B).
4. Control of variables. (a) We had to learn the hard way how to make
our variables vary or stay constant as needed in order to isolate each variable's
effects in turn. We learned how to start rumors only after many disappointing failures. Thus one juicy rumor, although carefully planted with a dependable gossip in a housing project, was known only to her in the 100 per cent
sample interviewed a week later. An offer implausibly made to five people
on the street to pay a dollar to anyone in town who reported hearing of the
offer brought no takers. Eventually we found that a civil defense topic commanded the most cooperation from officials and public and had the least
backfire on public relations. (b) With a "rumor" thus launched from person
to person, the mass media-chiefly radio, TV, newspapers, and wire services
-had to be sealed off to ensure that the message would spread by word of
mouth only. We learned to "control" the mass media for such scientific
experimenting in a censorless democracy during peacetime by the device of
publicity releases with release dates in the future. We called all the publicity
leaders together under official defense auspices and gave them well-prepared
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546 PUBLIC OPINION QUARTERLY
documents, pictures, etc.-but not to be released until several days after the
end of the experiment (B). This effectively controlled any premature presslaunched messages, even in a modern metropolis, and isolated the person-toperson diffusion. (However a press leak in one small town did require correction in one series of tests.)
SOME THEORETICAL FINDINGS ON LAWS OF DIFFUSION
This report to pollers is incomplete without a clear indication of the
Laboratory's contribution through Project Revere to building up a body of
scientific laws of human behavior. What generalizations, we ask in summary,
were discovered or developed which may apply in any culture, any time,
anywhere? Were any hypotheses tested which promise to transcend the usual
limitations of polls to local and transient findings particular to a situation
and a culture?
Our original prospectus in the Public Opinion Quarterly (12) promised
that among the generalized hypotheses7 to be tested were:
The Zipfian harmonic formula for a die-away curve relating diffusion to the inverse
of the distance traveled
The Weber-Fechner logarithmic formula for a diminishing returns curve relating
diffusion to the log of the strength of stimulation
The normal probability curve relating diffusion to multiple causes
The logistic S-shaped curve of interaction relating diffusion to the most probable
meetings of the interactors
The overall interactance formula in dimensional form relating diffusion to the
weighted product of its factors, namely the acts and actors, and their temporal
and spatial, motivational, stimulational, and residual conditions.
What then happened to this crop of hypotheses? Did any of them ripen
into social laws?
All these hypotheses and some others underwent (1) testing, (2) rationalizing, and (3) systematizing in Project Revere (25). Let us look at each
of these three research processes briefly. (For technical details the reader may
consult the appropriate articles in the journals or the twenty volumes of
working documents filed in the Laboratory.)
1. For testing these hypotheses, situations were designed, either in captive
populations or in open communities, so as to try to vary in isolation the one
factor which was hypothesized to influence the diffusion as predicted by its
curve. Thus one experiment let the distance that tellers of a message went to
7 The dimensional formulas in simplest form for the differential equations of these distribution curves are tabulated below. A dimensional formula shows the essential shape of the curve
ignoring local accidents of its slope and origin point on a graph. For these shapes, it shows only
the exponents of the variables ignoring their coefficients and addend parameters, which statistical
formulas also show. Dimensional formulas emphasize laws, i.e., the generalized properties freer
of what is particular to each set of data with which statistical formulas deal. Dimensional
formulas are useful for classifying phenomena but must be further specified-as by our standardized four "corner scripts"-to be used for statistical computing in a given case (see Table 1).
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FORMULAS FOR SPREADING OPINIONS 547
find hearers vary spontaneously while the time per
responding act of telling-or-not-telling, the motivatio
other conditions were fairly constant. The hypothesize
found to fit the distance data excellendy (correlating about r = .99 in
uncumulated data and showing non-significant discrepancies by the chisquare test at the 1 per cent level) (15, 16).
Similarly, our twenty-eight series of tests tended to confirm the other
hypothesized formulas listed above insofar as their specified pre-conditions
existed fully and solely in the situation observed. This proviso largely meant
"insofar as we knew the conditions and were able to control them cleanly
and fully." This proviso, of course, is what is meant by any scientific lawnamely a general statement of relation between entities which will always be
observed if and only if its preconditions are present, fully and solely. The
deeper problem becomes: What are the pre-conditions of a given law or
formula? These pre-conditions are the "rationale" which convert any empirical model into a rationalized model. The pre-conditions "explain" a model
by describing its immediate observable causes or necessary antecedent states.
Table 1 summarizes the pre-conditions for our models in Project Revere.
2. For rationalizing these hypotheses, Project Revere was fully successful
in the case of the logistic, exponential, and Gompertz curves of diffusion;
made some progress on the harmonic and its cumulated form, the logarithmic curve; needed no new rationalizing of the well-studied linear and normal
curves; and got nowhere with our explorations of the higher-power curves.
As an illustration, consider the logistic curve which had been much used to
describe spurts of growth in human and animal populations, in culture traits,
and much else of complex causation. We sought the social conditions, expressible mathematically as assumptions, whereby an all-or-none message
would spread according to the logistic S-shaped curve regardless of the population's differences in age or sex, income or color, language spoken or religion
claimed, tastes in gossip or childhood toilet frustrations. These culture-free
conditions were finally refined to just three, namely, the attribute must spread
by random, steady, pairi'ng. Insofar as molecules, mice, or men transfer any
all-or-none state to each other solely by steady pairing off, with some-
what equal opportunity for each molecule, mouse, or man, that state will
grow in a logistic curve no matter what the entities or attribute transferred,
their history or current situation otherwise. This general mathematical law
of dynamic probabilities can be demonstrated as a social law embodied in
human behavior any time anyone wishes. We can now set up a perfectly
controlled experiment in a classroom, plot an exact logistic growth curve on
the blackboard in advance as a prediction, start the rumor circulating, and
watch it follow closely (within sampling limits) that forecast (15, 16, 20, 25,
28).
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548 PUBLIC OPINION QUARTERLY
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17"C")IRMULAS FOR SPREADING OPINIONS 549
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550 PUBLIC OPINION QUARTERLY
We submit that when a science can thus predict and control behavior in
its field, it is maturing as an exact science. When a dimensional formula analyzes opinion spreading so well that such speech behavior can by resynthesized at will, that field of science is beginning to come of age.
3. Finally, for systematizing the set of hypotheses, in summarizing this
report, just what sort of system was developed?
We discovered that the dimensional formulas for this set of curves could
be simplified and rewritten with the aid of our standardizing script notation
in a single master formula (18, 23, 24, 25). T'his master formula constitutes
our formula system, or unified organizing of relevant symbols. The master
formula represents any attribute, A, when operated upon as specified by its
four corner scripts. This statistical attribute can be any item of human be-
havior if observed in all-or-none form so that it has only the two numerical
values of 1 or 0. It is thus the simplest possible act or unit for the behavioral
sciences.
This elemental act is always observed as an act-in-a-population and in-aperiod and embedded in other context. Such an act-in-context we call a
"transact," using Dewey's term. By our vanishing result test, the total act-incontext is a product of factors. These three factors, or dimensions of activity,
people, and time, are necessary and sufficient to describe these simple forms
of diffusion or transitive interacting. The master formula that generates the
formulas in Table 1 is simply
P= itAm the power formula for diffusion (Eq. 1)
where p is the proportion of attribute holders or message knowers in a
population
A is the attribute or act diffused (A = 1, 0)
a is the set of attributes, a in number (a 1 in the exponential and
logistic models)
m is the integer exponent or power (m 0 O, 1, +- 2, etc.)
i is the set of individuals or actors, i in number
t is the set of successive subperiods or time units, t in number
As the scripts vary, families of curves or forms of diffusion become specified.8
8 Thus equation 1 becomes the logistic case of diffusion of one attribute spreading from person to person when opportunities are equal, by letting a = 1 and m = 2. The exponential case
of diffusing of one attribute by mass media when opportunities are equal is equation 1 when
a = 1 and m = 1. The normal probability curve can be generated (for one of many ways)
when m = 0 (by thoroughly mixing a attributes among the i individuals in the t time periods).
These three forms of diffusion-the normal, the exponential, and the logistic-are simply probability growth curves which we call the "moments clan" of power models because they are
specified by the zeroth, first, and second statistical moments of an attribute (i.e., by m = 0, 1, or
2, with a = 1, in equation 1).
When a = 1 and t = 1 (i.e., when we have just one static attribute), equation 1 becomes
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FORMULAS FOR SPREADING OPINIONS 551
The generative operational definition of these probab
for predicting diffusion is specified by the script oper
wise sequence as follows: first, sum the attributes to g
cal variate X, X _ A a, which is just A when a -
the mth power; third, observe Xm in the set of ind
distribution curves, iXm, and go on to average Xm or
moments of A, iAm; fourth, sum each moment over t
the most probable growth distribution or moments cu
We call Equation 1 the "powers formula" for the
when observed in all-or-none form in any roughly hom
A "homogeneous" population here means any popula
miners of the opinion-spreading acts at issue tend to b
(relative to their total), and largely different for ea
the randomness condition which underlies all these
propose the name "powers formula" because the exp
is the chief feature in defining which family of sha
conditions specified in Table 1, will most probably
predict the spread of opinion.
BIBLIOGRAPHY OF REVERE-CONNECTED PAPERS
1. Bowerman, Charles, with Stuart C. Dodd and Otto N. Larsen, "Testing Message
Diffusion-Verbal vs. Graphic Symbols," International Social Science Bulletin,
UNESCO, Vol. 5, September 1953.
2. Catton, William R., Jr., "Exploring Techniques for Measuring Human Values,"
American Sociological Re'iew, Vol. 19, 1954, pp. 49-55.
3. , and Melvin L. DeFleur, "The Limits of Determinacy in Attitude Measurement," Social Forces, Vol. 35, 1957, pp. 295-300.
4. , and Stuart C. Dodd, "Symbolizing the Values of Others," in Symbols and
Values: An Initial Study, Thirteenth Symposium of the Conference on Science,
Philosophy, and Religion, New York, Harper, 1954, Chap. 34, pp. 485-496.
the statistical moments of an attribute and specifies the elementary laws of probability-simple
probability, complementary probability, alternative probability, null probability, and joint proba-
bility-according as m = 0, 1, or 2 (and according as zero or the mean is taken as origin).
These forms of probability simply reflect how we observe the attribute as in noting either its
presence, or its absence, or either one, or neither one, or both.
When a > 1, t = 1, and m < 0, i.e., with many static attributes combined into one
variable which has a negative exponent, equation 1 becomes the family of hyperbolas (and
harmonic series in discrete form). In this family if m = -2, equation 1 becomes the differential
equation for the harmonic curve or rectangular hyperbola; and if m = -1, equation 1 becomes
the differential equation for logarithmic curves; while if m = 0, equation 1 becomes the differential equation for a straight line. Thus the linear, logarithmic, and harmonic families of
models for diffusion which were all observed in Project Revere are systematized by the exponents of -2, -1, and 0 when t = 1 in equation 1. We call this set of families of curves the
"conics clan" of power models, since their equations represent in elementary form the subdivisions of the field of conic sections or coordinate geometry. The conics clan deal with plural
static attributes and so are identified by t = 1 in equation 1, while the moments clan deal
with singular changing attributes and so are identified by a 1 and t > 1. (The normal curve
is somewhat intermediate in becoming better approximated as the number of attributes enlarges.)
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552 PUBLIC OPINION QUARTERLY
5. , and Richard J. Hill, "Predicting the Relative Effectiveness of Leaflets: A
Study in Selective Perception with Some Implications for Sampling," Research
Studies of the State College of Washington, Proceedings of the Pacific Coast Socio-
logical Society, 1953, Vol. 21, pp. 247-251.
6. DeFleur, Melvin L., and 0rjar 0yen, "The Spatial Diffusion of an Airborne Leaflet
Message," American Journal of Sociology, Vol. 59, 1953, pp. 144-149.
7. Dodd, Stuart C., "The Interactance Hypothesis-A Gravity Model Fitting Physical
Masses and Human Groups," American Sociological Review, Vol. 15, 1950, pp. 245256.
8. , "Sociomatrices and Levels of Interaction-for Dealing with Plurels, Groups,
and Organizations," Sociometry, Vol. 14, 1951, pp. 237-248.
9. , "On Classifying Human Values-a Step in the Prediction of Human Valuing," American Sociological Review, Vol. 16, 1951, pp. 645-653.
10. , "On All-or-None Elements and Mathematical Models for Sociologists,"
American Sociological Review, Vol. 17, 1952, pp. 167-177.
11. and staff, "Testing Message Diffusing in C-Ville," Research Studies of the
State College of Washington, Proceedings of the Pacific Coast Sociological Society,
1952, Vol. 20, 1952, pp. 83-91.
12. , "Testing Message Diffusion from Person to Person," Public Opinion Quarterly, Vol. 16, 1952, pp. 247-262.
13. , "Controlled Experiments on Interacting-Testing the Interactance Hypo-
thesis Factor by Factor," read at the Sociological Research Association Conference,
Atlantic City, N. J., September 1952.
14. , "Human Dimensions-a Re-search for Concepts to Integrate Thinking,"
Main Currents in Modern Thought, Vol. 9, 1953, pp. 106-113.
15. , "Testing Message Diffusion in Controlled Experiments: Charting the Distance and Time Factors in the Interactance Hypothesis," American Sociological
Review, Vol. 18, 1953, pp. 410-416.
16. , "Can the Social Scientist Serve Two Masters-An Answer through Experimental Sociology," Research Studies of the State College of Washington, Proceedings of the Pacific Sociological Society, Vol. 21, 1953, pp. 195-213.
17. , "Formulas for Spreading Opinion-a Report of Controlled Experiments on
Leaflet Messages in Project Revere," read at A.A.P.O.R. meetings, Madison, Wis.,
Apr. 14, 1955.
18. , "Diffusion Is Predictable: Testing Probability Models for Laws of Interaction," American Sociological Review, Vol. 20, 1955, pp. 392-401.
19. , "Testing Message Diffusion by Chain Tags," American Journal of Sociology,
Vol. 61, 1956, pp. 425-432.
20. , "Testing Message Diffusion in Harmonic Logistic Curves," Psychometrika,
Vol. 21, 1956, pp. 192-205.
21. , "A Predictive Theory of Public Opinion-Using Nine 'Mode' and 'Tense'
Factors," Public Opinion Quarterly, Vol. 20, 1956, pp. 571-585.
22. , "Conditions for Motivating Men-the Valuance Theory for Motivating Behaviors in Any Culture," Journal of Personality, Vol. 25, 1957, pp. 489-504.
23. , "The Counteractance Model," American journal of Sociology, Vol. 63, 1957,
pp. 273-284.
24. -, "A Power of Town Size Predicts Its Internal Interacting-a Controlled Experiment Relating the Amount of an Interaction to the Number of Potential Inter-
actors," Social Forces, Vol. 36, 1957, pp. 132-137.
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FORMULAS FOR SPREADING OPINIONS 553
25. , with Edith D. Rainboth and Jiri Nehnevajsa, "Revere Studies on Interaction"
(Volume ready for press).
26. Hill, Richard J., "A Note on Inconsistency in Paired Comparison Judgments,"
American Sociological Review, Vol. 18, 1953, pp. 564-566.
27. , "An Experimental Investigation of the Logistic Model of Message Diffusion,"
read at AAAS meeting, San Francisco, Calif., Dec. 27, 1954.
28. , with Stuart C. Dodd and Susan Huffaker, "Testing Message Diffusion-the
Logistic Growth Curve in a School Population," read at the Biometrics Conference,
Eugene, Ore., June 1952.
29. Larsen, Otto N., "The Comparative Validity of Telephone and Face-to-Face Inter-
views in the Measurement of Message Diffusion from Leaflets," American Sociological Review, Vol. 17, 1952, pp. 471-476.
30. , "Rumors in a Disaster," accepted for publication in Journal of Communication.
31. , and Melvin L. DeFleur, "The Comparative Role of Children and Adults in
Propaganda Diffusion," American Sociological Review, Vol. 19, 1954, pp. 593-602.
32. , and Richard J. Hill, "Mass Media and Interpersonal Communication,"
American Sociological Review, Vol. 19, 1954, pp. 426-434.
33. Nehnevajsa, Jiri, and Stuart C. Dodd, "Physical Dimensions of Social Distance,"
Sociology and Social Research, Vol. 38, 1954, pp. 287-292.
34. Pence, Orville, and Dominic LaRusso, "A Study of Testimony: Content Distortion
in Oral Person-to-Person Communication," submitted for publication.
35. Rainboth, Edith Dyer, and Melvin L. DeFleur, 'Testing Message Diffusion in Four
Communities: Some Factors in the Use of Airborne Leaflets as a Communication
Medium," American Sociological Review, Vol. 17, 1952, pp. 734-737.
36. Rapoport, Anatol, "Nets with Distance Bias," Bulletin of Mathematical Biophysics,
Vol. 13, 1951, pp. 85-91.
37. , "Connectivity of Random Nets," Bulletin of Mathematical Biophysics, Vol.
13, 1951, pp. 107-117.
38. , "The Probability Distribution of Distinct Hits on Closely Packed Targets,"
Bulletin of Mathematical Biophysics, Vol. 13, 1951, pp. 133-138.
39. , "'Ignition' Phenomena in Random Nets," Bulletin of Mathematical Bio-
physics, Vol. 14, 1952, pp. 35-44.
40. , "Contribution to the Mathematical Theory of Mass Behavior: I. The Propagation of Single Acts," Bulletin of Mathematical Biophysics, Vol. 14, 1952, pp. 159169.
41. , "Response Time and Threshold of a Random Net," Bulletin of Mathematical Biophysics, Vol. 14, 1952, pp. 351-363.
42. , and Lionel I. Rebhun, "On the Mathematical Theory of Rumor Spread,"
Bulletin of Mathematical Biophysics, Vol. 14, 1952, pp. 375-383.
43. , "Contribution to the Mathematical Theory of Contagion and Spread of Information: I. Spread through a Thoroughly Mixed Population," Bulletin of Mathematical Biophysics, Vol. 15, 1953, pp. 173-183.
44. , "Spread of Information through a Population with Socio-structural Bias:
I. Assumption of Transitivity," Bulletin of Mathematical Biophysics, Vol. 15, 1953,
pp. 523-533.
45. , "Spread of Information through a Population with Socio-structural Bias:
II. Various Models with Partial Transitivity," Bulletin of Mathematical Biophysics,
Vol. 15, 1953, pp. 535-546.
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554 PUBLIC OPINION QUARTERLY
46. , "Spread of Information through a Population with Socio-structural Bias:
III. Suggested Experimental Procedures," Bulletin of Mathematical Biophysics, Vol.
16, 1954, pp. 75-81.
47. Shaw, John G., "Testing Message Diffusion in Relation to Demographic Variables:
an Analysis of Respondents to an Airborne Leaflet Message," submitted for publication.
48. Turabian, Chahin, and Stuart C. Dodd, "A Dimensional System of Human Values,"
Transactions Second World Congress of Sociology, International Sociology Association, 1954, pp. 100-105.
49. Winthrop, Henry, and Stuart C. Dodd, "A Dimensional Theory of Social Diffusion
-an Analysis, Modeling and Partial Testing of One-way Interacting," Sociometry,
Vol. 16, 1953, pp. 180-202.
Theses
50. M.A. Catton, William R., Jr., "The Sociological Study of Human Values," 1952.
51. M.A. 0yen, 0rjar, "The Relationship between Distances and Social Interactionthe Case of Message Diffusion," 1953.
52. Ph.D. Catton, William R., Jr., "Propaganda Effectiveness as a Function of Human
Values," 1954.
53. Ph.D. DeFleur, Melvin Lawrence, "Experimental Studies of Stimulus Response Relationships in Leaflet Communication," 1954.
54. Ph.D. Hill, Richard J., "Temporal Aspects of Message Diffusion," 1955.
55. Ph.D. Larsen, Otto N., "Interpersonal Relations in the Social Diffusion of Messages,"
1955.
56. Ph.D. Shaw, John G., Jr., "The Relationship of Selected Ecological Variables to
Leaflet Message Response," 1954.
57. M.A. West, S. S., "Variation of Compliance to Airborne Leaflet Messages with Age
and with Terminal Level of Education," 1956.
Monographs Published
58. DeFleur, Melvin L., and Otto N. Larsen, The Flow of Information, New York,
Harper, 1958.
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