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Beyond Dyads and Triads:
A Comparison of Tetrads
in Twenty Social Networks
Cassie McMillan1and Diane Felmlee2
Abstract
Social psychologists focus on the microlevel features that define interaction, often attending to
dyads and triads. We argue that there also is utility in studying how configurations of four
actors, or tetrads, pattern our social world. The current project considers the prevalence of
directed tetrads across twenty social networks representing five relationship types (friend-
ship, legislative co-sponsorship, Twitter, advice seeking, and email). By comparing these
observed networks to randomly generated conditional networks, we identify tetrads that occur
more frequently than expected, or network motifs. In all twenty networks, we find evidence for
six tetrad motifs that collectively highlight tendencies toward hierarchy, clustering, and
bridging in social interaction. Variations across network genres also emerge, suggesting
that unique tetrad structural signatures could define different types of interaction.
Keywords
bridging, clustering, hierarchy, motifs, social networks, tetrads
Social psychologists often focus on small
subsets of individuals to better under-
stand social and group processes, repre-
senting a form of ‘‘sociological miniatur-
ism’’ (Stolte, Fine, and Cook 2001). For
example, many studies within the broad,
interpersonal relationship literature
examine two-person groups, or dyads
(e.g., Fagundes and Diamond 2013), and
theoretical work regarding pairs of indi-
viduals, such as the norm of reciprocity
(Gouldner 1960), continues to be influen-
tial today. Dating back to Simmel’s
(1902) early work, groups of three people,
or triads, also receive extensive attention
within social scientific inquiry. Repeated
research documents the tremendous
power of microlevel processes within
these small subsets for shaping a variety
of social outcomes, such as friendship for-
mation (Krackhardt and Handcock 2006),
peer aggression (Felmlee and Faris 2016),
health (Pescosolido 2006), and delin-
quency (Kreager, Rulison, and Moody
2011). The examination of a group’s
microstructure is valuable, furthermore,
because small-scale patterns have impli-
cations for hierarchy and clustering at
the overall group or network level
1Northeastern University, Boston, MA, USA
2Pennsylvania State University, University Park, PA,
USA
Corresponding Author:
Cassie McMillan, Department of Sociology and
Anthropology, Northeastern University, 900
Renaissance Park, Boston, MA 02115, USA.
Email: c.mcmillan@northeastern.edu
Social Psychology Quarterly
2020, Vol. 83(4) 383–404
ÓAmerican Sociological Association 2020
DOI: 10.1177/0190272520944151
journals.sagepub.com/home/spq
(Holland and Leinhardt 1971; Johnsen
1985), although certain configurations
may fail to scale up to larger structures
(Martin 2009b).
Nevertheless, social psychologists and
network scholars alike often end their
analysis of microlevel patterns with sub-
groups composed of three people. Some
methodological work on the conditional
dependence of network structures consid-
ers the role of slightly larger groups of
four, or tetrads (e.g., Pattison and Robins
2002; Snijders et al. 2006; Yaverog
˘lu
et al. 2015). However, these four-person
groups receive less attention than dyads
and triads across the theoretical and
empirical literature (for exceptions, see
Bearman, Moody, and Stovel 2004; Cook
and Emerson 1978; Sarajlic
´et al. 2016;
Skvoretz and Willer 1991). We maintain
that this omission is unfortunate because
tetrads can provide additional insight
into structures of hierarchy, clustering,
and bridging that are not evident from
solely studying patterns of dyads and tri-
ads. Moreover, a focus on larger sub-
groups of four can enhance our under-
standing of why we occasionally observe
unexpected lower-level network patterns,
such as unreciprocated dyads and imbal-
anced triads.
Our purpose in the current project is to
demonstrate the ways in which tetrad
patterns in social networks can contrib-
ute to our understanding of microlevel
social processes and therefore be of value
to social psychology inquiry. First, we
compare the prevalence of all possible
directed tetrads across twenty social net-
works drawn from five unique types of
social groups. Specifically, we look for those
tetrads that occur more frequently than
expected, that is, network motifs (Alon
2007; Milo et al. 2002). By taking a compar-
ative social network approach, we identify
tetrad patterns within and across five dis-
tinct genres of social connections: friend-
ship, legislative co-sponsorship, Twitter,
advice seeking, and email communication.
To determine which tetrads are motifs, we
use a novel approach that relies on expo-
nential random graph models (ERGMs)
to generate comparative random networks
in our analyses. Our findings suggest that
new theoretical insight can be gained by
studying patterns of four-person sub-
graphs in social groups.
DYADS AND TRIADS
Extensive research within social psychol-
ogy examines patterns of dyadic interac-
tion, including within the social exchange
framework, the close-relationship litera-
ture, and other scholarship that focuses
on microlevel interactions between pairs
of people (e.g., Molm and Cook 1995;
Fagundes and Diamond 2013). The norm
of reciprocity, coined by Gouldner
(1960), applies to groups of two and sug-
gests that individuals feel obligated to
return or repay favors and acts of kind-
ness on the part of another. In social net-
work analysis, one of the ubiquitous
structural controls included in statistical
models is an indicator of whether or not
a tie is reciprocated, or mutual (Steglich,
Snijders, and Pearson 2010; Robins et al.
2007). This body of research tends to pro-
vide widespread support for reciprocity
across networks from many different
types of social interaction, with mutual
ties being overrepresented when com-
pared to random graphs (e.g., Block
2015; Desmarais and Cranmer 2012;
Felmlee and Faris 2016). In other words,
offered friendships tend to be returned,
emails are frequently responded to, legis-
lators co-sponsor each other’s bills, advice
giving can become a two-way street, and
people retweet each other’s posts on Twit-
ter (Felmlee, McMillan, et al. 2018).
Three-person groups, or triads, also
form the nexus of attention for consider-
able scholarship within the literature on
social networks. Simmel (1902) was the
384 Social Psychology Quarterly 83(4)
first to point to the triad as a constellation
of importance, noting that when a social
group grows from two to three actors,
new social arrangements arise. Some
third parties operate as mediators, or
arbitrators, between the two others, for
instance, while others stir up conflict
between their fellow group members and
attempt to drive a wedge through the
group. Simmel’s theoretical developments
regarding three-person subgroups con-
tinue to inspire scholarship on triads
today (Krackhardt and Handcock 2006).
Another influential development that
has stood the test of time is Heider’s
(1946) balance theory, which spawned
decades of research and debate regarding
triad formation. According to Heider, atti-
tude consistency leads people to try to
maintain a balanced state in which their
cognitions are internally similar. Balance
theory implies that in groups of three,
people will tend to like the friends of their
friends. If two friends do not like each
other, a state of stressful imbalance will
unfold. Previous work extends the
assumptions of balance to the concept of
transitivity (e.g., Holland and Leinhardt
1971; Rapoport 1963). According to the
principle of transitivity, or triadic closure,
if actor asends a tie to actor band actor
bsends a tie to actor c, actor ais expected
to send a tie to actor c. Triples that
exhibit this particular configuration of
transitive relations tend to occur more
frequently than by chance, as confirmed
by numerous studies on various genres
of social ties (e.g., An and Schramski
2015; Davis 1970; Desmarais and
Cranmer 2012; Hallinan 1974; McMillan
2019). On the other hand, many intransi-
tive triads—or groups of three that
include at least one triple such that actor
asends a tie to actor b, actor bsends
a tie to actor c, but actor adoes not
send a tie to actor c—are underrepre-
sented in empirical networks (Holland
and Leinhardt 1971; Wasserman and
Faust 1994).1
TETRADS
When compared to prior research on
dyads and triads, significantly fewer
studies consider patterns of four-actor
subgroups, or tetrads. Previous research
in social psychology, however, suggests
that there is value in considering these
slightly larger subgraphs. For instance,
social exchange theorists argue that to
study the social processes of power and
negotiation effectively, researchers need
to broaden their focus beyond the dyad
(Cook and Emerson 1978; Schwaninger,
Neuhofer, and Kittel 2019). Experimental
studies of exchange networks consider
small groups that frequently include
four participants to compare power differ-
ences and decision-making processes
within different relational configurations.
Such configurations include the ‘‘four-
actor line,’’ in which each participant
can make an exchange only with those
actors to whom they are adjacent, and
the ‘‘three branch,’’ or star tetrad, in
which three participants are connected
to one central individual (e.g., Lewis and
Willer 2017; Skvoretz and Willer 1991).
Studies consistently document links
between actors’ power and their positions
in these types of four-person exchange
networks, with central actors in the star
tetrad being advantaged over those
located on the branches, for example
(Lawler and Yoon 1996; Skvoretz and
Willer 1991). Yet it is important to note
that this experimental work considers
artificial tetrads that are not embedded
in larger observed networks. Few studies
examine tetrads in nonexperimental set-
tings, and almost none examine directed
1Note that triads also exist that include nei-
ther transitive nor intransitive triples. Holland
and Leinhardt (1971) define these configurations
as vacuously transitive triads.
Tetrads in Social Networks 385
tetrads, focusing instead on configura-
tions where all social ties are symmetric
(for an exception, see Sarajlic
´et al. 2016).
In spite of their relative neglect within
social psychology, we argue that a detailed
study of four-person groups, or tetrads,
has much to offer the field. Given the
noteworthy tetrads that emerge in biolog-
ical and engineering networks (e.g.,
Schreiber and Schwo
¨bbermeyer 2009), it
is likely that social networks produce
interesting tetrad patterns, as well. Fur-
thermore, by considering the slightly
larger, four-person group, we can gain
new insight into when unexpected config-
urations of dyads and triads are apt to
occur. For instance, although there are
exceptions, previous work finds that
many types of social networks are defined
by fewer intransitive triads than would
be expected by random chance (e.g., Davis
1970; Hallinan 1974; Holland and Lein-
hardt 1971). Analyzing those tetrads in
which intransitive triads are situated
can provide more context as to when
these uncommon, lower-order structures
are tolerated. In the current project, we
maintain that relations among four actors
facilitate the study of three formative
structural patterns: hierarchy, cluster-
ing, and bridging. We describe these
structures in further detail in the follow-
ing sections.
Tetrads and Hierarchy
Hierarchy in a network typically refers to
an arrangement in which some type of
path asymmetry exists between actors,
such that a particular path from actor
ato actor bis not reciprocated. According
to Harrington and Fine (2000), the small
group is where individuals come directly
into contact with systems of social hierar-
chy. In small groups of same-gender, ado-
lescent campers, for example, hierarchy
emerges in the form of dominance order-
ings, with special roles, such as a ‘‘top
boy’’ or ‘‘bottom girl’’ (Martin 2009a). Pat-
terns among dyads and triads provide
some information about hierarchy in
these very small subgraphs, where cer-
tain configurations suggest that one or
two individuals are ranked above others.
Transitivity, for instance, is routinely
measured to document unequal status or
hierarchy within broader social networks
(e.g., McFarland et al. 2014). Holland and
Leinhardt (1971, 1978) demonstrate that
a structure of hierarchically ranked clus-
ters of cliques emerges in networks when
intransitivity is avoided, noting that ten-
dencies toward balance and hierarchy
are inherently linked in social networks.
A network macromodel extends these
arguments by incorporating hierarchy
within, as well as between, cliques (John-
sen 1985).
Yet studying patterns of tetrads can
provide additional insight regarding hier-
archy and group status structure. Most
notably, tetrads enable one to determine
whether hierarchical triads reinforce
and expand upon one another in the
broader network. A common motif in biol-
ogy, the ‘‘bifan’’ (Schreiber and Schwo
¨b-
bermeyer 2009), for example, could reflect
hierarchy in a social context. The bifan
tetrad consists of two nodes that both
send unreciprocated ties to two other
nodes (see Figure 1). This configuration
includes two triads that are positioned
to reinforce possible differences in social
status between actors, suggesting that
the network could be characterized by
broad patterns of hierarchy. Other tetrad
configurations imply that the hierarchical
structures of certain networks expand
beyond the two or three levels of status
that can emerge in dyads and triads.
For instance, previous work considers
directed versions of the ‘‘four-actor line’’
(Skvoretz and Willer 1991), where each
actor in a group experiment is restricted
to sending no more than one unrecipro-
cated tie to a higher-status actor (see
386 Social Psychology Quarterly 83(4)
Figure 1). If such configurations occur in
observed social networks, this would sug-
gest the existence of a stratification system
that consists of at least four status levels.
One tetrad that has received signifi-
cant attention in the social sciences is
known as the ‘‘box’’ tetrad, and through
considering its prevalence—or absence—
researchers can draw inferences about
how status shapes the structure of social
networks (see Figure 1 for an example of
a box tetrad, or a ‘‘cordless 4-cycle’’). For
example, Bearman and colleagues (2004)
found that adolescents in a romantic
sexual network of a high school avoided
four-cycles, or symmetric formations of
the box tetrad, in which youth fail to
date the former (or current) partner of
their former (or current) partner. Pre-
sumably, involvement in such mutual,
four-person, heterosexual connections
would violate social norms and lead to
a public loss of status and esteem. Other
studies (Marcum, Lin, and Koehly 2016),
but not all (Stadtfeld, Hollway, and Block
2017), document additional evidence of
such an avoidance in certain sexual net-
works. In other relevant work, Coleman
(1988) argues that the box tetrad facili-
tates status reinforcement between two
higher-ranked actors, each of whom is
connected to a lower-status actor, such
as ties across generations of two families
(e.g., parents and children). The box tet-
rad also may lead to a process by which
two actors engage in stonewalling, that
is, the creation of alliances between the
two dyads at the cost of strong, nonhier-
archical, positive ties among all four.
Overall, we hypothesize that across
the networks in our sample, many of
the tetrads that occur more frequently
than expected will be characterized by
hierarchical structures (Hypothesis 1).
After accounting for lower-order pro-
cesses, we expect to uncover over-
represented four-person groups in which
patterns of asymmetric dyads and transi-
tive triads reinforce concurrent struc-
tures of hierarchy. In other words, we
suspect that transitive triads and asym-
metric dyads do not occur randomly
throughout individual networks. When
these configurations appear, we predict
that their structures will complement
one another such that the same actors
will continuously serve as the receivers
of asymmetric ties and endpoints of tran-
sitive triads (i.e., actor cwhen aàb, bàc,
and aàc). Note that tetrads represent the
smallest subgroup of actors in which we
can observe this level of reinforcement.
Tetrads and Clustering
Social psychologists and network scholars
also have a continued interest in studying
clustering in the form of reciprocity, net-
work closure, and small subgroups. For
instance, previous work argues that pro-
cesses of tie reciprocity and the develop-
ment of the completely connected triad
clique contribute to network clustering
(Wasserman and Faust 1994). One
advantage of studying tetrads is that we
can examine additional levels of cluster-
ing, such as that occurring in the ‘‘four-
clique,’’ or subgroups of four people who
are all connected to one another by recip-
rocated ties (see Figure 1).
Patterns of clustering relate to several
group-level processes of interest. First,
high levels of clustering tend to be associ-
ated with relationship stability. The
three-person clique, for example, often
forms an absorbing, or ending, state in
the development of friendship ties over
Figure 1. Tetrads of Interest
Note: Tetrads are numbered according to their iso-
morphic class (Csa
´rdi and Nepusz 2006).
Tetrads in Social Networks 387
time (Hallinan and Sørensen 1983). In
a longitudinal analysis of Facebook interac-
tions, Doroud and colleagues (2011) find
that the most common evolutionary trajec-
tory for triads began with an unconnected
set of three nodes and culminated in a com-
pletely connected three-person clique.
Additionally, focusing on patterns of clus-
tering can provide insight into processes
of homophily, or the tendency for individu-
als to group together with similar peers
(McPherson, Smith-Lovin, and Cook
2001). Individuals belonging to the same
social group tend to report similar behav-
ioral and demographic characteristics,
because they either select to associate
with similar others or are influenced to
adapt others’ behaviors (McMillan, Felm-
lee, and Osgood 2018; Osgood et al. 2013).
In sum, we hypothesize that patterns of
clustering will define those tetrads that
are overrepresented across our sample of
social networks (Hypothesis 2). Here, we
define clustering by groupings of symmet-
ric dyads. Specifically, we expect that the
fully connected four-clique will be overrep-
resented across our sample of networks,
even after accounting for reciprocity and
certain clustering patterns among triads.
This is because pockets of reciprocated
dyads and three-cliques should not occur
sporadically throughout a network but
instead cluster together to form the basis
of larger, cohesive subgroups.
Note, too, that clustering can operate
in tandem with structures of hierarchy.
By examining four-node subgraphs, it is
easier to detect the ways in which hierar-
chy and clustering can define network
structures simultaneously. Across many
networks, actors are likely to be situated
at different rungs of the social ladder,
with those on the same tiers clustering
together. Through studying patterns of
tetrads, we can gain additional insight
into whether this type of clustering is
more likely to connect those actors situ-
ated at high- or low-status levels. Are
there more likely to be lone individuals
or clusters of actors at the top versus
the bottom of the interpersonal ‘‘food
chain?’’ For instance, a tetrad that con-
sists of one uniquely high-status individ-
ual who is chosen by all actors in a group
of three connected others would suggest
a pyramid-shaped status hierarchy where
lower-status actors cluster together in
large numbers. Alternatively, a funnel-
shaped status hierarchy would be defined
by tetrads where all members of a com-
pletely connected three-person clique
receive social ties from a lower-status
‘‘hanger on’’ who is not chosen in return.
Tetrads and Bridging
A third process of interest in studying tet-
rads is that of bridging. According to
graph theory, a bridge represents an isth-
mus, or an edge whose deletion separates
the graph into disconnected components.
Bridges between different sectors of a net-
work tend to consist of weak, rather than
strong, ties, according to Granovetter’s
(1973) well-known thesis. Even though
weak ties are defined by lower levels of
interaction and intimacy, they often play
a crucial role in connecting the broader
network because these connections repre-
sent unique avenues for the spread of
novel information. Strong ties are less
likely to inspire this diffusion due to their
insular and redundant clustering tenden-
cies, although exceptions arise in complex
contagions (Centola 2018). Burt (2004)
further considers the implications of
brokers, or individuals who connect dis-
tinct groups and span a social network’s
‘‘structural holes.’’ He argues that
brokers tend to have greater levels of
power than nonbrokers, such as the abil-
ity to act as a gatekeeper in transferring
information from one group to the other.
Tetrads enable the examination of
local network bridges in a manner that
is not possible from solely considering
388 Social Psychology Quarterly 83(4)
dyads and triads. For example, the ‘‘kite’’
tetrad (Friedkin and Cook 1990) consists
of an interconnected triad in which one
of the three members reports a single tie
(reciprocated or directed) to a fourth actor
who is otherwise disconnected from the
cluster of three (see Figure 1) and linked
by a ‘‘cut edge.’’ It is likely that the fourth
actor has other connections outside the
tetrad from whom they can gather and
then diffuse information and ideas to
the connected triad. Extending Granovet-
ter’s (1973) logic to this tetrad configura-
tion, we expect that the bridging tie, or
cut edge, in a kite formation would be
weak in strength. For certain relation-
ships, such as friendship and patterns of
online communications, this could be
reflected in an asymmetric, rather than
a symmetric, bridging tie. Given the
importance of bridging in social relation-
ships, we expect that some ‘‘kite’’ tetrads
will occur more frequently than expected
by random chance (Hypothesis 3). We
argue that the tetrad represents the
smallest subgraph in which patterns of
bridging can be observed. For example,
subgroups of four allow for the examina-
tion of whether two-paths and other
intransitive triads bridge together dispa-
rate groups (see Figure 1, Tetrad 142)
or reinforce patterns of hierarchy (see
Figure 1, Tetrad 29).
NETWORK MOTIFS
In order to study tetrad configurations
systematically, we focus on those sub-
graphs that occur more frequently than
expected, or network motifs. Network
motifs refer to recurring, overrepre-
sented, small-scale patterns of interaction
between sets of nodes and represent the
essential building blocks of larger struc-
tures (Milo et al. 2002). Identifying net-
work motifs, including those with four
nodes, has provided insight into the func-
tioning of biological networks (e.g., Alon
2007). Despite the difficulties in ade-
quately capturing complex and messy
patterns of interpersonal relations, previ-
ous work finds that certain dyads, triads,
and symmetric tetrads are more likely to
occur than expected across a variety of
different types of social groups (e.g.,
Felmlee, McMillan, et al. 2018). These
network motifs point to the importance
of mutuality and transitivity in defining
interpersonal relationships. In the cur-
rent project, we extend upon earlier
work by seeking out those directed tet-
rads that are network motifs across all
20 social networks in our sample. While
some previous work considers the directed
tetrad census in single types of social net-
works (e.g., Sarajlic
´et al. 2016), to the
best of our knowledge, the current study
represents the first to consider the preva-
lence of directed tetrad subgraphs across
several genres of social ties.
Existing research on tetrad motifs
tends to use univariate, conditional dis-
tributions to generate comparison ran-
dom networks to identify overrepresented
subgraphs (e.g., Artzy-Randrup et al.
2004; Milo et al. 2002, 2004). Here, we
introduce a new technique for uncovering
patterns of tetrad prevalence that uses
ERGMs to generate random, comparable
graphs. We compare these simulated net-
works to the observed data to examine
the extent to which certain tetrads occur
more frequently than expected. This
approach builds on earlier statistical
work that highlights how tetrads relate
to endogenous dependencies in networks
(e.g., Pattison and Robins 2002; Snijders
et al. 2006) by allowing one to incorporate
controls for multiple, simultaneous struc-
tural processes, such as reciprocity and
transitivity.
STRUCTURAL SIGNATURES
Evaluating structural patterns across
different groups can provide insight
Tetrads in Social Networks 389
regarding fundamental social processes
(Faust 2010; Faust and Skvoretz 2002).
Hierarchy and status differences define
many types of social interaction, such as
friendship (McFarland et al. 2014) and
workplace interactions (Spinuzzi 2015).
However, it remains unclear how these
patterns vary across different types of
relationships. Thus, another purpose of
this research is to compare tetrad pat-
terns across various types of social net-
works. We hypothesize that tetrad pat-
terns within specific network genres will
be more alike than those between differ-
ent networks genres (Hypothesis 4), sug-
gesting that each type of social network
has a unique tetrad ‘‘signature’’ or ‘‘fin-
gerprint.’’ Such a finding would suggest
that certain types of graphs could be iden-
tified by their tetrad pattern alone.
METHODS
Data
We consider tetrad patterns across five
types of social networks: adolescent
friendship, U.S. Senate bill co-sponsor-
ship, Twitter online messaging, advice
seeking, and email communication. We
focus on these five network types because
they vary on key dimensions that are
likely to shape network structure, such
as whether ties represent formal or infor-
mal connections and in-person or online
interactions. Within each of the five social
network genres, we consider four distinct
networks to compare more systematically
whether each of the types exhibits its own
tetrad pattern or whether these patterns
overlap substantially across genres. In
total, our sample includes 20 social net-
work graphs.
For our adolescent friendship data, we
select four random school-based networks
from the in-school survey collected during
Wave 1 of the National Study of Adoles-
cent to Adult Health (Add Health).
During the first wave, Add Health sur-
veyed the entire student bodies from
over 100 U.S. middle and high schools.
Respondents were asked to nominate up
to 10 of their closest within-school
friends.2We use these nominations to
construct directed networks where nodes
are individual adolescents and a social
tie from node ato node bindicates that
adolescent anominated adolescent bas
a friend.
We construct four co-sponsorship net-
works using data on U.S. Senate co-
sponsorship patterns from the 1995,
2000, 2005, and 2010 congressional terms
(Fowler 2006). Each node represents an
individual senator. A directed edge from
senator ato senator bindicates that dur-
ing the congressional term of interest,
senator aco-sponsored at least one piece
of legislation for which senator bwas
the primary sponsor. If senators aand c
both co-sponsor senator b’s bill, this
action does not result in a tie between
senators aand c.
We analyze Twitter data that were
collected during a period of one week at
the end of February 2017. Tweets were
gathered from the Twitter application
programming interface by using a key-
word search function for aggressive,
harmful terms that targeted women and
minorities (i.e., curse words and racial
slurs) and downloading tweets and their
connected messages (Felmlee, Inara
Rodis, and Francisco 2018). Nodes repre-
sent individual users who engaged
with a tweet containing a keyword, and
edges represent retweets, likes, and men-
tions. Two of our networks represent
2Even though respondents were limited in the
number of friends they could nominate on the
Add Health survey, previous work finds that
most students nominated fewer friends than the
maximum (Goodreau, Kitts, and Morris 2009).
However, it is important to note that truncation
and out-of-design missingness remain limitations
of the data set.
390 Social Psychology Quarterly 83(4)
cyberbullying instances that surrounded
the use of a specific slur. The other two
networks consist of cyberbullying attacks
that either originated from, or targeted,
a celebrity. While many of the ties in
our Twitter networks represent negative,
harmful connections, others represent
positive ties of support.
The four advice networks were col-
lected from various types of surveys
administered to employees in four differ-
ent workplaces: a consulting firm (Cross
and Parker 2004), an information tech-
nology department in a Fortune 500 com-
pany (Almquist 2014), a law firm (Lazega
2001), and a high-tech company (Krack-
hardt 1987). In each survey, participants
were asked to nominate those coworkers
whom they sought for professional advice.
Using these nominations, we construct
directed networks where nodes are indi-
vidual employees and an edge between
two nodes indicates that employee aseeks
advice from employee b.
The four email communication net-
works also were collected from workplace
environments. One is from the company
Enron (Klimt and Yang 2004), and the
remaining three are extracted from dif-
ferent bureaucratic departments in the
European Union (EU) (Leskovec, Klein-
berg, and Faloutsos 2007). All four net-
works include email-sending patterns
over an eighteen-month period. From
this information, we construct directed
networks where nodes represent individ-
ual employees and directed edges indicate
that employee asent at least one email to
employee b.
Plan of Analysis
The current project is an exploratory
study that seeks to uncover those directed
tetrads that represent key building blocks
in broader network structures. To iden-
tify overrepresented tetrads in our
observed networks, our analysis includes
three steps. First, we estimate ERGMs
on each of our observed networks that
include parameters to account for a vari-
ety of structural phenomena, including
certain patterns of dyads and triads.
Then, we use the coefficient values from
each ERGM to simulate 1,000 conditional
graphs. Finally, we calculate zscores to
compare the prevalence of each directed
tetrad in the observed networks to their
prevalence in the random graphs.
Step 1: Estimate ERGMs. In order to con-
struct our sample of conditional graphs,
we first estimate ERGMs across our 20
observed networks. ERGMs are a statisti-
cal network method that compares the
patterns in an observed network to what
would be expected to occur by random
chance (Hunter et al. 2008; Robins et al.
2007). More specifically, we can define Y
as an n3nmatrix (where nis the num-
ber of actors) such that the (i, j) entry of
this matrix is 1 if there is a relational
tie between actors iand jor 0 if no such
tie exists. The ERGM specifies the proba-
bility that network Ywill occur given
a set of individuals:
PðY5yjXÞ5exp½uTgyðÞ
kðuÞ:
Here, Xrepresents a matrix of covariates
and uis a vector of all network coeffi-
cients that are hypothesized to relate to
the probability of the observed network’s
structure. A vector of network statistics,
g(y), is calculated using the observed
adjacency matrix, and k(u) is a normaliz-
ing factor that ensures that the result is
a legitimate probability distribution. We
present a discussion of ERGM conver-
gence and goodness-of-fit statistics in
the supplemental material (available in
the online version of the article).
In the current project, we estimate an
ERGM on each of our 20 observed net-
works that includes four parameters to
Tetrads in Social Networks 391
account for structural tendencies of inter-
est. First, we include an edges term to
control for the base log odds of a tie.
This variable accounts for the likelihood
that an edge will exist between any two
actors in the network and serves a similar
role as an intercept in a regression model.
Next, we include a mutual term to
account for the tendency toward reciproc-
ity in social networks (e.g., actor asends
a tie to actor band actor bsends a tie to
actor a).
Finally, we include a set of two terms
to account for theoretically relevant triad
patterns: the geometrically weighted
dyadwise shared partner (GWDSP) and
geometrically weighted edgewise shared
partner (GWESP). The GWDSP term
measures the tendency for two nodes to
be linked indirectly through one or more
shared partners, regardless of whether
the two nodes are tied directly. The
GWESP term measures the degree to
which two linked nodes have one or
more partners in common (Hunter
2007).3Both terms are assigned decay
parameters that adjust the extent to
which each additional shared partner
connection contributes to the measure.
We include the GWESP term to capture
tendencies toward triadic closure directly,
while GWDSP aids in reducing bias when
estimating the GWESP coefficient and
facilitates interpretation of the GWESP
parameter as transitivity (Goodreau
2007; Snijders et al. 2006). These triad
measures are used because an estab-
lished line of theoretical work focuses on
the significance of transitivity in social
interaction (e.g., Heider 1946) and previ-
ous empirical research finds that
transitive triads are overrepresented
across a variety of observed social net-
works (e.g., Davis 1970; Hallinan 1974).
Nevertheless, the inclusion of alternative
triad controls, such as those for cycles or
various closure patterns, could generate
different estimates and represents
a task for future work.
Step 2: ERGM simulations. After each
ERGM reached adequate convergence,
we used the estimated coefficients to
draw random, simulated networks that
are conditional on the structural phen-
omena parameterized in the ERGM (for
additional details, see Handcock et al.
2008). Given the controls included in
each ERGM, the random graphs we simu-
late are conditioned on the observed net-
work’s density as well as its tendencies
toward reciprocity and transitivity. Note
that the graphs we simulate are similar
to those generated using the classic
U|MAN null distribution (e.g., Faust
2010; Holland and Leinhardt 1975) but
additionally control for the triadic tenden-
cies discussed previously. Overall, we gen-
erate 1,000 random networks for each
observed network, which results in a final
sample of 20,000 simulated networks.
Step 3: Calculate z scores.For each tet-
rad, we calculate a zscore to determine
whether the subgraph is overrepresented
in each of the observed networks. More
specifically, the zscore for each itetrad
is calculated as follows:
zi5Nobservedi\Nrandomi.
SDrandomi1e
:
Here, Nobservediis the count of observed
itetrads in the network, Nrandomiis
the average count of such tetrads that
appear across the random networks, and
SDrandomiis the associated standard devia-
tion. Given that some tetrads never occur
across certain sets of random graphs and
3More technically, the GWDSP (geometrically
weighted dyadwise shared partner) and GWESP
(geometrically weighted edgewise shared part-
ner) terms estimated here consider each actor
bto be an ‘‘outgoing two-path’’ shared partner
of the pair of actors aand cin a triple where
aàb, bàc, and aàc(Butts 2008).
392 Social Psychology Quarterly 83(4)
this results in a standard deviation of 0,
we include a small error term, e, that we
set to 0.01. This error term ensures that
we can calculate z
i
for each tetrad.
While there exist 218 possible tetrads,
19 of these subgraphs are unconnected
(e.g., the null tetrad that consists of no
edges). Following previous work (e.g.,
Kashtan and Alon 2005; Krumov et al.
2011; Schreiber and Schwo
¨bbermeyer
2009; Shen-Orr et al. 2002), we consider
only the occurrence of subgraphs that
are weakly connected, resulting in a final
sample of 199 directed tetrads. We assign
all tetrads a numeric label according to
their isomorphic class (following Csa
´rdi
and Nepusz 2006). We define motifs as
those tetrads iwhere the average zscore
within each network genre is greater
than 1.645. In other words, we consider
a tetrad to be overrepresented significantly
if it appears within each genre of observed
graphs more frequently than would be
expected across comparable conditional
graphs (p\.05, one tailed).
RESULTS
Motifs
We find six tetrads that appear more fre-
quently than expected among the net-
works in our sample (see Figure 2 for
a graphical depiction of each motif and
Table 1 for a summary of average zscores
by network genre). As expected, many of
the asymmetric dyads that occur in these
overrepresented tetrads are organized in
a manner that reinforces structural pat-
terns of hierarchy, giving some support
to our first hypothesis. Many of the motifs
confirm the existence of at least three
status levels, and ties are patterned
to support—rather than contradict—
the hierarchical arrangements of their
embedded dyads and triads. For example,
Tetrad 89 occurs more frequently than
expected across all five network genres
and is defined by at least three levels of
status hierarchy. This tetrad includes
a pair of 030T transitive triads (i.e., actor
anominates actor b, actor bnominates
actor c, and actor anominates actor c)
that complement one another’s hierarchi-
cal structure. Those actors situated at the
lower rungs send ties to those actors at
the higher levels, but these ties are not
reciprocated. Tetrad 89 most commonly
occurs in our sample of Twitter (mean z
score = 35.19) and friendship networks
(mean zscore = 27.52)
Complementing the conclusions of
classic works on network structure (e.g.,
Holland and Leinhardt 1971; Johnsen
1985), we uncover several additional
motifs that suggest the co-occurrence of
clustering and hierarchy, such that those
actors situated in the same status groups
tend to be linked. For instance, Tetrads
90, 91, and 213 are all defined by a rigid
hierarchical structure that includes two
levels of status distinction. Those actors
that occupy the same level of the hierar-
chy are clustered together through pre-
dominately reciprocated ties, which gives
some support to our second hypothesis.
Actors at lower strata send unrecipro-
cated ties to those at higher levels; how-
ever, these tend to be embedded in transi-
tive triads that complement hierarchical
structures. All three of the aforemen-
tioned tetrads are especially likely to
occur in networks of advice, email, and
co-sponsorship.
Figure 2. Tetrad Motifs across the Sample of
Social Networks
Note: Tetrads are numbered according to their iso-
morphic class (Csa
´rdi and Nepusz 2006).
Tetrads in Social Networks 393
We identify one motif (Tetrad 214) that
similarly highlights the co-occurrence of
clustering and hierarchy but is also
defined by occasional instances of intran-
sitivity. Analogous to the three motifs dis-
cussed in the previous paragraph, Tetrad
214 indicates a hierarchical structure
consisting of two levels where clusters of
nodes located on the same strata are con-
nected by reciprocated edges. All nodes
occupying levels of lower status extend
ties to the individual at the higher level.
While most of these cross-level ties
remain asymmetric, there exists one
reciprocated social tie sent from the
higher-status node to a lower-status
node, resulting in two intransitive triads.
Tetrad 214 is most common in the net-
works of email (mean zscore = 48.18),
Twitter (mean zscore = 26.35), and co-
sponsorship (mean zscore = 21.08) from
our sample.
Our final tetrad that occurs more fre-
quently than chance is Tetrad 79, which
represents a variation of a kite tetrad
and suggests that the networks in our
sample exhibit certain patterns of bridg-
ing (Hypothesis 3). This subgraph is char-
acterized by a transitive triad in which
one of the three actors sends an unreci-
procated tie to a fourth actor. The fourth
individual has no ties to either of the
other two in the transitive triad. Tetrad
79 is particularly common in our net-
works of email (mean zscore =16.44)
and advice (mean zscore = 6.33).
Variations across Genres of Social
Networks
By comparing the correlations between
each network’s vector of zscores, it is
apparent that the networks in our sample
are more similar to those from their same
genre than those from different genres
(Hypothesis 4). As shown along the diago-
nal of Figure 3, correlations are highest
between graphs of the same type (e.g.,
between friendship networks), suggesting
that each variant of interaction in our
sample may exhibit a unique structural
signature. The average zscore correlation
between networks from the same genre
is 0.63, while the average correlation
between those of different types is 0.14.
However, some network types are more
alike than others. For instance, the aver-
age correlation between email and advice
networks is 0.46, while the average corre-
lation between friendship and email net-
works is 20.05. We further demonstrate
the variation in subgraph prevalence by
Table 1. Average zScores for Tetrads of Interest by Network Genre
Variable
Friendship Co-sponsorship Twitter Advice Email
M SD M SD M SD M SD M SD
Motifs
Tetrad 79 1.91 1.08*6.00 11.79*2.94 7.97*6.33 12.31*16.44 22.83*
Tetrad 89 27.52 22.24*14.42 8.88*35.19 47.29*7.70 8.55*18.84 26.67*
Tetrad 90 18.25 6.24*31.38 15.65*3.50 7.01*7.85 9.24*23.42 40.08*
Tetrad 91 6.35 3.58*95.07 55.74*6.88 7.96*19.92 26.01*27.30 36.12*
Tetrad 213 7.29 6.34*13.84 7.48*54.96 97.14*7.50 5.37*71.00 40.97*
Tetrad 214 9.26 7.92*21.08 21.50*26.35 49.17*13.49 14.04*48.18 51.77*
Additional tetrads
Tetrad 19 (bifan) 43.24 14.12*7.93 9.20*720.76 774.95*–0.33 1.73 1.27 4.93
Tetrad 203 (box) 7.64 5.40*–10.81 2.49 0.00 0.00 –1.62 2.87 –5.83 2.96
Tetrad 217 (four-clique) 3.12 2.79*67.60 33.71*0.70 1.39 28.53 35.58*80.36 60.44*
Note: Tetrads are numbered according to their isomorphic class (Csa
´rdi and Nepusz 2006).
*Tetrad occurs more frequently than expected, p\.05 (one-tailed test).
394 Social Psychology Quarterly 83(4)
plotting the zscores of each network for
several tetrads of interest (see Figure 4).
Patterns of zscores tend to be more alike
within each social network genre than
they are across genres.
It is also clear that some tetrads are
more likely to occur within certain genres
but not others. As hypothesized, the four-
clique tetrad (i.e., Tetrad 217) is more
common than we would expect by random
chance across many network genres.
However, contrary to our expectations,
the four-clique appears about as fre-
quently as expected in the Twitter net-
works. In the four-clique, all possible
ties are present and all ties are recipro-
cated, which gives strong evidence for
patterns of clustering. Tetrad 217 is espe-
cially frequent in networks of email
(mean zscore = 80.36) and co-sponsorship
(mean zscore = 67.60).
Additionally, the bifan tetrad (Tetrad
19), which is frequently observed in bio-
logical networks, varies in prevalence
across the social networks in our sample.
This tetrad, which includes two lower-
status actors who send unreciprocated
ties to two higher-status actors, appears
more frequently than expected in the Twit-
ter (mean zscore = 750.76), friendship
(mean zscore = 43.24), and co-sponsorship
networks (mean zscore = 7.93). In certain
advice and email networks, however, bifan
tetrads occur less frequently than expected.
We illustrate this variation in Figure 5
with a comparison of the relative preva-
lence of the bifan tetrad in each type of net-
work. Finally, we find that the symmetric
version of the box tetrad (Tetrad 203),
which was mentioned previously, is signifi-
cantly more likely to occur in the friendship
networks (mean zscore = 7.64). In this
symmetric box tetrad, four individuals
send reciprocated ties in a cyclical pattern,
and all four embedded triads remain
intransitive. However, other than the
friendship networks, the symmetric box
Figure 3. Correlation Plot for Tetrad ZScores
Figure 4. ZScores for Select Tetrads by Net-
work Type
Note: The y-axes correspond to the zscore for
each tetrad of interest. Tetrads are numbered
according to their isomorphic class (Csa
´rdi and
Nepusz 2006).
Tetrads in Social Networks 395
tetrad is less likely to occur across all gen-
res in our sample.
Differences in data collection strate-
gies both between and within network
genres could contribute to certain pat-
terns we observe in our data. However,
we believe it is unlikely that these varia-
tions can fully explain our results. Even
though the advice data sets in our sample
were all collected using different survey
items and data collection techniques, for
example, the average within-genre z
score correlation of these networks is
greater than the average cross-genre cor-
relation (r= .31 and .19, respectively).
Additionally, previous work that takes
a comparative network approach to study
dyadic and triadic properties of networks
finds that variations in data collection
technique (e.g., observed vs. self-
reported) account for relatively few of
the differences in network structure
(Skvoretz and Faust 2002).
DISCUSSION
Small groups represent a crucial link
between individuals’ actions and large-
scale processes or institutions (Harrington
and Fine 2000). While previous research
tends to end its analyses with groupings
of two or three individuals, we argue
that it is also useful to focus on larger sub-
groups of four, or tetrads. In our sample of
twenty networks representing five diverse
types of relationships—including friend-
ship, legislative co-sponsorship, Twitter,
advice seeking, and email—six tetrads
Figure 5. Prevalence of the Bifan Tetrad across Network Genres
Note: One representative graph has been plotted for each network genre. We do not display isolates, or
nodes that neither send nor receive ties.
396 Social Psychology Quarterly 83(4)
occur more frequently than expected.
These recurring tetrad motifs point to sev-
eral fundamental interaction structures
that are inherent to the social sphere,
including hierarchy, clustering, and bridg-
ing. Given that social phenomena tran-
scend our levels of analysis (Stole et al.
2001), patterns of tetrads can inform our
understanding of both individual-level
outcomes and broader group processes.
Many of the relational patterns high-
lighted by these tetrad motifs are not evi-
dent from solely analyzing configurations
of two or three individuals. For instance,
the hierarchical tetrad motifs suggest
that distinctions by unequal status are
reinforced, rather than contradicted, in
our sample of social networks. Processes
of network hierarchy do not limit them-
selves to local interactions between pairs
or groups of three, in other words. Cer-
tain hierarchical interactions in the social
world consist of multiple status levels;
and patterns of actors’ ties, particularly
their unreciprocated ties, tend to comple-
ment, rather than challenge, this system
of stratification. This finding is notable
because those actors who are situated on
the higher levels of the status hierarchy
are expected to have more influence, rep-
resent the most desirable connections,
and have access to the greatest amount
of information (e.g., Berger, Cohen, and
Zelditch 1972; Friedkin 1986; Ridgeway
2014).
We also uncover many motifs in which
processes of clustering and hierarchy
operate simultaneously. In several cases,
symmetric ties connect dyads or triads
that are situated on the same strata of
the status hierarchy, while these clusters
send and receive unreciprocated ties with
peers who occupy different status levels.
Furthermore, we find some tetrads that
evince a pyramid-shaped status hierar-
chy (e.g., more actors are situated on
lower status levels than on higher
levels) as well as others that suggest
a funnel-shaped system (e.g., more actors
occupy higher status levels versus lower
levels). Taken together, these patterns
suggest that the status hierarchy of our
social networks most likely exhibits a dia-
mond-shaped pattern, in which a small
number of actors are particularly elite or
of low rank but the vast majority are sit-
uated on a mid-tier level of the social sys-
tem. Finally, there is evidence of motifs
that preserve the rigid, stagnate struc-
ture of the social hierarchy as well as
others that appear to directly challenge
this system by encouraging social mobil-
ity or advancement. As a result, we con-
clude that the hierarchical system defin-
ing social networks is complex: the
status hierarchy is being challenged in
some sectors of the network while simul-
taneously receiving support to remain
intact in other locales. Future work could
benefit from investigating the explicit
implications of these patterns of hierarchy
and clustering in tetrads for the overall,
macrolevel structure of the network.
Building off prior work that relates triad
patterns to network-level properties (e.g.,
Holland and Leinhardt 1971), the over-
(or under-) abundance of certain tetrads
is apt to hold consequences for the devel-
opment of specific macrolevel structures.
In addition, tetrads provide unique
insights into bridging processes, particu-
larly how unreciprocated ties, which are
likely to be relatively weak as compared
to their reciprocated counterparts, can
connect the broader network. A type of
kite tetrad (Tetrad 79) that includes
a highly social, but low-status, bridging
actor appears in our sample of networks
more frequently than would be expected.
While these lower-status, bridging actors
may be undervalued by their peers, the
unreciprocated ties that they send play an
important role in connecting the broader
network. Due to their unique position,
these relatively weak connections may be
able to access novel types of information,
Tetrads in Social Networks 397
which could help them advance and gain
status within their social groups (Burt
2004; Granovetter 1973). Overall, findings
imply that a number of social networks
contain subgraphs made of both strong,
reciprocal ties that lead to clustering and
weaker ties that result in bridging.
Each of the five different network gen-
res in our sample also exhibits a relatively
unique structural fingerprint of directed
tetrad patterns. On the basis of tetrads
alone, key differences arise between the
various types of social networks, and these
variations indicate that the structures of
hierarchy, clustering, and bridging are
not always uniform across the types of
social groups. For instance, the bifan tetrad
(Tetrad 19), which indicates a rigid hierar-
chy without clustering, is likely to occur in
some networks but not others. The bifan is
most frequent in our Twitter networks,
which is unsurprising because it is unlikely
that members of such a large online com-
munity would have as many opportunities
to interact, even if they are located on the
same level of the status hierarchy.
Additionally, we find that the highly
clustered and completely connected four-
clique appears more frequently than
expected across all of the networks in our
sample, except for the Twitter networks.
The overrepresentation of the four-clique
is unsurprising since, compared to other
tetrad configurations, decision making in
a completely connected four-clique is more
likely to result in a groupwide consensus
(Friedkin 1986), and such agreement is
apt to enhance the subgroup’s stability.
Within the Twitter networks, completely
connected four-cliques were generally non-
existent in both the observed and compara-
ble random networks, which is likely the
result of the networks’ low tie densities
and perhaps the specific nature of these
Twitter interactions.
Furthermore, the symmetric box tet-
rad (Tetrad 203) occurs less frequently
than expected in all network genres, except
for those consisting of high school friend-
ship ties. This type of local structural con-
figurationappearstobeavoidedinseveral
types of social interactions, not only in ado-
lescent sexual relationships (Bearman et
al. 2004). The symmetric box tetrad is
likely to occur only when structural forces
and social norms produce barriers to the
formation of cliques and clustering. For
instance, we expect that the symmetric
box tetrad arises in our sample of friend-
ship networks because these relational
webs are embedded in high schools defined
by explicit grade levels. While certain
friendships cross grade levels (e.g., friend-
ships between members of clubs or teams),
norms and the lack of opportunities for
cross-grade interactions likely prevent
those connections from developing into
fully connected four-person cliques.
Several implications emerge from our
work that highlight the importance of
studying microlevel interactions within
groups of four people. To begin with, the
empirical study of directed tetrads can
be intimidating due to the large sample
size of possible tetrads. Here, we identify
six tetrad motifs that are common, rela-
tive to comparable random graphs, across
a sample of 20 observed networks, and
these motifs could provide guidance as
to which four-person subgraphs are espe-
cially worthy of further investigation.
These motif findings could be applied to
inform which tetrad measures to include
in multivariate statistical network mod-
els as controls for important lower-level
graph properties. If researchers are inter-
ested in studying variations across
diverse types of social interaction, on the
other hand, a promising avenue for
research could focus on those tetrads that
differ across network genres (e.g., the
bifan). In addition, the fact that patterns
of directed tetrads suggest that there could
be a largely unique fingerprint for each
type of network genre highlights the impor-
tance of this unit of study; perhaps tetrads
398 Social Psychology Quarterly 83(4)
can be used more generally to identify gen-
res of social interaction and to uncover fun-
damental group-level processes.
Second, status remains a key concept
of interest in past and present social sci-
ence inquiry (e.g., Berger et al. 1972;
Ridgeway 2014; Weber 1968), and here
we see the ways in which directed tetrads
facilitate the analysis of status differen-
ces in small groups. The previous work
that considers relational patterns among
tetrads almost always assumes that social
ties are undirected or symmetric (e.g.,
Krumov et al. 2011). Our findings sug-
gest that study designs involving four-
person groups could benefit from the use
of directed tetrads in order to gain addi-
tional information regarding the intrica-
cies of status and power processes. More-
over, the directed tetrad motifs identified
in our analyses are not theoretical
arrangements of connections among four
people. They represent those recurring
empirical patterns of relationship ties
forged in 20 social networks across five
variations of behavior. Future experimen-
tal research should consider how status
processes manifest in these particular
small-group configurations, given their
basis in empirical reality.
The results presented here also aid in
accounting for the occasionally puzzling
patterns that occur at lower levels of net-
work structure, such as the presence of
unreciprocated dyads and intransitive tri-
ads in friendship networks. Although
reciprocated dyads are most common in
our data, asymmetric dyads also occur.
One explanation for their presence is
that they represent instances where
mutuality is expected to develop over
time, a possibility that becomes apparent
when examining the larger, four-person
context in which dyads are embedded.
Other asymmetric dyads are incorporated
into the hierarchical structures of tet-
rads, and these unreciprocated ties
appear to reinforce the stable status
systems ingrained in larger networks.
Moreover, intransitive triads appear
occasionally in our sample of networks,
despite the general tendency toward bal-
anced, transitive triads. The tetrad motif
that implies opportunities for social
mobility (e.g., Tetrad 214), for example,
includes two intransitive triads. Since
previous theory (e.g., Heider 1958; Cart-
wright and Harary 1956) and empirical
research (e.g., Bearman and Moody
2004; Hallinan and Kubitschek 1988)
highlight the negative aspects of intransi-
tivity, we suspect that actors do not enter
these configurations randomly. Instead,
they form intransitive triads only when
these patterns offer the potential for par-
ticularly rewarding benefits, such as
increased social status. This explanation
for intransitivity becomes apparent only
when considering the context of the tet-
rad that encapsulates the triads.
Furthermore, theoretical implications
arise from our results. In his classic treat-
ment of small groups, for example,
Homans (1950) argues that two funda-
mental processes occur simultaneously:
‘‘standardization,’’ in which conformity
norms emerge and group members
become more alike, and ‘‘differentiation,’’
or the development of a status hierarchy.
Tetrads represent perhaps the smallest of
groups in which we can detect evidence of
both processes. In a number of key tet-
rads, for instance, we see both the unfold-
ing of clustering, which may emerge from
conformity pressures, and the develop-
ment of directed ties that imply a ranked
hierarchy. Both formal and informal
types of social ties, as well as those that
represent face-to-face and online interac-
tion, reproduce miniature systems of
standardization and differentiation.
This research has a number of strengths
but also limitations. One shortcoming is
that even though our sample of social net-
works includes graphs that represent
five distinct genres of interpersonal
Tetrads in Social Networks 399
interaction, it does not represent all
types of social networks. In addition, we
use a nonrandom sample of specific net-
works from each genre of network, each
with its own limitations. Patterns of tet-
rad motifs could vary depending on the
specific networks in the data set being con-
sidered, and future work should analyze
the occurrence of tetrads in other social
network data. In addition, data collection
strategies vary across the different net-
works in our sample, and future research
will need to examine how these variations
shape tetrad patterns.
Finally, our conclusions hold only for
the null models applied here. While
using other types of control distributions is
apt to yield slightly different conclusions,
we believe there is value in taking an
ERGM-informed approach to generate
multivariate conditional distributions for
subgraph research. In the current project,
we condition our graphs on observed
tendencies toward reciprocity and transitiv-
ity, given their high prevalence in social
interaction (Davis 1970; Diekmann 2004;
Felmlee, McMillan, et al. 2018; Gouldner
1960), but our models do not account for
all possible lower-order network tendencies.
Future research on tetrad patterns could
benefit from using an ERGM approach to
measure the frequency of these four-actor
configurations while simultaneously con-
trolling for alternative triadic patterns.
In sum, intriguing microstructural
processes do not end with groups of three.
Configurations of connections among four
people also provide insight into key social
processes that unfold within larger net-
works, especially regarding the develop-
ment of status systems, cliques of mutual
ties, and weak, bridging connections. We
highlight several frequently occurring
four-actor subgraphs across our sample of
networks, and when taken together, these
tetrad motifs suggest that a variety of dif-
ferent social interaction types are defined
by patterns of hierarchy, clustering, and
bridging simultaneously. At the same
time, we find that there are distinct ways
that tetrad patterns vary across these
particular networks of friendship, co-
sponsorship, Twitter, advice, and email,
lending a unique structural signature to
each type. Our findings have implications
for those interested in social network struc-
ture as well as for those scholars studying
interaction and exchange in four-person
groups. They also highlight the utility of
comparing the structures of multiple social
networks representing various types of
social interaction. More generally, we see
here the utility of branching beyond the
dyad and triad to gain further traction on
understanding the intriguing patterns
that define structures of interpersonal ties.
AUTHORS’ NOTE
The views and conclusions contained in this doc-
ument are those of the authors and should not
be interpreted as representing the official poli-
cies, either expressed or implied, of the U.S.
Army Research Laboratory, the U.S. government,
the U.K. Ministry of Defense, or the U.K. govern-
ment. The U.S. and U.K. governments are autho-
rized to reproduce and distribute reprints for gov-
ernment purposes notwithstanding any copyright
notation hereon.
ACKNOWLEDGMENTS
We are grateful for suggestions and comments
from David R. Hunter on earlier versions of this
work. Special acknowledgment is due Ronald R.
Rindfuss and Barbara Entwisle for assistance in
the original design.
FUNDING
The author(s) disclosed receipt of the following
financial support for the research, authorship,
and/or publication of this article: Research
reported in this manuscript was supported by
the Penn State Population Research Institute of
the National Institutes of Health under award
number 2P2CHD041025. This work was also sup-
ported by Pennsylvania State University and the
National Science Foundation under IGERT
award number DGE-1144860, Big Data Social
Science. This research was sponsored in part by
400 Social Psychology Quarterly 83(4)
the U.S. Army Research Laboratory and the U.K.
Ministry of Defense under agreement number
W911NF-16-3-0001. This project also uses data
from Add Health, a program project designed by
J. Richard Udry, Peter S. Bearman, and Kathleen
Mullan Harris and funded by grant P01HD31921
from the Eunice Kennedy Shriver National Insti-
tute of Child Health and Human Development,
with cooperative funding from 17 other agencies.
No direct support was received from grant
P01HD31921 for this analysis.
ORCID iDs
Cassie McMillan https://orcid.org/0000-0002-
4108-8112
Diane Felmlee https://orcid.org/0000-0003-
4211-0489
SUPPLEMENTAL MATERIAL
Supplemental material for this article is available
online.
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BIOS
Cassie McMillan is an assistant profes-
sor of sociology, criminology, and criminal
justice at Northeastern University. Her
research applies a social networks per-
spective to study processes related to
inequality, adolescent delinquency, and
immigration. Her recent work appears
in Demography, Social Networks, and
Journal of Quantitative Criminology.
Diane Felmlee is a professor of sociology
and demography at Pennsylvania State
University. Her research focuses on social
networks, dynamic modeling, and the
social processes involved in the relation-
ship ties of friendship, peer aggression,
and online harassment. Recent publica-
tions have appeared in Sociology of Edu-
cation, PLoS ONE, Socius, Journal of
Quantitative Criminology, and Sex Roles.
404 Social Psychology Quarterly 83(4)
