How Peer Pressure Shapes Consensus, Leadership, and Innovations in Social Groups

Abstract

What is the effect of the combined direct and indirect social influences-peer pressure (PP)-on a social group's collective decisions? We present a model that captures PP as a function of the socio-cultural distance between individuals in a social group. Using this model and empirical data from 15 real-world social networks we found that the PP level determines how fast a social group reaches consensus. More importantly, the levels of PP determine the leaders who can achieve full control of their social groups. PP can overcome barriers imposed upon a consensus by the existence of tightly connected communities with local leaders or the existence of leaders with poor cohesiveness of opinions. A moderate level of PP is also necessary to explain the rate at which innovations diffuse through a variety of social groups.

Full text

How Peer Pressure Shapes Consensus,
Leadership, and Innovations in Social
Groups
Ernesto Estrada
1,2
& Eusebio Vargas-Estrada
1
1
Department of Mathematics & Statistics, University of Strathclyde, Glasgow G1 1XH, UK,
2
Institute for Quantitative Theory and
Methods (QuanTM), Emory University, Atlanta, GA 30322, USA.
What is the effect of the combined direct and indirect social influences—peer pressure (PP)—on a social
group’s collective decisions? We present a model that captures PP as a function of the socio-cultural distance
between individuals in a social group. Using this model and empirical data from 15 real-world social
networks we found that the PP level determines how fast a social group reaches consensus. More
importantly, the levels of PP determine the leaders who can achieve full control of their social groups. PP can
overcome barriers imposed upon a consensus by the existence of tightly connected communities with local
leaders or the existence of leaders with poor cohesiveness of opinions. A moderate level of PP is also
necessary to explain the rate at which innovations diffuse through a variety of social groups.
The social group’s pressure on an individual—peer pressure (PP)—has attracted the attention of scholars in a
variety of disciplines, spanning sociology, economics, finance, psychology, and management sciences
1–4
.In
analyzing PP we should consider not only those individuals directly linked to a particular person, but also
those who exert indirect social influence over other persons as well
5–8
. Although PP is an elusive concept, it can be
considered a decreasing function of a given individual’s socio-cultural distance from the group. Thus, an indi-
vidual’s opinion may be influenced more strongly by the pressure exerted by those socio-culturally closer to her.
Consensus is well documented across the social sciences, with examples ranging from behavioral flocking in
popular cultural styles, emotional contagion, collective decision making, pedestrians’ walking behavior, and
others
9–12
.
We can model consensus in a social group by encoding the state of each individual at a given time tin a vector
u(t). The group reaches consensus at tR‘when uitðÞ{ujtðÞ
?0 for every pair of individuals, and the collective
dynamics of the system is modeled by
dutðÞ
dt ~{Lu tðÞ,u0ðÞ~u0,ð1Þ
where Lis a linear operator (Laplacian matrix) capturing the topology of the social network
9
.
Decisions in groups trying to reach consensus are frequently influenced by a small proportion of the group who
guides or dictates the behavior of the entire network. In this situation a group of leaders indicates and/or initiates
the route to the consensus, and the rest of the group readily follows their attitudes. The study of leadership in
social groups has always intrigued researchers in the social and behavioral sciences
13–17
. Specifically, the way in
which leaders emerge in social groups is not well understood
18
. Leaders may emerge either randomly in response
to particular historical circumstances or from the individual having the most prominent position (centrality) in
the social network at any time.
Results
Emergence of leaders and PP.To capture the influence of PP over the emergence of leaders in social groups, we
consider that the pressure that an individual preceives from qdeteriorates proportionally with the social distance
between pand q. The social distance is captured by the number of links in the shortest path connecting pand q.
Mathematically, we model the mobilizing power between two individuals at distance das D
d
,f(d)
21
, where f(d)
represents a function of the social distance (see Methods equations (11) and (12)). The collective dynamics of the
network under peers’ mobilizing effects is described by the following generalization of the consensus model
OPEN
SUBJECT AREAS:
COMPLEX NETWORKS
APPLIED MATHEMATICS
APPLIED PHYSICS
Received
10 September 2013
Accepted
19 September 2013
Published
9 October 2013
Correspondence and
requests for materials
should be addressed to
E.E. (ernesto.estrada@
strath.ac.uk.)
SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 1

dutðÞ
dt ~{ X
d
DdLd
!
utðÞ,u0ðÞ~u0,ð2Þ
where L
d
captures the interactions between individuals separated by
dlinks in their social network, D
d
,1/d
a
where the parameter a
accounts for the strength of the PP pulling an individual into the
consensus.
We now compare the hypotheses about the random emergence of
good leaders—those who significantly reduce the time for reaching
consensus in a network—to those in which leaders emerge from the
most central individuals. Let us examine the emergence of leadership
from five centrality criteria: degree, eigenvector, closeness, between-
ness, and subgraph (see Supplementary Information equations (S1)–
(S5)). In general, we observe that the leaders emerging from the most
central individuals are better in leading the consensus than those
emerging randomly. However, when there is certain level of PP over
the actors, the situation changes dramatically (Fig. 1a, b). First, the
time to reach consensus significantly decreases to less than 20% of the
time needed when no PP exists. Second, a leader emerging randomly
in the network could be as good as one emerging from the most
central actors when PP exists in the system. Due to the recent results
about the role of low-degree nodes in controlling complex networks
19
we have also tested the role of PP over these potential drivers. Our
results show again that good leaders emerge regardless of their cent-
rality in the network when PP exists in the system (Supplementary
Information). In other words, under the appropriate PP any indi-
vidual in a social group could emerge as a good leader independently
of her position in the network. This result adds a new dimension to
the problem of network controllability
19–22
by demonstrating that PP
is a major driving force in determining how potential controllers can
emerge in the network independently of their centrality (Supplemen-
tary Fig. S1) and — in contrast with previous results
19,23,24
— of the
degree distribution of the network (Supplementary Fig. S2).
In roughly half of the 15 social networks studied (Supplementary
Information) we observe the following anomalous pattern. Leaders
randomly emerging in the network are better in leading the con-
sensus than some emerging from the most central individuals (see
Fig. 1c). This situation appears when the network has the leaders
distributed through diverse communities in the network. A com-
munity is a group of individuals who are more tightly connected
among themselves than with the other actors in the network
25
.
Actors in one of these communities reach consensus among them-
selves easily, but it is difficult to reach consensus between different
communities. Most central actors in such networks are frequently
located in a single community. When they emerge as leaders, they
Figure 1
|
Random and centrality-based emergence of leaders. The emergence of leaders is analyzed according to randomness (Rnd), betweenness
(BC), closeness (CC), degree (DC), eigenvector (EC), and subgraph (SC) centrality. The peer pressure is modeled by D
d
,d
a
, with aequal to 21.5 and
22.0. The third line corresponds to no peer pressure. (a) Communication network among workers in a sawmill. (b) Elite corporate directors.
(c) Friendship network of injected drug users in Colorado Springs. (d) Random network having communities.
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SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 2

drive consensus only in their community but not in the global net-
work. In contrast, when leaders emerge randomly, they more likely
emerge simultaneously in different communities, a situation that
favors global agreement in the network. Constructing a random
network with communities as illustrated in Fig. 1d corroborates this
hypothesis (Supplementary Tables S6 and S7). These results suggest
the necessity of considering community leaders in social networks as
effective mobilizers of actors throughout the network. We have
observed that the leaders emerging on the basis of their community
positions exhibit greater success in reaching consensus than those
randomly emerging in the network. However, when appropriate PP
exists, leaders who effectively reach consensus emerge regardless of
their position in their communities.
The leaders in a social group do not always exhibit a high level of
cohesiveness. We posit that the leaders’ capacity to lead the consensus
in a network depends on their divergence of opinions. A cohesive
group of leaders can more effectively lead the social group than
leaders with larger divergences among their opinions. To model lea-
der cohesiveness we introduce the divergence parameter =
L
,whichis
the circumradius of the regular polygon comprising all the leaders.
=
L
50 indicates a very cohesive group of leaders. We now examine
the influence of the leaders’ cohesiveness on consensus. Figure 2
Figure 2
|
Leaders’ cohesiveness and consensus. Analysis of the influence of leaders cohesiveness on the time to reach consensus in the
communication network among workers in the sawmill without (left plots) and with (right plots) PP. The leaders’ divergences used are: 0.0 (top), 0.1
(middle), and 0.2 (bottom). The time to reach consensus (in blue) relative to a total time of 1,500 units (Insets).
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SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 3

illustrates the results for the friendship network of workers in the
sawmill with either no PP (left plots) or with PP modeled by D
d
,1/
d
2
(right plots). The values of leader divergence range from 0.0 to 0.2.
The lack of leader cohesiveness significantly increases the time to
consensus when there is no PP. In fact, the time increases more than
33% when the divergence changes from 0.0 to 0.2 (it grows to 80.2%
for =
L
50.5, see Supplementary Figs. S3 and S4 and Supplementary
Tables S1, S3–S5). In addition, the cohesiveness of the group—mea-
sured by the standard deviation at consensus =
G
—is very poor for
large values of =
L
(=
G
5154.6, 183.6, and 226.9 for =
L
50.0, 0.1, and
0.2, respectively), which indicates highly heterogeneous group opi-
nions. However, when PP exists, the situation dramatically changes.
First, the time to consensus does not increase as drastically with the
decrease of leader cohesiveness. Second, group cohesiveness at the
consensus is very high even for the lowest leader cohesiveness (=
G
5
27.0, 35.4, and 33.0, for =
L
50.0, 0.1, and 0.2, respectively). In short,
when PP is absent, leader cohesiveness plays a fundamental role in
the time needed to reach consensus and in group cohesiveness at the
consensus. When PP is present, the time needed to reach consensus
and group cohesiveness are largely independent of the degree of
divergence in the leaders’ opinions, and the consensus is driven prim-
arily by the influence of the nearest neighbors and PP.
Diffusion of innovations and PP.Another area that has received
great research attention is the diffusion of innovations
26–29
. The
diffusion of innovations refers to the process through which new
ideas and practices spread within and between social groups. Here
we consider the hypothesis that PP plays a fundamental role in
innovation adoption or rejection. To test our hypothesis, we study
two datasets in which diffusion of innovations was followed for
different periods of time (Supplementary Information). The first
study analyzed the diffusion of a modern mathematic method
among the primary and secondary schools in Allegheny County
(Pennsylvania, USA). Results revealed that innovation diffused
through the friendship network of the superintendents of the
schools involved. The study was followed for a period of six years,
1958–1963. The second dataset represents the second phase of a
longitudinal study about how Brazilian farmers adopted the use of
hybrid seed corns, examining personal factors influencing farmers’
innovative behavior in agriculture. We consider here the social
network of friendship ties and the cumulative number of adopters
of the new technology in three different communities of the Brazilian
farmers study (Supplementary Fig. S5). The study was conducted
over the course of 20 years and we consider only the individuals in
the largest connected components of the networks.
Figure 3 depicts the number of actors that adopted the respective
innovations at different times. These values correspond to the num-
ber of adopters observed empirically in field studies. To simulate the
process of innovation adoption, we study the consensus dynamics
with equation (2), assuming D
d
,d
a
: no PP, moderate PP (26.0 #a
#25.0), high PP (24.0 #a#23.0) (see Supplementary
Information). The simulations follow perfect sigmoid curves, as
Fig. 3 illustrates. Observe that when there is no PP effect, the dif-
fusion curves predict slower rates of adoption than those empirically
observed. For example, the empirical evidence demonstrates that
50% of schools adopted the new math method in roughly three years,
whereas the simulation without PP predicts a period of four years of a
total of six years. In the case of the Brazilian farmers, the empirical
time for 50% of the farmers to adopt the innovation is roughly 12
years, whereas the simulation without PP predicts 16 years of a total
of 20 years. When the model uses strong PP, the diffusion curves
display very rapid adoption rates, which are far from the reality of the
empirical evidence in both cases. However, using a moderate PP
predicts very well the outputs of the empirical results in both studies.
These PP values are found by a reverse engineering method, but the
important message is that a certain PP level is necessary to describe
the empirical evidence on the diffusion of innovations in social
groups (see also Supplementary Information).
These results demonstrate that interpersonal communication
alone cannot sufficiently explain the process of innovation adoption
in a social group. The pressure exerted by the social group plays a
fundamental role in shaping this important social phenomenon. Our
model describes effectively PP’s role in these and other important
phenomena, consistent with our intuition and with the existing
empirical evidence.
Discussion
In this work, we have presented a methodology to address the prev-
iously unexplored influence of the combined action of direct and
indirect peer pressure on social group dynamics. The developed
model considers that the consensus dynamics is controlled not only
by the agreement between directly connected peers, but also by the
influence of those peers which are socially or culturally close to them.
The results obtained with this generalized consensus model highlight
the important role played by the indirect peer pressure on the
Figure 3
|
Diffusion of innovations under PP. (a) Adopters of a new mathematical method among US colleges in a period of 6 years. (b) Adopters of
the use of hybrid seed corns among Brazilian farmers for a period of 20 years. Experimental values are given as stars and the simulation with no
(broken red line), moderate (continuous blue line) and strong (dotted green line) PP are illustrated.
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SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 4

processes of consensus, emergence of leadership and diffusion of
innovations in social groups.
Consensus is known to be influenced by a small group of leaders
who guides the behavior of the whole network
13–18
. The role of these
drivers in the system controllability, and in particular their status or
position in the complex network, has received great importance
recently
19–24
. As expected the presence of these leaders reduces sig-
nificantly the time for consensus in the network. In terms of control-
ling the system we show here that appropriate levels of indirect peer
pressure allows that randomly emerging leaders could be as good as
those occupying special positions or centrality in the network.
We also explore the role of two factors that have been previously
ignored in the analysis of network controllability. The first is the role
played by the presence of tightly connected groups or communities
of nodes. The other is the cohesiveness of the leaders trying to drive
the consensus of the whole network. In both cases we show here that
if the level of indirect peer pressure is relatively weak, local leaders
and leaders with strong cohesiveness are the best in controlling the
network. However, as the indirect peer pressure increases the barriers
imposed by the communities and leader cohesiveness vanish, and the
networks are easily controlled even by leaders emerging from ran-
dom positions.
Another area in which we have found that the indirect peer pres-
sure plays a fundamental role is in the diffusion of innovations. In
this case we show, with the help of real-world data about the diffusion
of innovations in two different scenarios, that a moderate indirect
peer pressure is needed in order to reproduce the rates of diffusion of
these innovations independently of the social scenario in which they
take place.
Our results not only offer a new perspective for the analysis of
consensus in social groups, but also raise questions about the role of
indirect peer pressure in the controllability of social networks. Future
researches must explore how indirect peer pressure influences social
activities in networks with very different topologies. Other models,
apart from the consensus dynamics, can also be adapted to account
for indirect peer pressure, opening new avenues in the analysis of
these networked systems.
Methods
Consensus dynamics model.We consider a social group of nactors who will
accomplish a certain goal or reach an agreement. Every actor in the group is
represented by an element of the node set V5{1,…,n} of a network G5(V,E), in
which links (edges) E(fV|Vgrepresent the relationships (friendship, any form of
communication) among the actors. The set of neighbors of the actor iis denoted by
Ni~j[V:(i,j)[Efg.LetA~½aij[Rn|nand L(G)~½lij[Rn|nbe the adjacency
matrix and Laplacian matrix, respectively, associated with graph G. The Laplacian
matrix is defined as L5K2A, where Kis the diagonal matrix of node degrees of G
and Ais the adjacency matrix.
The information states of the actors evolve according to the single-integrator
dynamics given by
dui(t)
dt ~gi,i~1,...,n,and ui(0)~zi,ð3Þ
where ui[Ris the information state at time t,gi[Ris the information control input,
and zi[Ris the initial state of actor i, which is always considered to be selected at
random. A continuous time consensus algorithm is given by
gi~X
j[Ni
aij(uj(t){ui(t)), i~1,:::,n,ð4Þ
where a
ij
is the (i,j) entry of the adjacency matrix A. The information state of each
actor is driven toward those of her neighbors. Equations (3) and (4) describe the
collective dynamics of the social group and can be written in matrix form as
du(t)
dt ~{Lu,ð5Þ
where u5[u
1
,…,u
n
]
T
is the vector of the states of the actors in the system. The
consensus among the actors is achieved if, for all u
i
(t) and all i,j51,…,n,
ui(t){uj(t)
?0astR‘.
When the interaction among agents occurs at a discrete time, the information state
is updated using a difference equation, and a discrete time consensus algorithm is
then given by
ui(tz1)~ui(t)zeX
j
aij(uj(t){ui(t)), i~1,:::,n,ð6Þ
where a
ij
is as before and eis the time step. The information state of each actor is
updated as the weighted average of her current state and those of her neighbors.
Equation (6) is written in matrix form as
u(tz1)~Pu(t):ð7Þ
The matrix pis known as the Perron matrix, which is obtained as P5I2eL, for
0vev1
=
kmax
,
where k
max
is the maximum of the degrees of the nodes of G. The
entries of the Perron matrix satisfy the property P
j
pij~1 with p
ij
$0, mi,j, and hence,
it is a valid transition matrix
9
.
Consensus with leaders–followers.We consider that there exist one or multiple
leaders who guide the entire group to the consensus through the effect produced by
the rest of the group, which follows them
30
. In a leaders–followers structure with a
single leader, actors attempt to reach an agreement that is biased to the state of the
leader, whereas in the case of multiple (stationary) leaders, all followers converge to
the convex hull formed by the leaders’ states.
An actor is called a stationary leader if her opinion is available for the other actors
but is not modified during the process. Then, the set of all actors can be divided into
two subgroups: leaders and followers. As a result, the vector of the states of all
actors can also be divided into two parts: the states of leaders, u
l
, and the states of
followers, u
f
.
For a system with multiple stationary leaders, all the nodes can be labeled such that
the first n
f
represents the followers and the remaining n
l
represent the leaders. The
total number of actors in the system is n5n
f
1n
l
, such that the Laplacian matrix
associated with the social network Gis partitioned as
L(G)~
Lflfl
lT
fl Ll
"#
,ð8Þ
where Lf[Rnf|nf,Ll[Rnl|nl, and 1fl[Rnfxni.
Because the leaders are stationary, their dynamics are given by u
i
(t)50, i5n
f
1
1,…,n. Then, the dynamics of the system are expressed by
:
uf
:
ul

~{Lpu~{ Lflfl
00

uf
ul

:ð9Þ
The discrete version of equation (9) is given by
u(tz1)~(In{eLp)u(t), ð10Þ
where u(t)~u1(t),:::,un(t)½
T,I
n
is the identity matrix of size n3n,andL
p
is the
Laplacian matrix of network G, with each entry of the jth row equal to zero for j5n
f
1
1,…,n.
Modeling peer pressure.The consensus dynamic modeling assumes that the actors
only interact with their directly connected neighbors to cooperatively achieve an
agreement in the system
31
. However, in many real-world situations, the actors are
exposed not only to their closest contacts but also to individuals who are socio-
culturally close to them despite not being directly connected. For instance, this
situation appears in actors’ attitudes toward copying others. The predisposition of an
actor to copy a behavior depends not only on her friends’ adoption of such behavior
but also on other, socio-culturally close people having a positive predisposition to that
behavior. For instance, adolescents adopt ‘‘binge drinking’’ not only by copying their
mates but also by observing similar behavior among others of a similar age, education,
and social class. Then, we argue that this socio-cultural distance can be captured in a
model by considering the shortest path distance between two actors in their social
group. The shortest path distance is the number of steps in the shortest path
connecting the two actors. The influence that an actor receives/produces from/for
others in her social network, i.e., peer pressure, decays as a function of this socio-
cultural distance, which separates the two actors
32
.
Peer pressure can then be modeled by considering the generalized Laplacian
matrix
33
. Consequently, the consensus dynamics model of equation (6) can be written
as
u(tz1)~In{eX
d
DdLd
! !
u(t), ð11Þ
where P
d
DdLdinvolves the d-Laplacian matrices and the coefficients D
d
indicate the
strength of the interactions at distance d#d
max
(G), with d
max
(G) being the maximum
distance between two nodes or the diameter of graph G.Thed-Laplacian matrix is
defined as
33
Ld(i,j)~
{1
ud(i)
0
dij~d
i~j
otherwise
8
>
<
>
:
,ð12Þ
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SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 5

where the expression u
d
(i) is the d-path degree of node idefined as the number of non-
redundant shortest paths of length dhaving ias an endpoint.
The coefficients D
d
should account for the decay in peer pressure for the socio-
cultural distance between the actors of D
d
,f(d)
21
, where f(d) represents a function of
distance d. In this study, we consider three different decay behaviors described by the
following equations:
1) Power-law decay: D
d
5d
2a
,
2) Exponential decay: D
d
5e
2bd
, and
3) Social interactions: D
d
5dd
d21
,
where a,b, and dare parameters to be adjusted to consider the different strengths of
peer pressure.
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Acknowledgements
EE thanks the New Professor’s Fund, University of Strathclyde, and EPSRC grant EP/
I016058/1, MOLTEN: Mathematics Of Large Technological Evolving Networks for partial
financial support as well as QuanTM, Emory University for warm hospitality during
September-December 2013.
Author contributions
E.V.E. collected data, performed research and analyzed data. E.E. designed and performed
research, analyzed data and wrote the paper. Both authors discussed the results and
commented on the manuscript.
Additional information
Supplementary information accompanies this paper at http://www.nature.com/
scientificreports
Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Estrada, E. & Vargas-Estrada, E. How Peer Pressure Shapes
Consensus, Leadership, and Innovations in Social Groups. Sci. Rep. 3, 2905; DOI:10.1038/
srep02905 (2013).
This work is licensed under a Creative Commons Attribution 3.0 Unported license.
To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0
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SCIENTIFIC REPORTS | 3 : 2905 | DOI: 10.1038/srep02905 6