# Indirect social influence and diffusion of innovations: An experimental
approach

Manuel Miranda  Instituto de Física Interdisciplinar y Sistemas Complejos
IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain  María Pereda  Grupo de
Investigación Ingeniería de Organización y Logística (IOL), Departamento
Ingeniería de Organización, Administración de empresas y Estadística, Escuela
Técnica Superior de Ingenieros Industriales, Universidad Politécnica de
Madrid, Madrid 28006, Spain  Grupo Interdisciplinar de Sistemas Complejos
(GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911
Leganés, Spain  Angel Sánchez  Grupo Interdisciplinar de Sistemas Complejos
(GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911
Leganés, Spain  Instituto de Biocomputación y Física de Sistemas Complejos,
Universidad de Zaragoza, Zaragoza 50018, Spain  Ernesto Estrada  Instituto de
Física Interdisciplinar y Sistemas Complejos IFISC (UIB-CSIC), 07122 Palma de
Mallorca, Spain

11footnotetext: Corresponding authors. E-mails<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>
## Abstract

A fundamental feature for understanding the diffusion of innovations through a
social group is the manner in which we are influenced by our own social
interactions. It is usually assumed that only direct interactions, those that
form our social network, determine the dynamics of adopting innovations. Here,
we put this assumption to the test by experimentally and theoretically
studying the role of direct and indirect influences in the adoption of
innovations. We perform experiments specifically designed to capture the
influence that an individual receives from their direct social ties as well as
from those socially close to them, as a function of the separation they have
in their social network. The results of 21 experimental sessions with more
than 590 participants show that the rate of adoption of an innovation is
significantly influenced not only by our nearest neighbors but also by the
second and third levels of influences an adopter has. Using a mathematical
model that accounts for both direct and indirect interactions in a network, we
fit the experimental results and determine the way in which influences decay
with social distance. The results indicate that the strength of peer pressure
on an adopter coming from its second and third circles of influence is
approximately 2/3 and 1/3, respectively, relative to their closest neighbors.
Our results strongly suggest that innovation adoption is a complex process in
which an individual feels significant pressure not only from their direct ties
but also by those socially close to them.

##  1 Introduction

In a world of traditions, an innovation is an idea, practice or object that is
perceived as new by an individual [1, p.12]. The adoption of an innovation is
not a trivial process, sometimes requiring a lengthy period of time. Once an
innovation of interest arises, individuals and organizations often need to
accelerate their adoption [2, 3, 4, 5] for reasons that go from public health
policies [6, 7] to marketing ones [8, 9]. Therefore, understanding the
diffusion of innovations, i.e., how innovations are “communicated through
certain channels over time among the members of a social group” [1, p.5] is of
vital importance in many areas of social sciences research [10, 11, 8, 12, 13,
14]. Starting with the seminal book of Rogers, first published in the 1960’s
[1], several works have attempted to find mechanisms for accelerating
innovation diffusion, either from the theoretical or the experimental point of
view [3, 15, 16, 17, 18].

A specific, but quite generic case of innovation diffusion relies on
communication channels formed by interpersonal communication, by means of
which an individual persuades others to accept a new idea [19, 20, 21]. In
principle, this kind of communication channel may be understood as a social
network in which pairs of individuals are connected if they share an
interpersonal communication [3, 22], be it face-to-face exchange or via
e-communication, such as email or social media [23]. However, as far as
diffusion of innovations is concerned, the communication structure may be so
complex that it goes beyond the interpersonal channels of communication
recorded in the social network structure [24, 25]. Indeed, as Rogers put it,
“even the members of the system may not understand the communication structure
of which they are part.” [1, p.337]. One of the reasons for which the network
of interpersonal ties does not capture the totality of the communication
channels is that individuals can learn from observation of other people’s
behavior by means of non-verbal exchange of information. This mimicry of
other’s behaviors conforms a phenomenon known as “social” or “observational”
learning [26, 27, 28]. In fact, even knowledge about certain statistics can
act as a trigger of observational learning. One example is provided by Åberg
[29] who cited the case of local demographic as an important influence on the
risk of divorce. That is, knowing that some people not different from me have
a certain behavior makes me copy them [30, 31, 32, 33]. Several similar
examples are given in [34].

A fundamental research problem is then how to account for the combination of
interpersonal channels and observational learning into a unified network
representation of communication channels for the diffusion of innovations.
Here we take advantage of the seminal ideas of Granovetter [35] who assumes
that all decision makers are influenced by everyone else in an “all see all”
network. However, not everyone is equally considered in these interactions
[36] as people often respond most to the behaviors of those similar to them in
terms of common beliefs, education, socioeconomic status, and so forth [37].
Thus, Granovetter [38] proposed that there is a range of ties with different
strengths, where “the strength of a tie is a (probably linear) combination of
the amount of time, emotional intensity, the intimacy (mutual confiding) and
the reciprocal services that characterize the tie”. The problem is again how
to quantify these strengths of ties. In this work, we use the ideas of Simmel
about social distance [39]. According to Simmel, social distance measures the
nearness or intimacy that an individual or group feels towards another
individual. Therefore, it is natural to associate the concept of social
distance to that of communication proximity, due to their natural equivalence.

In this work, we analyze innovation diffusion by considering a situation in
which an agent, who is influenced by their interpersonal ties in a social
network, is also influenced by any other individual in the network with a
strength that decays with the social distance separating them. Here, we
consider the social distance to be exactly the number of ties that separate
two individuals in their network of interpersonal channels. That is, the
individuals connected to a given agent form their first circle of influences
(see panel A in Fig. 1). Those individuals separated by two ties form the
second circle of influences, and so forth. Then, the traditional view of the
process of diffusion in which an innovation is communicated through several
iterations between agents directly interconnected by interpersonal ties, for
instance, from A to B and then from B to C and from C to D (see panel B in
Fig. 1), is confronted with the model in which the diffusion process occurs
via direct plus indirect influences, where the last take place via the second,
third, etc. circles of influences (see panel C in Fig. 1), such as A receives
direct influence from B, but also indirect influences from C and D. The
framework for our research is a study of the adoption of a drug between
physicians in a hospital [40]: based on this work, we designed and conducted a
series of experiments to empirically measure how long-distance connections
affect the adoption of innovations in a social network. Using the mathematical
framework developed in [41, 17, 42, 43], we can measure the strength of such
non-direct connections compared to the effect of direct friends in such
adoption process. Therefore, we are following Rogers’ suggestion that
“Alternative research approaches to post hoc data gathering about how an
innovation has diffused should be explored. ”[1]. While some recent
experiments have been done in this field [44, 25, 6, 45, 13], the effect of
non-direct relationships has only been studied through data collection of
studies not specifically designed to such purpose [24, 46]. Therefore, our
work fills the gap of experimental research explicitly designed to address the
issue of non-direct connections on innovation diffusion.

Figure 1: Representation of the cone of influences of an individual.

##  2 Experiment and model

### Experimental Setup

As stated above, to investigate the existence and strength of
influences–direct influences only or direct plus indirect ones–on the adoption
of an innovation, we designed the following experiment. A group of
participants is placed in all nodes of a network. At each round of the
experiment, every participant has to choose between two colors, one which is
assigned to the majority of participants and another one assigned only to a
small number of them (but the respective fractions are not known to the
participants). The participants receive a monetary reward (see Methods section
5.1 for details) if they reach a global consensus in one of the two colors.
The incentive is inversely proportional to the number of rounds they need to
reach this steady state. In this scenario, in the initial round, the color of
the majority represents the “tradition”, while the color of the minority
represents an “innovation”. To stimulate the adoption of the innovation,
participants receive a greater incentive if a consensus is reached on the
initial minority color. Although the proportion of vertices with each of these
colors evolves with time, we will refer to “majority” and “minority”
throughout the experiment. In each session of the experiment we randomly
assigned some pairs of participants to role as friends. Pairs of friends form
the edges which are created by imitating one network previously studied for
the diffusion of an innovation in the real-world [40].

Let us now discuss how the initial choice of colors is assigned to the
participants. Assuming that only 13% of the experimental subjects were early
adopters, we initialized all sessions with 27 participants having the majority
color and only 4 with the minority one. Once the participants have been
assigned a color, the experiment takes place in four different settings. In
each one of them, every subject sees a picture of the network, presented as an
ego network centered at themselves, with vertices at longer distances being
smaller than the ones which are nearer to them. The picture they observe is
similar to the one in Fig. 2 where only limited information about the colors
that every other vertex has is provided to them. The exact screenshots of the
experiment can be found in the Supplementary Material. A round finishes when
all participants have made their choice. At the end of every round,
participants see again a similar picture with updated information about the
colors of the corresponding neighbors.

In setting I, every subject had information about the colors of two of the
agent’s nearest neighbors. In setting II, such information consisted of the
colors of two of the agent’s nearest neighbors and two of the participants who
are at distance two from the target. Similarly, for settings III and IV, the
information provided was about two nearest neighbors and two in layer three or
four, respectively (cf. Fig. 2). Every experimental session consists of a
sequence of the four settings, each one comprising in turn of 13-15 rounds. A
round is defined as the step in which every participant makes a decision,
either keeping or changing the color currently assigned to that agent.
Although participants knew that settings would end after at most 15 rounds,
they did not know the exact maximum number of rounds used in each setting. In
each session, the order of the settings was randomized to avoid order effects.
Further details of the experimental design can be found under Methods, section
5.1, and the Supplementary Material.

Figure 2: Schematic illustration of the experimental setup used in this work.
The first setting consists in showing the participants information about the
color chosen by two other subjects directly connected to them in the network.
In the $k$-th settings ($k=2,3,4$ for settings II, III, IV, respectively) the
subjects observe the colors of two participants connected to them and two
others in their $k$-th influence circle. The colors in every setting change to
about bias in the color selection: setting I (blue and yellow); setting II
(magenta and green); setting III (orange and red); setting IV (purple and
lilac). The first color in every pair is always the one of the “minority ”. In
the figure, the vertices observed by the subject are “illuminated”, while the
others in the same circle are shown with pale colors (they are not observed at
all by the subjects).

### Theoretical Model

In order to understand and analyze our experiments, we will compare them with
the following analytical model [41, 17, 42, 43]. Let us assume that every
agent $i$ has a propensity $u_{i}(0)=u_{i}^{0}$ to adopt an innovation at an
initial time $t=0$. Then, the adoption of this innovation in a network is a
consensus process in which the change of the state of subject $i$ at a given
time, $\dot{u_{i}}(t)$, is determined by

$\dot{u_{i}}(t)=\gamma_{NN}\sum_{NN}\left[u_{j}(t)-u_{i}(t)\right],$ (1)

where $\gamma_{NN}$ represents the “strength” of the nearest neighbors (NN)
interactions, i.e., those vertices in the network directly connected among
them. If we represent the states of every individual at a given time in the
vector $u(t)$, we can write

$\dot{u}(t)=-\gamma_{NN}\mathcal{L}_{NN}u(t);\>u(0)=u^{0},$ (2)

where $\mathcal{L}_{NN}$ is the Laplacian matrix of the network operating over
the pairs of NN individuals. This is understood as an operator on a Hilbert
space on the set of vertices of the network acting on a function $f$ defined
in the same space and evaluated on the vertex $v$ as:
$(\mathcal{L}_{NN}f)(v):=\sum_{NN}\left[f(w)-f(v)\right]$, where the sum is
over all the NN of $v$.

The solution of this equation is:
$u(t)=e^{-t\gamma_{NN}\mathcal{L}_{NN}}u^{0}$, and the steady state is the one
in which every vertex has a state equal to the average of the initial
condition. This equation represents the diffusion of the adoption of the
innovation across the network, assuming that the process is continuous in time
as well as in which the state of the individuals may take a continuous range
of values. However, in an experimental setup as the one described before the
time is discrete as it is determined by the rounds taken to reach the
consensus and the results are binary, i.e., an agent either adopt or does not
adopt the innovation. Therefore, here we discretize time as follows. For any
given time $T$ and a number of rounds $r,$ we equidistribute $r$ points in the
interval $\left[0,T\right]$, such that the discretized solution is equal to
the continuous solution at those times. We also proceed to discretize the
output of the model by introducing a threshold parameter: $\triangle\cdot
u_{i}(t_{c})$, where $\triangle\in\left[0,1\right]$ and $u_{i}(t_{c})$ is the
state of the vertex $i$ when the consensus was reached, which is equal to the
average of the entries of $u^{0}$. This means that when an agent has a
propensity to adopt the innovation larger than this threshold it is assumed
that the agent adopts the innovation. Otherwise, it is assumed that it has not
adopted the innovation.

To account for the influence of the individuals in the second circle of
influence of the agent $i$ we can define the Laplacian operator
$(\mathcal{L}_{NNN}f)(i):=\sum_{NNN}\left[f(j)-f(i)\right]$, where now the sum
is carried out over all next nearest neighbors (NNN) of $i$, i.e., those
separated by two edges in the network. Similarly, we can extend this
definition to the third, fourth and so for NN of a given agent, such that we
can write the innovation diffusion model as:

$\displaystyle\dot{u}(t)=-\gamma_{NN}\mathcal{L}_{NN}u(t)-\gamma_{NNN}\mathcal{L}_{NNN}u(t)-\cdots-\gamma_{D}\mathcal{L}_{D}u(t);\qquad
u(0)=u^{0},$ (3)

where $\gamma_{NNN}$ is the “strength” of the interactions between next
nearest neighbors and $D$ designates the diameter of the network, i.e., the
longest shortest path between any pair of vertices. The intuition dictates
that the strength of the interaction decays with the separation between the
pairs of agents in the network, i.e.,
$\gamma_{NN}>\gamma_{NNN}>\cdots>\gamma_{D}$. In the experiments designed in
this work the diameter of the network is five, i.e., $D=5,$ and we can use the
following notation accordingly: $c_{1}=\gamma_{NN};c_{2}=\gamma_{NNN};\ldots$.
Similarly, we designate $\mathcal{L}_{1}=\mathcal{L}_{NN}$;
$\mathcal{L}_{2}=\mathcal{L}_{NNN}$, etc., where, as defined before,
$(\mathcal{L}_{d}f)(v):=\sum_{d(v.w)=d}\left[f(w)-f(v)\right]$. Let us fit
$\gamma_{NN}=c_{1}=1$, such that we can write:

$\dot{u}(t)=-(\mathcal{L}_{1}+\sum_{d=2}^{D}c_{d}\mathcal{L}_{d})u(t);\>u(0)=u^{0}.$
(4)

##  3 Experimental results

We recruited 592 participants from the IBSEN subject pool at Universidad
Carlos III de Madrid (UC3M) to participate in a series of 21 experimental
sessions. The research was approved by the Ethics Committee of UC3M and was
carried out with the approved plan. The average age was 30.4 years (median 25,
mode 22). The gender representation was 63.8% female, 35.9% male, and 0.3%
non-binary. The distribution of gender and age through the experimental
sessions is shown in the Supplementary Information in Table S1 and Figs. S14
and S15.

To analyze the experimental results from a realistic perspective, we
considered that a subject who has adopted the “minority” color becomes an
adopter of the innovation from that round on. This definition is intended to
take into account the differences in time between the experimental settings
and the real adoption of an innovation. While the first takes minutes, the
second can take years, and once a subject has adopted an innovation in the
real world, it will take long times until they can abandon it, in case they
ever do so. In 14 of the 21 experimental sessions, the participants reached
consensus in setting I, and for settings II-IV, the global consensus was
reached in 16 of the 21 sessions (see Fig. S9 in the Supplementary
Information). This means that in some of the 21 experiments there was at least
one individual (stubborn) who did not join the consensus of the group for the
duration of the setting. In total there were 11, 6, 10 and 8 stubborn
individuals in settings I, II, III and IV, respectively, which clearly points
out to the lack of any bias in the number of such individuals in relation to
the type of social interactions considered in the experiments.

Figure 3: Average percentage of adopters (including bots) as a function of the
experimental rounds, per treatment. Shaded areas represent standard
deviations.

In Fig. 3 we illustrate the cumulative distributions of the proportion of
adopters of the innovation versus round for each of the four settings
considered here and averaged over the 21 experimental sessions.In sessions
where most participants reached consensus in round x but some were stubborn,
we adjust the Fig. 3 plot to show global consensus at round x+2 for aesthetic
purposes. In general, the percentages of adopters in the second round are
approximately the same for the four settings (for round I all the percentages
are exactly the same as we initialize all the experiments with this percentage
of adopters). However, for rounds 3-5 these percentages show the largest
differences between the four settings. In round 3, the percentage of adopters
in setting I is about 84.5%, while for settings II and III it grows to 88.8%,
and for setting IV it is 85.4%. In round 4 these percentages are: 89.2%,
92.9%, 93.4% and 91.1%, respectively, and for round 5 they are: 92.9%, 96.2%,
96.2% and 94.2%. Although these values are average percentages that may be
hiding the specifics of each session, (see further analysis), they clearly
indicate an acceleration in the number of adopters in settings II-IV relative
to setting I, particularly for settings II and III. That is, these results
seem to point to the fact that the second and third neighbors of an agent
significantly influence their decision in choosing an innovation. Such
influence seems to drop for the fourth circle of influences.

Let us now discuss the fit of the experimental results to our model. To that
end, we proceed by considering the individual experiments. For each setting,
we fit the results of each of the experimental sessions to find the parameters
$c_{2},$ $c_{3}$ and $c_{4}$ as well as to find the values of $\triangle$ (the
value of $u(t)$ that triggers the adoption of the innovation) and $T$ (the
time equivalent to the number of rounds in the experiment) that best fit the
data as detailed in Methods section 5.2.

In Fig. 4 we illustrate the results of the fitting procedure for the four
settings in the 21 experiments. The experimental data is visualized as points
of colors representing each experimental session. The best fits obtained by
the procedure described previously are illustrated as curves of the same
colors as those of the data points. As can be seen in the plots, the fittings
are much better for the initial times of the time evolution of the adoption
procedure than for the final ones. The reason is that, as mentioned before, in
several experimental sessions, there were stubborn participants who never
joined the consensus state followed by the large majority of subjects. On the
contrary, the diffusion model assumes that every participant is predisposed to
reach the consensus state. In any event, we have analyzed our experimental
data by removing outliers that are basically coincident with the presence of
stubborn subjects, and the results are approximately the same (see the Methods
section 5.4). As there are theoretical models that take into account the
presence of stubborn participants, we maintain the general idea of using a
diffusion model as our main goal here is to investigate the role of indirect
peers pressure in the adoption of innovations. Further studies can be designed
to design models in which stubborn participants are explicitly considered.

Figure 4: Proportion of adopters (dots) and their theoretically expected
behavior (solid lines) as a function of the rounds for all the experimental
sessions (in colors) and the different settings: setting I in upper left
subplot, setting II in upper right subplot, setting III in lower left subplot
and setting IV in lower right subplot.

From the perspective of accounting for the direct and indirect influences of
peers on the adoption of an innovation, the parameters $c_{d}$ are the most
relevant. In Fig. 5 we illustrate the distributions of the parameters $c_{2}$
(setting II), $c_{3}$ (setting III), and $c_{4}$ (setting IV), obtained from
the best fittings of the experimental data to the models of direct plus
indirect influences on the network. The values of the mean and standard
deviations of these coefficients are as follows: $c_{2}=0.651\pm 0.354$;
$c_{3}=0.373\pm 0.427$; $c_{4}=0.513\pm 0.420$. We recall that the strength of
direct influences is $c_{1}=1$.

We then check whether the differences between the means of these coefficients
are significant according to their $P$-values, i.e., the probability of
obtaining the observed difference between the samples if the null hypotheses
were true. The null hypothesis states that the difference between the averages
is 0, that is, there is no difference. We obtained: $p(c_{2},c_{3})=0.0269$,
$p(c_{2},c_{4})=0.256$, $p(c_{3},c_{4})=0.290$. Therefore, the only
significant difference, i.e., $p<0.05$, is between the coefficients that
represent the influences of the second and third circles, but not between the
second or third with the fourth, where there is no empirical evidence to
reject the null hypothesis.

This lack of significant difference between the means of $c_{4}$ and the other
two coefficients could be due to several experimental factors that cannot be
explained with the information that we have obtained from them. Consequently,
we eliminate the results concerning the influence of the fourth circle of
influence and focus on the fact that our results indicate that there is a
relatively large influence of the second NN on the adoption of an innovation
by an agent, which is on average 65% as strong as the direct influence of
peers, and a relatively small, but not negligible, influence of the third NN,
which is on average 37% as strong as the direct interaction. We can then write
an approximate model that describes the results of our experiments as follows:

$\displaystyle\dot{u}(t)\approx$ $\displaystyle-(\mathcal{L}_{1}+(0.651\pm
0.354)\mathcal{L}_{2}+(0.373\pm 0.427)\mathcal{L}_{3})u(t);\qquad u(0)=u^{0}.$
(5)

The empirical model (5) reflects the fact that the strength of the influences
decays with the increase in the social distance (measured here as the shortest
path) between the subjects. This model can be approximated very well by
considering that the coefficients $c_{d}$ are indeed a linear function of the
distance, such that we can write:

$\dot{u}(t)\approx-\sum_{d=1}^{D}\left[(\dfrac{4-d}{3})\mathcal{L}_{d}\right]u(t);\qquad\>u(0)=u^{0}.$
(6)

Using this approximation we can say that the strength of the interactions
between a subject and its second circle of influences is about 2/3 of that
with their closest neighbors, and those in the third circle have an influence
which is about 1/3 of the ones between NN. Whether this is a general
expression for other cases of diffusive adoption of innovations is something
which should be taken with prudence and analyzed in individual cases. Linear
decay models have previously been used to consider social effects, such as
rumor transmission in a network, where an exponentially truncated linear decay
function is used to characterize the decay such that if the acceptance time is
small, the decay function is dominated by a linear decay function [47] (see
also [48, 49]). Other kinds of decay are also studied in the Supplementary
Information.

Figure 5: Violin plot of the distribution of fitted parameters for each of the
Settings that include indirect peer-pressure. Black lines are the individual
value of the fitted parameter for each of the sessions, while the red lines
are the interquartile ranges of each distribution.

To gather further evidence on the role of non-direct influences, we now focus
on the individual decisions. We want to unveil which pieces of information are
people using to make their decisions (whether to choose the innovation color
or not). In order to do so, we study the problem as a classification problem,
where our aim is to predict the color a person chose as function of the
information available: whether people had the innovation as their initial
color, whether the innovation color is the majority color seen, the percentage
of their first neighbors with the innovation color, and the percentage of
$n$-distance neighbors with the innovation color. Then, the chosen color is
the dependent variable and the four pieces of information are the features or
independent variables. We use Random Forests as classification technique and
the feature importance analysis (see the section of Methods 5.3 for details).

Our first result is that the color that a person chooses in each round can be
predicted with high accuracy ($84\%$). Subsequently, when we study the feature
importance of this classification problem, i.e., the contribution of each
piece of information to predict people’s decisions, we see that the importance
of the four pieces of information is different depending on the experimental
setting (see Fig. 6).

Figure 6: Feature importance analysis per experimental setting. The features
are: (1) innovation as initial color, (2) innovation as majority color seen,
(3) percentage of first neighbors with the innovation color, and (4)
percentage of n-distance neighbors with the innovation color.

As can be seen in the plot, the initial color assigned to the participants
(feature 1) is irrelevant to people’s decisions. In the first setting,
participants have no information of their neighbors at distances bigger than
one, and hence feature is being unimportant in this setting. The most
important feature is whether the direct neighbors have acquired the innovation
or not. It is twice as important as the majority opinion (feature 2). In
settings II to IV, the information of $n$-distance neighbors is relevant for
the decisions, and, notably, this information becomes very important. In
setting II, with direct neighbors and neighbors at distance two, the most
important variable is still the direct neighbors information; four times more
important than the 2-distance information. In this setting, the information of
the majority (feature 3) is irrelevant. In settings III and IV, there is a
decrease in the importance of the first neighbors information in favor of the
importance of the majority and the $n$-distance neighbors information. This
suggests that people are taking the whole picture into account when making
their decisions. In general, the analysis suggests that the $n$-distance
information influences the adoption of innovations, and this is more relevant
as the information present is from higher distances.

##  4 Discussion

In this paper, we have provided solid evidence pointing out to the fact that
indirect influences play a fundamental role in innovation diffusion, a key
process in a globalized technological society such as the current one. The
results of an experiment specifically designed to probe into this question
demonstrate that, as summarized in Fig. 7, the adoption of the innovation by
about 60% of participants in our experiments may take around five times less
steps if we allow them to see the influence of those socially close but not
connected to them. The situation is even more dramatic if we consider the
times at which 80% of the experimental subjects adopted the innovation. In
this case, the reduction of time is more than 10-fold under the indirect
influence of peers: In practical terms, this means that an innovation which
would take around a year to be adopted under direct influences only, would be
adopted in about one month under the joint effect of direct and indirect
influences. We note also that the diffusion of innovation goes faster in the
first stages of the diffusion process if information on long-range distance is
present; see setting III and IV curves in Fig. 3. Further, independent
evidence that information is indeed the mechanism behind the acceleration of
innovation diffusion comes from a feature analysis that reveals the way
participants weigh their knowledge of the social context. All in all, the
experimental evidence sends a clear message with practical implications:
innovation diffusion can only be properly understood if the influence of
people at different levels of social distance is taken into account.

As an additional illustration of how such indirect influences can change the
rate of diffusion of an innovation, we do the following theoretical
experiment. By considering the same network studied here experimentally, we
tune the indirect influences without changing the direct interactions between
the agents. We tune these influences simply by changing the coefficient
$c_{d}$ which determines the weight that the not-direct influences have on a
given agent. In Fig. 7 we illustrate the results where we also plot the
experimental results obtained here for no indirect influences as well as for
direct+indirect ones, in which the coefficient fitted to the experimental data
is $c_{d}=\frac{4-d}{3}$. When we increase such indirect influences to
$c_{d}=\frac{5-d}{4}$ or to $c_{d}=\frac{2}{1+d}$, the results are obvious: a
significant increase of the diffusive dynamics in which the times to adopt the
innovation are significantly reduced.

One interesting question arising from our experimental results is the lack of
differences between the coefficients for the fourth neighbor influence and the
coefficient for the second and third neighbors. While, as already mentioned,
this may be an effect of sample size or, perhaps, of the network size, it may
also be the case that the weight we give to the influence of socially distant
contacts saturates, meaning that beyond the first two or three layers of
contacts we take in the input of further ones in the same manner. This might
arise as a consequence of limited cognitive capabilities: in a general
situation in the population at large, we will have many more contacts as
social distance increases, and we are thus led to consider them in a less
specific manner. Remarkably, these results coincide with those reported by
Christakis and coworkers who observed that the risks of spreading obesity
[50], smoking behavior [51], happiness [52], and alcohol consumption behavior
[53] are influenced by individuals up to three degrees of separation between
each other. They observed that by the fourth degree of separation there was no
excess relationship between individuals in the large social network analyzed
over a period of 32 years. Further experiments in larger networks would help
clarify this point, although it must be taken into account that that would
require a large sample of subjects that would play simultaneously with the
logistic challenge that implies [54].

In terms of real-world impact, our results indicate that acting on indirect
influences could change very significantly the adoption rates of innovations.
Mass media has been frequently identified as a principal actor of indirect
influences. By this means, for instance, teenagers in one country can observe
the attitudes and behaviors of others in a different one, copying them for
good or for bad. Therefore, mass media can act as a modulator of indirect
influences, which may change significantly the dynamics of innovation
adoption. While this is a source of influence which is not really amenable to
use as an intervention, other approaches may lead to specific actions in order
to increase, e.g., the adoption of socially desirable behaviors. Akin to
behavioral interventions in which people are informed of the expectations of
others on their behavior [55], we could think of campaigns in which subjects
of interest receive information on what other, socially distant, people do in
the relevant context. Our experimental results indicate that giving only a
limited amount of information about second- or third-order contacts could
already lead to highly increased rates of behavior adoption.

Figure 7: Comparison between different laws for indirect peer pressure,
averaging over 1000 initial conditions. Blue line represents the only direct
pressure diffusion. Dark green line corresponds to considering also indirect
pressure from next nearest neighbors, while pink line considers pressure from
all neighbors at distance less or equal than 3. Light green line corresponds
to the law $c_{d}=\frac{4-d}{3}$ commented in section 2.A. Yellow and brown
lines represent two laws that accelerate the spreading of the innovation,
using the laws $c_{d}=\frac{5-d}{4}$ and $c_{d}=\frac{2}{1+d}$, respectively.

In closing, the results found in this work clearly point out to the fact that
when adopting an innovation we are not only influenced by peers directly
connected to us in our social networks, but that we are also significantly
influenced by people socially close but not directly connected to us in any of
our social networks. This work paves the way to the development of further
experimental and theoretical setting which will allow us a better
understanding of the innovation diffusion dynamics.

##  5 Methods

###  5.1 Experimental methods

The experiment consisted of four treatments, which we refer to as “settings”
to highlight our interest in informational settings, to study the influence of
peer pressure on consensus. Here we summarize a few additional details that
are needed to complete the definition of the experimental setup. The colors in
each treatment were different to avoid learning biases. The eight colors were:
setting I (blue and yellow); setting II (magenta and green); setting III
(orange and red); setting IV (purple and lilac). The first color of each pair
being the majority. In case one or more subjects did not make a choice in a
given round, we declared them “inactive”. Then, to avoid any change in the
structure of the underlying network, we replace that player(s) with a “bot”,
which is programmed to have 50% probability of choosing color at random and
50% of following the majority of the color they would see. Nonetheless, bots
were clearly marked as inactive subjects and were not shown to active
participants if possible. The original network from the study [40] was
slightly modified so that every node had at least two nodes at distance one
and two nodes at distance four. This was done by removing the edge from node
10 to node 30 and by adding an edge between nodes 2 and 17.

Participants received points that, at the end, were converted into money. Each
participant received 1 point per active round (if they made a decision before
the timeout occurred). Then, if consensus was achieved, all of them received 5
points per each round left until round 15 (maximum possible number of rounds)
if they were active at least in one of the two last played rounds of that
setting.

These experiments were programmed using the Python package oTree [56] version
5.10.3, using Cytoscape.js [57] version 3.1.0 for graph visualization. The
code is available in [58]. Data results are available in [59]. Snapshots of
the webpages presented to the participants are shown in the Supplementary
Information.

###  5.2 Fitting method

For a given experiment and a specific setting, we obtain the vector $u_{exp}$
with the percentages of adopters in each round, as well as the initial
condition vector $u_{0}$. Then, for each triplet of parameters
$(c_{d},\triangle,T)$, we produce a prediction vector $u_{pred}$ based on our
model using the following method. From the interval $[0,1]$, discretized in
intervals of size $0.01$, we choose $c_{d}$ and calculate the solution of the
dynamical system, obtaining the solution
$u(t)=\left[\exp(-t(\mathcal{L}_{1}+c_{d}\mathcal{L}_{d}))\right]u_{0}$. Now,
we fix a value for $T$ taken between $10$ and $1000$ in steps of size $10$,
and we identify each round $i$ of the experiment with time $t_{i}\in[0,T]$
such that these times are equally distributed throughout the interval.
Finally, we use one of the threshold values
$\triangle\in\left\\{0.4,0.5,0.6,0.7,0.8,0.9,0.99\right\\}$ and calculate each
entry of the vector $u_{pred}$ by summing all the entries of $u(t_{i})$ above
the threshold ($u_{pred}(i)=\emph{sum}(u(t_{i})>\triangle)$. Hence, for each
combination of $(c_{d},\triangle,T)$, we obtained the prediction $u_{pred}$,
which we can compare to the real proportion of adopters during the experiments
($u_{exp})$ by calculating the mean square error (MSE) and the Pearson
correlation coefficient between the two vectors. The best fit is the
combination of parameters that minimize MSE.

###  5.3 Random Forests and feature importance analysis

Random Forests (RF) [60] are one of the most effective algorithms, excelling
in predictive performance in various application domains, while demonstrating
robustness against overfitting and internal correlations among explanatory
variables. Random Forests employ decision trees with a unique ensemble
technique called bootstrap aggregation (bagging). Unlike traditional decision
trees, RF combines the results of multiple weak learners using bagging, which
aggregates results through averaging in regression tasks and a voting system
in classification tasks. One notable feature of RF is its utilization of ”Out-
Of-Bag” (OOB) data, which comprises approximately one-third of the original
dataset that is not used in constructing each tree. OOB data serve as a test
data set to estimate misclassification error and can also be used to analyze
the relative importance of each feature in the classification problem. For
every tree in the forest, the $jth$ feature of the OOB sample is randomly
permuted, and the resulting increase in OOB error is computed. This increase
serves as a measure of the importance of the $jth$ feature for correct
classification: the greater the increase in OOB error, the more critical the
variable is for achieving accurate classification. This analysis provides
valuable insights into the contribution of each variable to the classification
process, helping in feature selection and interpretation. To estimate the
accuracy of the classifier, we used nested cross-validation (NCV) [61]. NCV
operates by employing cross-validation (CV) within two sequential loops: an
inner loop for hyperparameter selection and an outer loop for computing test
error. In our experiments, both the inner and the outer loops utilized five-
fold CV.

###  5.4 Analysis of outliers and stubborn individuals

It is evident that, during the realization of experiments, several
uncontrollable factors may produce outliers which deviate from the statistical
behavior of the majority. In order to detect such individuals, we tested the
results against three different methods for outlier detection: $Z$-score,
Tukey method, and mean regression (MR). The first two methods are well
documented in the literature [62, 63], so we need to explain the last method.
MR consist in measuring how much the mean changes when we remove the current
value. High scores corresponds to values that change notoriously the mean and
can be considered an outlier.

In Table 1 we give the number of experiments that the corresponding method
detected as an outlier. In parentheses, we give the number of stubborn
individuals that were present in such experiments. Then, in setting II all
outliers identified by Tukey method coincide with those in which there is at
least one stubborn individual. In setting III, Tukey and MR identify six
outlier sessions all of which have stubborn participants (6 out of 8 in this
setting). Finally, the eight stubborn participants that appear in setting IV
are present in the experiments identified by MR as the outlier sessions.
Therefore, the statistics of the results clearly point out the identification
of those experiments in which there are stubborn participants as outliers,
which is what it should be expected by considering that the assumption of the
diffusion model is the whole predisposition of agents to reach the consensus
state.

setting |  $Z$-score | Tukey | MR
---|---|---|---
II | 5 (1); 9 (1) | 2 (1); 5 (1); 9 (1); 16 (2); 20 (1) | 2 (1); 5 (1); 9 (1); 16 (2); 20 (1)
III | 7 (1) | 1 (3); 2 (1); 5 (3); 7 (1) | 1 (3); 2 (1); 5 (3); 7 (1)
IV | none | none | 3 (1); 6 (1); 9 (0); 16 (2); 20 (3)
Table 1: Table showing the outlier rounds detected by the different methods.
The numbers in parentheses represent the number of stubborn subjects for that
session and Setting.

Once we have identified the statistical outliers in the experiments, we
proceed to recalculate the models after their removal. The new coefficients
are: $c_{2}=0.707\pm 0.319$; $c_{3}=0.431\pm 0.453$; $c_{4}=0.503\pm 0.411$.
We also recalculate the $P$-values for the three pairs of coefficients and
obtain: $P(c_{2},c_{3})=0.0275$, $P(c_{2},c_{4})=0.0798$,
$P(c_{3},c_{4})=0.589$, which indicates that the means of $c_{3}$ and $c_{4}$
are less different than before and, although $P(c_{2},c_{4})$ is significantly
smaller than without eliminating outliers, it still is not significant at 95%
of confidence. The empirical model without the statistical outliers (excluding
those detected by MR) is then:

$\displaystyle\dot{u}(t)\approx-(\mathcal{L}_{1}+(0.707\pm
0.319)\mathcal{L}_{2}+(0.431\pm 0.453)\mathcal{L}_{3})u(t);\qquad u(0)=u^{0}.$
(7)

which increases slightly the strength of the influence of the second and third
circle of influence in relation to the case where the outliers were not
removed. All in all, the results indicate that the statistical outliers do not
significantly affect the general findings that long-range influences play a
fundamental role in the diffusion of innovations across a network of social
interactions. Guided by the fact that most of the outliers are those having
stubborn individuals, we conducted a final calibration of the model by
removing all experiments in which there was at least one of these individuals.
The results are very similar to those obtained when removing the outliers
detected by MR and are not reproduced here.

#### Acknowledgements:

A. S. acknowledge support from grant PID2022-141802NB-I00 (BASIC) funded by
MCIN/AEI/10.13039/501100011033 and by ‘ERDF A way of making Europe’, and from
grant MapCDPerNets—Programa Fundamentos de la Fundación BBVA 2022. M.M and
E.E. acknowledge support from Project OLGRA (PID2019-107603GB-I00) funded by
Spanish Ministry of Science and Innovation as well as by the Maria de Maeztu
project CEX2021-001164-M funded by the MCIN/AEI/10.13039/501100011033.

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