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ORIGINAL RESEARCH
published: 13 April 2017
doi: 10.3389/fpsyg.2017.00583
Edited by:
José Carlos Núñez,
Universidad de Oviedo, Spain
Reviewed by:
Ana Miranda,
Universitat de València, Spain
Mercedes Inda-Caro,
University of Oviedo, Spain
*Correspondence:
Xiaozhe Peng
pengxz@szu.edu.cn
Specialty section:
This article was submitted to
Educational Psychology,
a section of the journal
Frontiers in Psychology
Received: 22 December 2016
Accepted: 29 March 2017
Published: 13 April 2017
Citation:
Jiao C, Wang T, Liu J, Wu H, Cui F
and Peng X (2017) Using Exponential
Random Graph Models to Analyze
the Character of Peer Relationship
Networks and Their Effects on
the Subjective Well-being
of Adolescents.
Front. Psychol. 8:583.
doi: 10.3389/fpsyg.2017.00583
Using Exponential Random Graph
Models to Analyze the Character of
Peer Relationship Networks and
Their Effects on the Subjective
Well-being of Adolescents
Can Jiao1, Ting Wang1, Jianxin Liu2, Huanjie Wu1, Fang Cui1and Xiaozhe Peng1*
1College of Psychology and Sociology, Shenzhen University, Shenzhen, China, 2The Faculty of Humanities and Social
Sciences, City University of Macau, Macau, Macau
The influences of peer relationships on adolescent subjective well-being were
investigated within the framework of social network analysis, using exponential random
graph models as a methodological tool. The participants in the study were 1,279
students (678 boys and 601 girls) from nine junior middle schools in Shenzhen,
China. The initial stage of the research used a peer nomination questionnaire and
a subjective well-being scale (used in previous studies) to collect data on the peer
relationship networks and the subjective well-being of the students. Exponential
random graph models were then used to explore the relationships between students
with the aim of clarifying the character of the peer relationship networks and the
influence of peer relationships on subjective well being. The results showed that all
the adolescent peer relationship networks in our investigation had positive reciprocal
effects, positive transitivity effects and negative expansiveness effects. However, none
of the relationship networks had obvious receiver effects or leaders. The adolescents in
partial peer relationship networks presented similar levels of subjective well-being on
three dimensions (satisfaction with life, positive affects and negative affects) though
not all network friends presented these similarities. The study shows that peer
networks can affect an individual’s subjective well-being. However, whether similarities
among adolescents are the result of social influences or social choices needs further
exploration, including longitudinal studies that investigate the potential processes of
subjective well-being similarities among adolescents.
Keywords: peer relationships, subjective well-being, exponential random graph models, social network analysis
INTRODUCTION
In adolescence, and with increasing physical and cognitive development, a child’s psychological
awareness begins to resemble that of an adult. Adolescents spend more and more time with their
contemporaries, especially their peers. A peer relationship is the relationship of a common activity
and mutual cooperation among children in the same or similar age group, but mainly refers to a
relationship between peers or individuals at a similar level of psychological development, which
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Jiao et al. Peer Relationship Networks and Well-being
is built and developed through communication (Yang, 2008).
And peer relationships are the main sources of social support
for adolescents and the main driving force in enhancing
an adolescent’s self-concept and well-being (Furman and
Buhrmester, 1992). Good peer relationships cannot only promote
the development of an adolescent’s social cognition and social
skills, but also improve their physical and mental health and
enhance their subjective well-being. Negative peer relationships
not only hinder an adolescent’s academic performance and
personality development, but might lead to emotional problems
such as anxiety, depression, and mental illness (Chen and
Zhou, 2007). Adolescent school-based networks are important
for developing these peer relationships (Haas et al., 2010).
Researchers have argued that peer relationships may promote
the development of an adolescent’s self-identity through social
comparison and symbolic evaluation (Brown and Lohr, 1987).
Adolescents see their peer groups as typical models for their
views and behaviors, and use the identity of the peer groups to
regulate their own behavior. In this way, adolescents develop
similarities to others in their groups. According to social
communication theory, in a social network, a person’s emotions,
opinions, and behaviors are like an epidemic and can spread
by social interaction (Kiuru et al., 2012). And according to
similarity theory, individual similarities in values, characteristics
and behaviors increase predictably, which enables them to share
the same feelings and develop a sense of belonging, and makes
them easy to get along with (Batool and Malik, 2010).
The developmental literature has long emphasized the strong
role of peer groups in determining our inclination toward social
behaviors (Brown, 2004). Many researchers have carried out a
social network analysis of adolescents especially in relation to
their social behaviors. Researchers have studied the relationship
between peer-related physical activity social networks (Voorhees
et al., 2005) and peer aggression (Low et al., 2013); health (Haas
et al., 2010); obesity (Marqués et al., 2015); smoking (Ennett and
Bauman, 1993;Lakon et al., 2013;Lambert, 2014); substance use
(Ennett et al., 2006); and drinking (Mundt, 2011;Deutsch et al.,
2014). However, whether they like their lives is rarely explored
through this method. Our investigation aims to satisfy curiosity
about a child’s inner state.
Subjective well-being is a personal evaluation of an individual’s
overall living conditions. In other words, subjective well-being
is how much a person likes his or her life (Veenhoven, 2013)
and in colloquial terms is sometimes labeled “happiness.” This
is a multidisciplinary research field. The social psychologist
Diener, one of the few internationally recognized academic
authorities in this area, pointed out that the subjective well-
being of the individual produces a positive attitude and positive
feelings by comparing the actual state of life with ideal life.
Subjective well-being is characterized by subjectivity, initiative
and comprehensiveness (Diener, 2000). The evaluation criteria
are made by an individual’s own standards without reference
to any external evaluation criteria. The evaluation criteria have
the characteristics of subjectivity, stability and integrity. External
factors such as gender, age, income and life events, as well
as internal factors such as personality, self-esteem, self-efficacy,
and self-concept, all have an impact on subjective well-being
(Diener et al., 1995, 1999;McCullough et al., 2000;Schimmack
and Diener, 2003;Ferrer-i-Carbonell, 2005;Gutiérrez et al.,
2005;Gilman and Huebner, 2006;Karademas, 2006;Zhou et al.,
2012;Ulloa et al., 2013). Since subjective well-being is the main
aspect of living quality and has a close relationship with mental
health, studies of subjective well-being have been highly valued
(Diener, 1984). Meanwhile, various studies have shown that
peer/interpersonal relationships and subjective well-being are
related to a certain degree (Hussong, 2000;Liu and Gong, 2000;
Dai, 2005;Demir and Weitekamp, 2007;Litwin and Shiovitz-
Ezra, 2010). In addition, the former has a strong predictive ability
on the latter (Dai, 2005;Chen, 2006;Demir et al., 2007;Xia, 2007;
Wu, 2008;Zhang and Zhu, 2012).
The period known as adolescence is a critical period for
psychological development. Psychological symptoms such as low
subjective well-being have been recognized as being common
(La Greca and Lopez, 1998). These obstacles can last a long
time, often beginning in adolescence and extending to adulthood
(Devine et al., 1994), and are likely to become risk factors for adult
mental disorders (Devine et al., 1994;Aalto-Setälä et al., 2002).
Therefore, studies of the relationship between adolescent peer
relationships and subjective well-being have a certain practical
significance.
Although there has been much fruitful research on the
relationship between peer relationships and subjective well-
being, as outlined above, there is a common problem with
these studies; that is, their methodological premise in applying
statistical analysis. Specifically, researchers have presented
relational data in a simplified form as attribute data. Peer
relationships in essence are relational attributes and reflect
interpersonal relationships as well as interdependencies between
individuals. However, if peer relationships are regarded as
attribute data, then the methods used to conduct statistical
analysis on the basis of this assumption would be those that are
relevant to attribute data only, such as correlation analysis and
regression analysis. The statistic type is clearly against the premise
of these methods in this context, and the conclusions drawn from
such research may not be valid (Jiao et al., 2014).
Social network analysis is an effective method of solving this
problem. As a method for dealing with relational data, social
network analysis fully considers the impacts of the social situation
on individual behaviors and focuses on the relationship between
individuals. The social context is constituted by the relationship
between individuals. Relationships constitute a network. In
social networks, the points (or nodes) represent the units such
as individuals, families, organizations, and social groups. The
edges represent whether a relationship between points exists
and its strength. By network analysis, the relationships between
individuals can be described and measured. Additionally, the
resources and information within the relationship can also be
described and measured. Furthermore, a model can be built for
these relationships, which can be used to study the interactions
between these relationships and individual behaviors (Liu, 2004).
Exponential Random Graph Models (ERGMs) are a method
of social network analysis for building complex social network
structures (Robins et al., 2007). The model assumes that the
emergence of a relationship might be influenced by the presence
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or absence of other relationships and/or individual attributes
(Robins et al., 2007). Compared with other social network
analysis models, an ERGM focuses on the interaction between
the structures of a relationship network (such as reciprocity,
transitivity, and popularity) and individual attributes (such
as gender and education level). In order to understand the
inner mechanism of peer relationships and subjective well-being
more clearly, this study used an ERGM model to explore the
relationships between network structures in adolescent peer
relationships and the individual’s subjective well-being.
Our purpose is to clarify the character of a peer relationship
network and the mechanism of the peer relationship influence on
subjective well-being. Studies have shown that there are effects
of reciprocal structure, transitivity structure, popularity structure
and expansiveness structure in peer networks (Schaefer et al.,
2010;Daniel et al., 2013). Thus, this study assumes the following.
•Hypothesis 1: There are significant reciprocal structure effects
in adolescent peer networks.
•Hypothesis 2: There are significant transitivity structure effects
in adolescent peer networks.
•Hypothesis 3: There are significant popularity structure effects
in adolescent peer networks.
•Hypothesis 4: There are significant expansiveness structure
effects in adolescent peer networks.
Adolescents and their friends might have similarities in a
variety of social, behavioral, and psychological characteristics
(Prinstein and Dodge, 2008). Numerous studies have found that
adolescent friends have similarities in externalizing problems
such as attacks (Sijtsema et al., 2010); internalizing problems such
as depression (Van Zalk et al., 2010); health risk behaviors such as
smoking (Mercken et al., 2010); “happy” emotions (Fowler and
Christakis, 2008); and prosocial behaviors (Barry and Wentzel,
2006). Thus, this study also assumes the following.
•Hypothesis 5: Adolescents show similar levels among peers in
each dimension of subjective well-being (satisfaction with life,
positive affects and negative affects).
Receiver effects build relationships between individual
attribute variables and popularity. Popularity refers to the
nominated numbers that individuals receive from other
individuals in their network class. More nominated numbers
indicate that the individual is more popular. Studies have shown
that popular individuals prefer interactions with their peers and
have more positive attitudes. They are seldom isolated, refused
or repelled by peers. They experience more positive affects and
have fewer negative experiences (LaFontana and Cillessen, 2002;
Wen, 2011). Thus, this study further assumes:
•Hypothesis 6: There is a significant positive receiver effect
in the positive affective dimension of subjective well-being;
namely, individuals who have higher levels of positive affect
will have more friends and be more popular, and vice versa.
•Hypothesis 7: There is a significant negative receiver effect
in the negative affective dimension of subjective well-being,
namely, individuals who have higher levels of negative affect
will have fewer friends and be less popular, and vice versa.
MATERIALS AND METHODS
Participants
Nine junior middle schools were selected at random from
Shenzhen, China. Twenty-nine classes were then selected at
random from these nine schools (mean classes 3.22, standard
deviation 0.44). There were 1,497 students altogether. All
students were given questionnaires which were handled as
follows. First, the questionnaires that did not meet requirements
were excluded from the study. These included cases of: (a)
multiple answers (a participant provided more than one option
for an item); (b) no answers (a participant failed to provide
an option for an item); and (c) regular answers (a participant
provided the same option or a regular array of options for a series
of items). Secondly, the subjects that obtained subjective well-
being (SWB) scores exceeding three standard deviations were
removed from the study, by which the influence of extreme values
was eliminated. Thirdly, the subjects who were at the edge or on
the periphery of a class network were excluded from the study as
these subjects did not associate with other members in their class
network and became isolated points. Finally, there were 1,279
subjects remaining, including 678 boys and 601 girls. The study
was conducted in accordance with the Declaration of Helsinki
and was approved by the Academic Committee of the College of
Psychology and Sociology, Shenzhen University. All participants
(or the parents of participants who were under the age of 16)
provided written informed consent of participation in the study.
Research Tools
The Questionnaire: Peer Nomination
The questions in the questionnaire were designed as follows. As
regards peer nomination, participants were asked: “Please write
the name of your best friends in the class (at least three)” (Lubbers
and Snijders, 2007). In measuring peer relationships, if a member
nominated another member it meant there was a relationship
between them, which was then recorded as 1 in the relationship
matrix; otherwise it was recorded as zero.
Subjective Well-being Scale
The SWB scale was built from two constituents: the Satisfaction
with Life Scale (SWLS) and the Affect Balance Scale (ABS). The
SWLS scale was compiled by Diener et al. (1985). It has been
used to measure the cognitive dimension of subjective well-being
(Zhu et al., 2012). The scale contains five items, each of which
employs 7 score grades from “strongly disagree” to “strongly
agree.” Strongly disagree is recorded with a score of 1, while
strongly agree is recorded with a score of 7, the scores increasing
in sequence. Higher scores indicate a higher satisfaction with
life (SWL), whilst a low score means a low SWL. The Alpha
coefficient of the scale is 0.78. The split-half reliability is 0.70. The
fit index of confirmatory factor analysis is, respectively, the ratio
of (chi-square/degrees of freedom) that is 6.71, RMSEA =0.071,
GFI =0.97, and CFI =0.96, which shows that the structure of
the scale has a good validity (Xiong and Xu, 2009). The internal
homogeneity coefficient αof the measurement is 0.77 and meets
metrological standards.
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The ABS scale was compiled by Bradburn and Noll (1969).
The reliability estimates of Bradburn’s original study showed
acceptable reliability coefficients (Bradburn and Noll, 1969). ABS
has been used to measure the affective dimension of subjective
well-being such as positive affect and negative affect (Yue et al.,
2006). Our study uses the Chinese version of the Mental Health
Assessment Scale, which was collected by Wang et al. (1999). The
scale contains 10 items, which are used to describe a person’s
feelings in the past few weeks. Among them, five items describe
positive affects and five items describe negative effects. The
answer “Yes” is recorded with a score of 1; the answer “No”
with a score of 0. A higher score indicates a higher frequency
of experiencing positive or negative effects (Zhang et al., 2007).
In the scale, the retest reliability of positive affects and negative
effects are both above 0.80; additionally, the correlation values
between these two subscales is less than 0.10, which indicates
the validity of the scale and its reliability (Ou et al., 2009). In
the measurement, the internal homogeneity coefficient αof the
positive affective dimension is 0.60, and the internal homogeneity
coefficient αof the negative affective dimension is 0.63. The ABS
has been translated into many languages including Cantonese,
Vietnamese, and Laotian, and a cultural equivalence has been
found (Devins et al., 1997). The original version of ABS and the
Chinese version have both been shown to be reliable, and the
measurement meets metrological standards.
Exponential Random Graph Models
Exponential Random Graph Models (ERGMs), also known as P∗
models, were proposed by Frank and Strauss (1986) to explain
a series of statistical models in social networks. An ERGM can
infer how network relationships are formed. The model does not
focus on predicting individual outcome variables in the network
but focuses on the formation of deductive relations. It takes
the network as a graph constituted by nodes (actors) and edges
(relationships). It inspects the probability distribution of the set
of all graphs with a fixed number of points (or nodes) (Ma et al.,
2011). The model assumes that the network was generated at
random. The probability of the observed graph depends on the
number of occurrences of various structures in the model (Wang
et al., 2009). Its basic form is as follows:
Pr(X=x)=exp{θ0z(x)}
k(θ) =exp{θ1z1(x)+... +θrzr(x)}
k(θ)
In the above formula, the meaning of each expression is
as follows (Ma et al., 2011). Pr(X=x) is the probability
of some actual relationship between individuals. θis a series
of network structure parameter vectors, including reciprocal
structure parameters, transitivity structure parameters and star-
shaped structure parameters. z(x) is a series of network
statistic vectors. The series contains not only particular network
structure parameters (such as reciprocal parameters, transitivity
parameters and star-shaped structure parameters), but also
attribute parameters of the actors in the network (such as grade,
gender, and attitudes). kis a constant and guarantees that the
probability distribution is a normal distribution.
The dependence assumption is the basic theoretical
assumption of ERGMs. “My friend’s friend is my friend” is
a typical dependence assumption in a social network. The
assumption is that the existence of some relationships will
produce, maintain or destroy other relationships (Robins, 2011).
If a relationship does not rely on other relationships, it can be said
that the existence of some relationships will affect the existence
of other relationships to some extent, and that they have no
incentive to form the structure. The dependence assumption is
presented by specific structures which reflect how relationships
are generated in the network. The network structure is a small
network mode, which is constituted by points (or nodes) and
the relationships among points (Brughmans et al., 2014). Table 1
presents the common structural parameters of ERGMs for a
directed network.
The reciprocity assumption is that if actor A selects actor
B, then actor B will also select actor A. Reciprocity is a basic
characteristic of social life and has been proved in peer groups
(Snyder et al., 1996;Strayer and Santos, 1996). Popularity means
that an individual has a higher “in-degree” compared with other
individuals in the network (“in-degree” counts the total number
of actors who select a particular individual). Higher in-degrees
show that some people are more attractive than others, and the
popularity assumption is that an individual will select the one
actor that others have all selected to make friends (Barabási
and Albert, 1999;Gould, 2002). Transitivity measures triangular
closure trends in the network; namely, “my friend’s friend is also
my friend.” If transitivity appears in peer groups, the reason
might be that more and more individuals are willing to share
each other’s friends, or might be due to a psychological need for
balance (Daniel et al., 2013).
TABLE 1 | The common structural parameters of exponential random
graph models for a directed network.
Parameter Structure of point Description of structure
Reciprocity Two points select each other as friends,
which creates a reciprocal relationship.
Transitivity My (black point) friend’s (white point)
friend (white point) is also my friend.
Popularity The points on both sides all select the
central point as a friend. The central
point is the most popular one.
Expansiveness The central point selects the points on
both sides as friends. The central point
is actively making friends with other
points.
Receiver The relationship is sent from any point
(white point) to some specific attribute
point (black point).
Sender The relationship is sent from some
specific attribute point (black point) to
any point (white point).
The point represents an “actor”; the directed line represents a “directed relationship
from sender to receiver.”
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Based on various dependence assumptions, ERGMs contain
several kinds of models. The simplest one is the Bernoulli Graph
Model, which assumes that the edges and binary relations are
independent in all networks. Therefore, there are only edges and
no other structures in a Bernoulli Graph Model. A Dyadic Model
assumes that the dyadic relations in a directed network graph are
all independent. If the relationship between actor A and actor B is
irrelevant to the relationship between actor B and actor C, there
are two structures in the model: edges and reciprocated edges
(Wang et al., 2009). A reciprocated edge means that if actor A
selects actor B, then actor B will also select actor A.
However, the two models above are unrealistic, both from a
theoretical standpoint and from practical experience. Therefore,
Markov independence was introduced by Frank and Strauss
(1986); this assumes that the relationship between actor A
and actor B depends on any other relationships related
to A or B. Under this condition, if there is a common
actor in two relationships, then the relationships should be
considered as conditionally independent (Robins et al., 2007).
The Markov Random Graph Model was proposed based on
this Markov dependence assumption. A Markov Random Graph
Model contains not only the structural parameters of edges
and reciprocated edges, but also various “two-star” structural
parameters (where two actors both have relationships with a
third actor). The structural parameters of “two-out-star” (an
actor simultaneously selects two other actors as friends) are
related by expansiveness. The structural parameters of “two-in-
star” (an actor is selected by two other actors simultaneously
as a friend) involves popularity. The simplest Markov Random
Graph Model is a two-star model, which has only edges and
two-stars within its structure. Researchers subsequently noted
the importance of transitivity and cyclicity, and brought further
structural parameters into the Markov Random Graph Model
(Newman, 2003). The expanded model contains higher star-
shaped statistics such as three-star structures (where three actors
all have relationships with a fourth actor).
However, this model can only fit the data in quite limited
circumstances. Additionally, many studies have shown that
the model will have gradual degradation problems when it is
estimated and simulated. A Markov Random Graph Model was
therefore not considered to be a good model for observing social
networks (Pattison et al., 2007). Researchers then introduced
the concept of partial conditional dependence and proposed a
Realization-Dependent Model (e.g., Snijders et al., 2006). This
model assumes that if there is a relationship between two actors,
then it can be regarded as partially conditionally dependent.
The model has three new statistics: namely, alternating k-star
(k actors all have a relationship at the same time with actor
k+1); alternating k-triangles (two actors with a relationship
build triangular relationships with k actors); and alternating
independent two-paths (two independent actors build two-path
relationships with multiple third party actors). The convergence
of the model is effectively improved with these additions.
In summary, the Bernoulli assumption is unsuitable for real
network data. Although the Markov independent assumption
broadens the network structure, the model might degrade for
smaller networks. Partial conditional dependence assumptions
enable us to build network aggregation effects and are closer
to real social networks. Consequently, the realization-dependent
model has been widely applied by scholars.
The common parametric estimation methods of ERGMs are
the Maximum Pseudo-Likelihood Estimation (MPLE) method
and the Markov chain Monte Carlo maximum likelihood
estimation (MCMC MLE) method. The MPLE method transfers
the model into logit form, and then applies logistic regression
techniques to conduct likelihood fit tests. The core of MCMC
MLE is designed to simulate random graph distribution from a
set of parameter values. It adjusts parameter values by comparing
the distributions of corresponding random graphs and observed
graphs, and then repeats the process until the estimated value
becomes stable. Studies have shown that the MCMC MLE
method works better than the MLE method, especially when the
network has a strong dyadic dependence.
Statistical Analyses
Statnet’s ERGM R software package was used to conduct
statistical analysis. The study applied MCMC MLE to conduct
parameter estimations. First, peer relationship networks in each
class were built according to the measurement of peer networks
(Lubbers and Snijders, 2007). They were then stored in relational
data files by matrix form. To build a relationship network,
each student in the class was regarded as a network actor;
the connections between them formed the relationships in the
network. The relationship network in each class was composed
of a square matrix, in which the rows and columns were all
students in the network. The elements/data in the square matrix
represented whether there were connections between students.
If a student nominated another student, this indicated they had
a connection which was recorded as 1; otherwise, it would be
recorded as 0. The square matrix was not symmetrical; that
is, although A nominated B, B might not nominate A. The
attribute data files for each network were then built according
to demography variables. The data files were in SPSS format.
A list of data corresponded to an attribute variable. Having built
an attribute data file, all individual attribute variables needed to
be standardized in order to compare the data. For the initial
ERGM model, the effects of each attribute variable were assessed
separately. Finally, the three dimensions of subjective well-being
were brought into the model to build the final ERGM, and the
effects of multiple attribute variables assessed simultaneously.
In order to control structure effects when inspecting attribute
variables effects, the model contained both attribute variables
effects and structure effects. To further explore the mode
of peer relationships in the network, the study incorporated
four common structure effects: namely, reciprocal structure,
transitivity structure, popularity structure and expansiveness
structure. With regard to attribute variables effects, in order
to test Hypothesis 5 the study considered differential effects.
Differential effects were based on the absolute difference in
some attribute variables between individuals who had relations
with each other. If the estimated result of the differential effects
parameter was negative and had statistical significance, it showed
that, under the invariable condition of other effects in the
model, individuals with relations tended to have similarities
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in their attribute variables. In order to test Hypotheses 6 and
7, the study also estimated receiver effect parameters, which
were based on the interactions between attribute variables and
network structures. The study also analyzed the relationship
between subjective well-being and popularity structure. If
there was a positive correlation between them, it indicated
that an individual with high scores in SWB would be more
popular.
STATISTICAL RESULTS
Model Fitting Degree
The t-ratio is defined as the estimate of a parameter divided
by its standard error, with reference to a standard normal null
distribution (Snijders, 2001), and it is often applied to balance the
fitting degree of each parameter, which is calculated by taking
the observed values minus the sample mean, then dividing by
the standard error. Table 2 shows the estimated t-ratio of each
parameter in our ERGMs, which were based on the relationship
networks in each class. Every parameter of the ERGM in the 15
networks was between −1 and 1, which indicates that the model
built from the 15 classes was an acceptable fit that reflected the
features of the network.
Structure Effects
The estimated values and standard errors of network structure
effects are presented in Table 3. The results show the following:
(1) Reciprocal effect: The 15 networks all had obvious reciprocal
effects, which meant individuals tended to select each other as
friends.
(2) Transitivity effect: The transitivity parameters of the 15
relationship networks were all greater than 0.05 and had
statistical significance, which meant a friend’s friend tended to
be a friend.
(3) Popularity effect: The popularity effects in most classes
were not obvious, which meant the in-degree (the total
number of actors selecting an individual) of individuals in
the class networks had little difference. The distribution of
relationships was average. Among them, the parameters of
popularity effects in classes D, G, L, and M were negative
and had statistical significance, which showed that the actual
appearance probability of popularity structure was lower than
that of a random level in the four networks.
(4) Expansiveness effect: All classes presented obvious negative
expansiveness effects, which showed that an individual’s social
circle was stationary in the network; i.e., an individual would
not take the initiative to make friends with people outside the
circle.
Differential Effects
Absolute differential effect parameters were applied to inspect the
assumption “Adolescent peers have similarities in subjective well-
being.” If the estimated results were negative and had statistical
significance, then they supported the assumption. The estimated
results of differential effect parameters for the initial model (the
effects of each attribute variable were separately assessed) and
the final model (the effects of multiple attribute variables were
simultaneously assessed) are shown in Table 4. The final model
shows adolescent friends had some obvious similarity tendencies
in each dimension of subjective well-being.
In the initial model which only considered satisfaction with
life, the differential effect parameters of eight networks A, B, E, G,
H, K, L, and M were negative and had statistical significances.
Among them, when considering the three dimensions of
subjective well-being in the final model, the differential effects of
five classes E, G, H, K, and L were negative and had statistical
significances, which shows that there was a similar satisfaction
with life among friends in five networks. A, B, and M did
not have statistical significances in the final model, although
the initial model showed a similar satisfaction with life among
friends in the three networks. The observation that they did not
have similarities when affected by other variables in the final
model shows that individuals in the networks might not form
friendships based on the similarity of satisfaction with life but
on other correlated variables. The differential effects in the other
seven networks were not obvious.
For positive affective dimensions, only the differential effect
parameters of eight networks A, B, E, G, K, L, M, and
O in the initial model were negative and had statistical
significances. Among them, when simultaneously considering
three dimensions of subjective well-being in the final model, the
differential effects of five classes A, B, E, L, and M were also
negative and had statistical significances, which showed that there
were similar positive affects among friends in five networks. G,
K, and O did not have statistical significances in the final model
although, in the initial model, there were similar positive affective
levels among friends in these three networks. The observation
that they did not have similarities when affected by other variables
in the final model shows that individuals in these networks might
not form friendships based on the similarity of positive affects but
on other correlated variables. The differential effects in the other
seven networks were not significant.
For negative affective dimensions, the differential effect
parameters of seven networks B, E, H, K, L, M, and N in the initial
model were negative and had statistical significances. Among
them, when simultaneously considering three dimensions of
subjective well-being in the final model, the differential effects
of all seven classes B, E, H, K, L, M, and N were also negative
and had statistical significances, which shows that there were
similar negative affects among friends in these seven networks.
The differential effects in the other eight networks were not
significant.
Receiver Effects
Receiver effects are based on the relationship between subjective
well-being and the popularity of individual attribute variables.
The results of receiver effects in each relationship network
are shown in Table 5. Only life satisfaction and popularity
in class D and class M had significantly positive correlations
(r=0.39, p<0.05; r=0.41, p<0.05), which showed that
an individual with higher satisfaction with life would be more
popular.
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TABLE 2 | The estimated t-ratio of each parameter in the exponential random graph model.
Class Reciprocity Popularity Expansiveness Transitivity Life Satisfaction Positive Affection Negative Affection
Class A 0.47 0.04 −0.44 −0.11 0.08 −0.02 0.20
Class B 0.95 0.05 0.05 0.60 −0.07 −0.24 0.08
Class C 0.58 −0.18 −0.25 0.04 −0.13 −0.05 −0.24
Class D 0.18 −0.33 −0.55 −0.55 −0.21 0.02 −0.08
Class E 0.26 −0.38 −0.28 −0.30 0.11 0.02 0.10
Class F 0.54 −0.16 −0.33 −0.01 −0.38 −0.09 0.03
Class G 0.70 −0.01 −0.04 0.15 −0.33 −0.17 0.16
Class H 0.92 −0.23 −0.41 0.17 −0.20 −0.06 −0.10
Class I 0.76 −0.33 −0.65 −0.23 −0.10 −0.25 −0.19
Class J 0.98 0.00 −0.08 0.20 −0.02 0.15 0.17
Class K 0.36 −0.14 −0.49 −0.31 −0.42 −0.04 0.03
Class L 0.89 −0.16 −0.16 0.09 −0.22 −0.18 −0.17
Class M 0.45 −0.33 −0.42 −0.20 −0.03 −0.08 −0.18
Class N 1.00 −0.04 −0.25 0.33 −0.13 −0.15 −0.02
Class O 0.79 0.02 −0.13 0.09 −0.13 −0.06 −0.01
TABLE 3 | The estimated values (standard errors) of network structure
effects.
Class Reciprocity Transitivity Popularity Expansiveness
Class A 4.14 (0.27)∗∗ 2.04 (0.15)∗∗ 0.02 (0.03)−2.14 (0.19)∗∗
Class B 4.07 (0.20)∗∗ 1.46 (0.10)∗∗ −0.03 (0.02)−0.67 (0.06)∗∗
Class C 3.31 (0.24)∗∗ 1.02 (0.16)∗∗ −0.04 (0.02)−0.43 (0.06)∗∗
Class D 3.73 (0.33)∗∗ 1.41 (0.14)∗∗ −0.11 (0.06)∗−0.85 (0.13)∗∗
Class E 3.05 (0.30)∗∗ 1.14 (0.13)∗∗ −0.03 (0.03)−0.47 (0.08)∗∗
Class F 3.42 (0.30)∗∗ 1.29 (0.14)∗∗ 0.02 (0.01)−0.76 (0.09)∗∗
Class G 3.31 (0.28)∗∗ 1.29 (0.11)∗∗ −0.08 (0.04)∗−0.47 (0.07)∗∗
Class H 3.12 (0.36)∗∗ 0.78 (0.38)∗−0.24 (0.13)−1.01 (0.26)∗∗
Class I 3.75 (0.33)∗∗ 1.29 (0.22)∗∗ −0.01(0.04)−1.18 (0.18)∗∗
Class J 3.50 (0.31)∗∗ 1.37 (0.21)∗∗ 0.02 (0.02)−1.31 (0.15)∗∗
Class K 2.74 (0.25)∗∗ 1.09 (0.07)∗∗ −0.01 (0.01)−0.30 (0.03)∗∗
Class L 3.13 (0.33)∗∗ 1.04 (0.17)∗∗ −0.24 (0.08)∗∗ −0.18 (0.08)∗
Class M 2.84 (0.26)∗∗ 0.85 (0.13)∗∗ −0.11 (0.03)∗∗ −0.12 (0.03)∗∗
Class N 3.56 (0.28)∗∗ 1.90 (0.18)∗∗ −0.03 (0.05)−1.92 (0.21)∗∗
Class O 3.45 (0.33)∗∗ 1.42 (0.16)∗∗ −0.00 (0.03)−0.86 (0.13)∗∗
∗p<0.05, ∗∗p<0.01.
CONCLUSION AND DISCUSSION
This study examined the network features of adolescent peer
relationships and then applied ERGMs to inspect the impacts
of adolescent peer relationships on subjective well-being. As
regards reciprocal effects, transitivity effects and expansiveness
effects, the findings were in line with previous research on peer
relationship networks. However, the findings on receiver effects
and differential effects show interesting differences from previous
studies. We discuss each of these in turn.
Reciprocal Effects
All the peer relationship networks in our study showed positive
reciprocal effects. Reciprocity is the most fundamental and
common behavior in human activities (Blau, 1964). It is also
a significant part of friendship and plays a significant role in
friends’ selections (Snyder et al., 1996). Adolescents are no
exception. Mutual friends have more opportunities to influence
each other and form similarities between each other (Mercken
et al., 2010). A reciprocal relationship also improves the quality
of friendship and enhances intimacy (Hinde et al., 1985;Dishion
et al., 1996). Reciprocity is a major feature of adolescent
friendship (Rubin et al., 1998), and a reciprocal relationship
between friends is the significant factor in adolescent peer groups
(Pearson and Michell, 2000).
Transitivity Effects
All the peer relationship networks in our study showed positive
transitivity effects: that is, my friend’s friend might become
my friend. The results confirm previous findings on adolescent
friend networks (Espelage et al., 2007). The main reason why
transitivity exists in a network is that the actors attempt to reduce
the contradictions and uncertainties in social and cognitive
situations and make efforts to establish a balance in interpersonal
relationships. In a tripartite relation between friends, for instance,
unbalanced relations occur when actor E likes actor R, actor
R likes actor V, but actor E does not like actor V. This
might cause emotional stress and uncertainty (Batjargal, 2007).
Therefore, adolescents might tend to build transitive relations
with other peers to establish an equilibrium in a tripartite
relationship.
Expansiveness Effects
All the peer relationship networks in our study showed negative
expansiveness effects. The network circles in a class are relatively
stationary. The reason is because adolescence is a psychologically
sensitive period and adolescents fear rejection, so they have
a lower initiative to make friends. Meanwhile, in order to
maintain a stable friendship and optimize groups, the individual
relationship circles are often exclusive, which makes it difficult
for people outside the circle to enter and leads to less volatility for
each circle.
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TABLE 4 | The estimated values of differential effects parameters.
Class Life satisfaction Positive affection Negative affection
Initial Final Initial Final Initial Final
Class A −0.16 −0.12 0.75 −0.15 −0.10 −0.08
Class B −0.18 −0.10 −0.14 −0.13 −0.26 −0.25
Class C −0.09 −0.04 −0.13 −0.12 −0.06 −0.05
Class D 0.14 0.15 −0.08 −0.07 −0.02 0.10
Class E −0.32 −0.30 −0.18 −0.15 −0.20 −0.18
Class F −0.09 −0.10 −0.14 −0.13 −0.13 −0.12
Class G −0.22 −0.20 −0.18 −0.14 −0.10 −0.08
Class H −0.34 −0.30 −0.10 −0.03 −0.30 −0.24
Class I −0.16 −0.14 −0.07 −0.07 0.01 0.03
Class J 0.02 0.02 −0.01 0.01 0.03 0.03
Class K −0.21 −0.15 −0.16 −0.11 −0.22 −0.21
Class L −0.34 −0.26 −0.25 −0.20 −0.24 −0.24
Class M −0.25 −0.15 −0.44 −0.35 −0.25 −0.17
Class N −0.12 −0.12 −0.10 −0.06 −0.19 −0.17
Class O −0.12 −0.07 −0.18 −0.16 −0.18 −0.12
The black font indicates that the results of parameter estimations are obvious.
Leaders and Receiver Effects
The analysis of popularity effects indicates that the peer
relationship networks in our study have no leaders and have no
significant receiver effects. No leader emergence might be due to
the fact that the adolescents surveyed in our investigation all came
from urban schools and do not live in the school (as is the case
with boarding schools). Learning is the main task for adolescents
and they have little distractions beyond learning, which means
there is little ground for the emergence and growth of adolescent
leaders. Besides, the generations after 2000 have grown into
adolescence in the Internet age. Social media networks expand
their social horizons and enhance their cognitive levels which, to
a certain extent, would obviously hinder the emergence of leaders
in adolescent peer networks.
Meanwhile, whilst our study found that receiver effects
were not significant, there were no correlations between
popularity in the network and each dimension of subjective
well-being among most friends. There might be two reasons
for this. Firstly, popularity was the result of evaluation from
others, while subjective well-being was self-evaluated. Secondly,
subjective well-being has multiple sources, while the classroom
environment is only one factor.
Differential Effects
The results of differential effects show that adolescents in
partial peer networks in our study exhibit similar levels in
SWB dimensions (satisfaction with life, positive affects and
negative affects), which is consistent with the results of previous
studies (Ryan, 2001), but not all network friends presented such
similarities. The possible reasons for these mixed results are as
follows.
(1) Our study overcame the lack of peer self-reports in previous
studies which increases the objectivity and credibility of the
results. In previous studies, participants were only required
TABLE 5 | The estimated results of receiver effect parameters.
Class Life satisfaction Positive affection Negative affection
Class A 0.08 0.07 0.06
Class B 0.18 0.20 0.11
Class C 0.26 0.22 0.29
Class D 0.39∗0.25 0.27
Class E 0.15 0.15 0.19
Class F 0.14 0.07 0.08
Class G 0.25 0.23 0.28
Class H 0.28 0.26 0.24
Class I 0.19 0.20 0.22
Class J 0.17 0.13 0.14
Class K 0.01 0.19 0.04
Class L 0.28 0.27 0.27
Class M 0.41∗0.21 0.28
Class N 0.13 0.11 0.09
Class O 0.21 0.16 0.16
∗p<0.05.
to report the attribute variables of their friends. Adolescents
might overestimate the similarities among friends. In our
study, all individuals in the network were required to do
self-reporting so that the results would be more reliable.
(2) The structural features in this study were different
from previous studies, and therefore different results are to
be expected. Previous studies have been based on binary
relationship structures (such as reciprocal structures) while
this study was based on structural features such as transitivity,
expansiveness, and popularity. These ternary or multiple
structures were constituted by the interdependence of multiple
dual structures which are closer to real situations and have a
greater practical significance. The results therefore are more
reliable and have a greater accuracy.
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(3) This investigation studied the variables of peer
relationships only and did not take into account other factors
which could affect the conclusions (see below).
Subjective Well-being and Future
Research
Studies have shown that subjective well-being among friends will
lead to an effect on each other and tends to reach a similar level
(Fowler and Christakis, 2008). Social Communication Theory
and Similarity Theory could explain the similarities among
friends. For instance, these similarity effects might be the result
of social influences: that is, adolescents are affected by their
friends and take on similar behaviors. But they might also be
the result of social choice: that is, forming friendships based
on similar attitudes and behaviors such as a similar SWB.
Whether the similarities among adolescents are the result of
social communication and influences or social choices needs
further exploration, and future research could investigate the
potential processes of SWB similarities among adolescents by
longitudinal studies.
This study shows that peer networks can affect an individual’s
subjective well-being. The peer environment in school plays a
role in the process of maintaining groups of friends. However,
individual behaviors in the groups tend to promote each
individual and tend to be consistent. Since similar friends gather
together into smaller peer groups, so we could apply group
counseling to intervene in specific groups of friends. We could
also through influential individuals to intervene with other
peer in the network, by which we could promote the healthy
development of the individual and ultimately promote social
progress.
From the methodological point of view, this study adds to the
current literature on ERGMs, and also provides a platform for
future research. As one of our social network analysis models,
ERGMs are concerned not only with the relationship between
individual and individual, but also with a more in-depth study of
the dependent relationships between individuals. One advantage
of an ERGM is the ability to apply a simple graphical structure
to present selected local structure variables. Additionally, the
selection of structure variables is quite flexible and can easily be
corrected. A further advantage is that the potential statistics in
an ERGM enables a more in-depth exploration of the dependent
relationships between individuals compared to other network
models. (Hossain et al., 2015). In future, it could be extended
from binary random variables to classified relational variables
or multiple relational variables. It could also be put into use in
the fields of sociology, economics, and psychology and promote
interdisciplinary collaboration in the study of peer relationships.
LIMITATIONS
There are some aspects of our study that need to be improved:
Firstly, we did not measure other variables that could influence
the results such as age, gender and social economic level.
Secondly, we cannot completely exclude the possibility that the
lack of a relationship between popularity and subjective well-
being (the receiver effect) is related to the lack of a popularity
structure effect. Thirdly, longitudinal research is needed to
explore the potential changes in the similarity effect of peers’
subjective well-being. The similarity effect may be the result of
social influence, as adolescents can be influenced by their friends
and act out similar behaviors and performance; but it may also
be the result of social choice, as adolescents can select friends
on the basis of similar behaviors and attitudes such as subjective
well-being.
AUTHOR CONTRIBUTIONS
CJ, FC, JL, and XP developed the concepts for the study. HW
collected the data. CJ, TW, JL, XP, and HW analyzed the data. CJ,
JL, XP, and HW wrote the manuscript. All authors contributed to
the manuscript and approved the final version of the manuscript
for submission.
FUNDING
This research was supported by the Outstanding Young Faculty
Award of Guangdong province: YQ2014149.
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Conflict of Interest Statement: The authors declare that the research was
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be construed as a potential conflict of interest.
The reviewer MI-C and handling Editor declared their shared affiliation, and the
handling Editor states that the process nevertheless met the standards of a fair and
objective review.
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