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Title
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The Micro–Macro Link in Social Networks
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Author
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Stadtfeld, Christoph
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Research Area
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The Individual and Society
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Topic
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Social Networks
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Abstract
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Important questions in the social sciences are concerned with the link between micro‐level behavior and aggregate macro‐level outcomes. This essay proposes that studies of the micro–macro link in social systems can utilize conceptual representations and analytical strategies from the field of social networks. In particular, statistical network models and research strategies from agent‐based network modeling can be combined to investigate dynamics and the emergence of structure. An empirical case study illustrates how stochastic actor‐oriented models can be applied as empirically calibrated agent‐based simulations. The fruitfulness of this approach is demonstrated by a Schelling‐inspired case study on the emergence of segregation in social networks. It is shown that even individuals without homophilous preferences may find themselves in segregated structures due to the complex interaction of different network mechanisms. The example thereby illustrates how social networks can serve as a conceptual and analytical framework to study the micro–macro link in dynamic, interdependent, and multi‐mechanistic social systems.
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Identifier
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etrds0463
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extracted text
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The Micro–Macro Link in Social
Networks
Christoph Stadtfeld
Abstract
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Important questions in the social sciences are concerned with the link between
micro-level behavior and aggregate macro-level outcomes. This essay proposes
that studies of the micro–macro link in social systems can utilize conceptual
representations and analytical strategies from the field of social networks. In
particular, statistical network models and research strategies from agent-based
network modeling can be combined to investigate dynamics and the emergence
of structure. An empirical case study illustrates how stochastic actor-oriented
models can be applied as empirically calibrated agent-based simulations. The
fruitfulness of this approach is demonstrated by a Schelling-inspired case study on
the emergence of segregation in social networks. It is shown that even individuals
without homophilous preferences may find themselves in segregated structures due
to the complex interaction of different network mechanisms. The example thereby
illustrates how social networks can serve as a conceptual and analytical framework
to study the micro–macro link in dynamic, interdependent, and multi-mechanistic
social systems.
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INTRODUCTION
The observation that societies are more than the sum of their parts, and
that complex phenomena can emerge from the actions of many interdependent social actors is a central motif of Sociology, ranging from Émile
Durkheim’s work (Sawyer, 2002) to recent developments in the field of
analytical sociology (Hedström & Bearman, 2009). Many aspects of complex
social systems can be well represented by social networks in which the
dynamically changing relations between social units give rise to processes
and structural outcomes that cannot merely be described by an aggregation
of the units’ desires, beliefs, and actions. Complex dependence between
individual and relational observations as found in social networks (Lusher,
Johan, & Garry, 2013; Robins, 2015) can, in fact, constrain the set of action
Emerging Trends in the Social and Behavioral Sciences.
Robert A. Scott and Marlis Buchmann (General Editors) with Stephen Kosslyn (Consulting Editor).
© 2018 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.
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opportunities of social actors, and social networks can affect their desires
and beliefs, for example, through processes of social influence or knowledge
diffusion (Friedkin, 1998). Social networks have been used to represent
societally relevant macro-level outcomes such as sub-groups, social distances, network segregation along attributes, or hierarchies (Robins, 2015).
At the same time, the behavior of individual actors in a network can be well
described with sociological, economic, and social-psychological micro-level
theories (Kadushin, 2012; Robins, 2015). I propose that the link between
those two levels, the micro-level of individual desires, beliefs, and actions
and the macro-level of complex network phenomena, can be explored by
applying newly developed statistical and computational techniques from
the field of social network analysis.
Section titled “A Micro–Macro Framework for Social Networks” explains
a conceptual and analytical framework for the study of micro–macro links
in social networks. It starts with Coleman’s micro–macro model and links
it to the dynamic, interdependent, and multi-mechanistic nature of social
networks. The discussed analytical strategy builds upon the seminal work
of Snijders and Steglich (2015) who propose the study of micro–macro
links in social networks through empirically calibrated simulation models.
Section titled “Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration” exemplarily shows how the full Coleman cycle can be explored with a combination of novel statistical and
computational network analysis techniques. I will illustrate how individuals without preferences for homophily may find themselves embedded
in homogeneous clusters of similar nodes. This case study is related to
Schelling’s model on neighborhood segregation (Schelling, 1978), but
uses the formulation of a multi-mechanistic network process in which the
dynamic relation between three interdependent mechanisms—homophily,
reciprocity, and transitivity—explains the emergent macro-level outcome of
network segregation. Section titled “Network Representations and Outlook”
broadens the scope of the paper by discussing how advanced network
representations could be utilized to address more complex micro–macro
problems and concludes with a discussion of promising future research
directions.
A MICRO–MACRO FRAMEWORK FOR SOCIAL NETWORKS
SOCIAL NETWORK DYNAMICS IN COLEMAN’S MODEL
Coleman’s (1990) model can serve as a starting point in an attempt to
develop a conceptual framework for the study of the micro–macro link
in social networks (Figure 1). The analytical goal is to explain how social
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Social network
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Emergence
Dependence
constraints
Aggregation
Individual
Desires
opportunities
beliefs
Actions
Figure 1 A social network perspective on Coleman’s micro–macro model.
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networks change through time and how macro-level phenomena emerge
(the dashed arrow). This is achieved by investigating the underlying
individual micro-level dynamics (the three solid arrows). First, by understanding how macro-level structures affect individuals’ desires, beliefs,
and opportunities (Hedström, 2005) and, in particular, how dependence
and opportunity constraints can limit and structure the set of possible
behavioral actions. Second, by understanding how individuals’ actions
are based on their desires, beliefs, and opportunities, for example, when
they decide to create or modify interpersonal relations, or when their
behavior is related to their current network position. Third, by understanding how the multi-mechanistic and interdependent actions of many
individuals aggregate and thereby shape social networks on the macro
level.
While Coleman’s framework is conceptually appealing, it is hard to study
on the whole. Empirical network studies, behavioral research studies in
small settings, and studies of theoretically motivated models are exemplary
approaches that have been utilized to examine parts of it. In observational
network studies, it is possible to examine how individuals’ actions—like
the creation and dissolution of network ties (the lower-right box)—are
associated with their structural network position (the upper-left box). The
desires, opportunities, and beliefs of individuals (the lower-left box) are
important in constructing meaningful theoretical frameworks but they often
remain unobserved. Detailed behavioral studies, for example, situated in
experimental lab settings, can aim at understanding the causal link between
individuals’ desires, opportunities, and constraints and their relational
and individual actions (the lower solid arrow). Owing to the typically
limited scale of experimental studies it is, however, hard to represent
network dependence and network constraints. Theoretical studies can, for
example, by using mathematical formulations or simulation frameworks,
aim at establishing the link between the actions of many interdependent
individuals and aggregate macro-level observations. However, due to the
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complexity of network processes is difficult to calibrate the theoretical
empirically.
THE DYNAMIC, INTERDEPENDENT, AND MULTI-MECHANISTIC NATURE OF SOCIAL NETWORKS
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Social networks are dynamic systems that are constructed and changed by the
social actions of a potentially large number of actors. They formally consist
of two types of entities. First, one or more sets of nodes, representing social
actors (e.g., individuals or organizations) or nonsocial entities (e.g., social
foci, social settings, affiliations, or the internal state of social actors). Second,
one or more sets of relations that connect pairs of nodes (e.g., kinship between
individuals, collaboration between organizations, or individuals’ affiliation
with a specific social setting). Social networks are flexible in how they represent social actors and can express their behavior, desires, or beliefs either as
node attributes or as network affiliations.
Social networks are a way to describe dependence between individuals’ relation and actions explicitly and to investigate how dependence between social
actors can enable emergent phenomena. The study of dependence is indeed
at the core of social network analysis (Robins, 2015). Within a dyad (a pair of
nodes) dependence may arise because of mutual exchange processes (Emerson, 1976). Dependence between more than two nodes can originate from
transitive processes in which two nodes are more likely to connect if they link
to the same third node. Transitivity can be explained, for example, by cognitive balance mechanisms (Heider, 1958). Dependence that involves more
than three nodes can, for example, be related to degree popularity mechanisms (Merton, 1968) or to in-group cohesion and intergroup conflict (Tajfel
& Turner, 1979). Social network researchers have discussed the translations
of a variety of theoretically motivated micro-level mechanisms into mathematical representations of network dependence (Lusher et al., 2013; Robins,
2015; Snijders, 2017).
Importantly, none of these dependence mechanisms in social networks
operates in isolation, but typically, multiple mechanisms operate simultaneously within one network and jointly affect the actions of social
actors. This multi-mechanistic nature of social network processes is one
possible explanation of complex emergent phenomena (as illustrated in
section titled Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration). State-of-the-art statistical network models have sophisticated ways of expressing dependence in
multi-mechanistic systems (Robins, 2015). Some of these models have
been developed to be fit to dynamic data explicitly (Snijders, 2017; Stadtfeld, Hollway, & Block, 2017), however, also network models that are
applied to cross-sectional data typically assume that the static network
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observations emerge from dynamic network processes (Lusher et al.,
2013).
INTEGRATING TWO APPROACHES
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Social networks research so far mostly takes one of two approaches to studying the link between the micro and the macro level. The first approach aims
at inferring micro-level models from empirically observed (macro-level)
network data. The most prominent methods in the field are exponential random graph models and stochastic actor-oriented models (SAOMs) (Lusher
et al., 2013; Snijders, 2017). Block, Stadtfeld, and Snijders (2016) discuss their
similarities and differences. The second approach aims at understanding
how theoretically grounded micro-level mechanism that describe actors’ (or
agents’) behaviors, strategies, and actions lead to the emergent macro-level
phenomena through agent-based simulation models (Bianchi & Squazzoni,
2015). A related research area is work by computational scientists and
physicists who explain widely observed network level structures with
straightforward micro-level models (e.g., small world patterns, Watts &
Strogatz, 1998).
Statistical network models tend to be more complex than agent-based
models in terms of the number of parameters and mechanisms that are
considered simultaneously. At the same time, they often have more rigorous assumptions than agent-based models. For example, micro-level
interpretations of exponential random graph models and SAOMs tend to
think of agents as myopic and assume that individuals with equivalent
attributes and network position will follow the same behavioral patterns.
Agent-based network models, in comparison, often aim at expressing
strategic (forward-looking) behavior and actor heterogeneity. A promising
approach to integrating the two research traditions is to develop computational network simulation models that are empirically calibrated (as
discussed by Hedström & Åberg, 2005) while acknowledging the dynamic,
interdependent, and multi-mechanistic nature of social networks. Such
models could capitalize on solutions to problems discovered in the statistical
network literature that relate to near-degeneracy of simulation models
(Snijders, Pattison, Robins, & Handcock, 2006), and extend findings on the
multi-mechanistic nature of social networks explored in a variety of empirical studies. At the same time, empirically calibrated models can be extended
to express, for example, strategic considerations and actor heterogeneity.
The proposed combination of statistical and computational models is in line
with paradigms discussed in the field analytical sociology (Coleman, 1990;
Hedström & Bearman, 2009). The application of such models could open
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new insights into the emergence of macro-level phenomena in complex
social systems.
SCHELLING’S MODEL OF SEGREGATION AS A MULTI-MECHANISTIC
NETWORK MODEL – AN ILLUSTRATION
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I illustrate how to investigate the micro–macro link with a straightforward
example that is inspired by Thomas Schelling’s foundational model of residential segregation (Schelling, 1978). In one formulation of Schelling’s model,
agents (imagined as coins of two colors) are placed on the 64 squares of a
checkerboard, but move their position if they are unsatisfied with the composition of their neighborhood. The neighborhood of an agent i is defined by
the coins positioned on the (max. eight) squares that are adjacent to its current square. Coins are satisfied with their position if the ratio of neighbors
of the other color is below a threshold ?. If nodes collectively act according to these rules, they will typically find themselves in neighborhoods in
which the ratio of differently colored nodes is eventually much lower than
?—the segregation that emerges on the macro-level thus that cannot merely
be explained with the individuals’ preferences. I translate Schelling’s model
into a multi-mechanistic network model in which the size and the structure
of each agents’ neighborhood is modeled with endogenous network mechanisms. Other than in Schelling’s original formulation I allow that some of
the actors have no preference for homophily at all—their emerging neighborhoods will be of specific interest.
In a first analytical step, I fit a SAOM (Snijders, 2017) to data of friendship
relations collected among school children. The resulting micro-level model
is cross-sectional rather than longitudinal, thus describes the underlying actor-oriented processes under the assumption that the empirically
observed network is drawn from the model’s stationary distribution
(Snijders & Steglich, 2015).
In a second analytical step, I use the micro-level model as an agent-based
simulation model (Snijders & Steglich, 2015) to investigate the emergence
of network segregation. In the following, I use the term homophily only for
the micro-level preferences of individual actors, while network segregation
refers to the (macro-level) ratio of network relations that connect nodes
of the same attribute. By randomly varying the preferences of actor types
within the agent-based simulation, I illustrate how individuals’ position in
a network depends on the preferences of others. In particular, I show that
even actors who have no preference for homophily will find themselves
positioned in networks where the majority of their network neighbors is of
their type. This outcome will be observed because actors are assumed to have
additional preferences—they prefer maintaining ties that are embedded
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intransitive and reciprocal structures. This illustrates the importance of
considering the multi-mechanistic nature of social networks.
INFERRING MICRO-MECHANISMS FROM EMPIRICAL DATA
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The empirical starting point is a single friendship network collected in the
“Wired into Each Other” study by Károly Takács and colleagues (Pál, Stadtfeld, Grow, & Takács, 2016; Vörös & Snijders, 2017). It is shown in Figure 2.
The data are fit to a cross-sectional SAOM. These models conceive of
network structure as drawn from a stationary distribution of an underlying
network process in which individuals (the “actors”) create and dissolve
network ties according to a set of shared preferences. The model that I fit
is specified with four parameters—a relatively small number compared to
many empirical network studies. The first parameter is concerned with the
number of times that individuals have to others. It indicates that individuals
with many friendship ties will be more likely to drop one rather than
creating an additional tie. This “outdegree” effect can be linked to the fact
that network ties are often costly to maintain. It is important to include
in SAOM specifications and is often understood as the model intercept
(Snijders, 2017). The second parameter expresses that individuals will be
more likely to create and maintain mutual network ties rather than one-sided
relationships. The “reciprocity” mechanism is a fundamental mechanism
in social networks (Emerson, 1976). The third parameter is concerned with
Figure 2 An empirically observed friendship network with a high proportion of
in-group (same-gender) relations.
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individuals’ preference to be embedded intransitive structures, meaning that
if individuals A–B are friends and B–C are friends then A and C will have a
higher tendency to be friends as well. This “transitivity” mechanism in social
networks is again considered important in many contexts and can be linked
to some of the most prominent sociological and social-psychological theories within social networks research (Heider, 1958; Feld, 1981; Granovetter,
1973). Transitivity is operationalized in terms of “geometrically weighted
edge-wise shared partners” (Snijders et al., 2006), indicating that individuals
have a preference to maintain ties in transitive structures, but the number
of shared friends (the individuals B in the example above) has a sub-linear
effect on the tendency of A and C to connect. The fourth parameter models
the preference to form ties to individuals with the same attribute—gender in
the empirical example. “Homophily” is considered an important social force
in many social networks as well (McPherson, Smith-Lovin, & Cook, 2001). A
“rate parameter” (Snijders, 2017) is not estimated in cross-sectional SAOMs
as the model is a stationary distribution and the rate is infinite in theory
(and set to a very high value in practice). Modern specifications of SAOMs
typically include a number of additional effects that, for example, also
consider additional attributes, degree-related effects, different triadic effects,
or interactions between different mechanisms (Block, 2018; Snijders, 2017).
Estimation results are shown in Figure 3. The outdegree parameter is
negative, indicating that the overall network density is low, while reciprocity, transitivity, and gender homophily have a positive effect on the
creation and maintenance of friendship ties. The fit of the model is good
in terms of the structures that are explicitly included—the micro model
generates networks with the correct density, reciprocity, transitivity, and
level of gender segregation. The fit in terms of other macro-level statistics,
such as degree distributions, or specific types of triads will be unsatisfying,
but can be corrected by considering additional network mechanisms. The
straightforward model specification is, however, purposeful in this illustration, as it is comparable to Schelling’s model of residential segregation. Both
Outdegree (1)
Reciprocity (2)
Transitivity (GWESP) (3)
Homophily (gender) (4)
−2
0
2
Figure 3 Results from the stochastic actor-oriented model with a straightforward
four-parameter specification. Point estimates and 95% confidence intervals are
shown.
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models express preferences for similarity in neighborhoods and personal
networks, respectively. The network models explicitly consider individuals’
preferences to maintain a positive number of ties (i.e., having neighbors
at all in Schelling’s view), that are reciprocal and embedded in transitive
structures. The three mechanisms together define and structure individuals’
personal networks—their neighborhoods. These endogenous network
mechanisms serve a similar purpose to the exogenous space constraints on
Schelling’s checkerboard that enforce the formation of neighborhoods in the
first place.
STUDYING MACRO-STRUCTURES FROM MICRO-MODELS
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In the next step, the empirically estimated model is utilized as an agent-based
simulation model. However, I vary the percentage of nodes with homophily
preferences—this extension deviates from the assumption of the SAOM
that nodes with the same attributes and network positions will express the
same behavior. To be able to generate different preference compositions
with meaningful subsets, the model is applied to a larger network of 200
nodes (100 males, 100 females), rather than simulating networks as small
as the empirical network in Figure 2 (33 nodes). However, in principle, the
findings can be replicated with a network of the original size. The estimated
parameters from Figure 3 remain unchanged. Only for the nodes without
homophily preference, the homophily parameter is set to zero so that
only outdegree, reciprocity, and transitivity matter for these individuals’
relational actions.
Figure 4 shows prototypical networks simulated from the empirical model.
The percentage of nodes with a preference for gender homophily is 10% (20
(a) 10%
(b) 50%
(c) 90%
Figure 4 Networks simulated from the empirically calibrated micro-model on a
set of 200 nodes. The percentage of nodes with preference for gender homophily
is 10% (20 nodes), 50% (100 nodes), and 90% (180 nodes).
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Percentage of ties to similar nodes...
0.9
0.8
...Of nodes with homophily preference
...Of nodes without homophily preference
0.7
0.6
0.5
0.25
0.50
0.75
Proportion of nodes with homophily preference in the network
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Figure 5 The more nodes with a homophily preference in the network (x-axis),
the larger is the share of ties to nodes of the same type (y-axis). This holds both for
the nodes with a homophily preference (blue line) and the nodes without
homophily preference (red line). Values were estimated from 200 simulations each.
nodes), 50% (100 nodes), and 90% (180 nodes) in the three panels. Nodes with
homophily preference are indicated as circles, those without as squares. It is
evident that the more nodes with homophily preference are in the network,
the higher is the level of network segregation. The last of the three networks is
indeed highly segregated and it appears that also nodes without homophily
preferences (the squares) are mostly linked to nodes of their own color.
I now investigate the level of homogeneity in the personal networks of
homophilous and non-homophilous nodes. Results are shown in Figure 5.
The more homophilous nodes are in the network, the more homogeneous
will their personal networks be—this is the case both for homophilous (blue
line) and non-homophilous nodes (red line). The reason for this phenomenon
of unintended individual consequences lies in the multi-mechanistic social
network process. All individuals strive for reciprocal and transitive relations.
For a small minority of non-homophilous nodes, these needs will be easiest
to satisfy when they “accept” reciprocal connections to others of the same
color or embed themselves in dense clusters with homophilous nodes of the
same color. The level of homogeneity in individuals’ personal networks thus
does not only depend on their own preferences, but also on the preference of
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others in the network. This effect also operates in the reverse direction: The
more non-homophilous nodes are in the networks, the more diverse will the
networks of homophilous nodes be.
The study illustrates how the level of network segregation is partly
explained by the amplifying effect of transitivity on homophily (Stadtfeld &
Pentland, 2015). The complex interaction of these fundamental micro-level
network processes has been investigated in empirical network studies
(Block, 2018; Stadtfeld & Pentland, 2015; Goodreau, Kitts, & Morris, 2009),
but to my knowledge not in agent-based simulation studies.
NETWORK REPRESENTATIONS AND OUTLOOK
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I believe that the combination of statistical and computational network analysis techniques can open new insights into the dynamic, interdependent, and
multi-mechanistic nature of social networks. Many societal phenomena can
be expressed with social networks, and recently proposed network representations have the potential to importantly extend the scope of micro–macro
network studies in the social sciences.
Recent publications have emphasized how multivariate, two-mode,
and weighted network representations can be linked to detailed network
mechanisms. Multivariate and weighted networks can be used to express
the co-evolution between positive and negative relationships (Labianca &
Brass, 2006; Pál et al., 2016), different strength of network ties (Elmer, Boda,
& Stadtfeld, 2017), or the associations between networks of different types
(Boda, 2018; Lazega & Pattison, 1999). Two-mode network can describe
the affiliations of social actors to nonsocial entities. Such representations
are very powerful, as they allow to explicitly model actors’ preferences,
beliefs, activities, social foci, internal structures, or affiliation with social
settings (Snijders, Lomi, & Torlo, 2013; Stadtfeld, Mascia, Pallotti, & Lomi,
2016). Statistical network models have further been developed to express
the coevolution of individual outcomes and network change (Niezink &
Snijders, 2017; Steglich, Snijders, & Michael, 2010). Thereby, they can connect
to agent-based simulation studies that explore processes of polarization or
social influence (Mäs, Flache, Takács, & Jehn, 2013).
In this essay, I discussed how the conceptual micro–macro model of Coleman can be utilized in the study of social networks that are characterized by
their dynamic, interdependent, and multi-mechanistic nature. I proposed
to integrate recent advances in statistical network models and agent-based
simulations. The application of this approach was illustrated in a straightforward case study on the emergence of segregation in social networks.
Other macro-level outcomes can be explored similarly. For example, how
individual behavior relates to the emergence of sub-groups, hierarchical
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status systems, social distances, or to the polarization of political opinions.
This essay provides a conceptual framework to approach such questions on
the micro–macro links in social networks.
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Christoph Stadtfeld is an assistant professor of Social Networks at ETH
Zürich, Switzerland. He develops methods for the statistical analysis of
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dynamic social network data and publishes theoretical-empirical work on
network dynamics in different subfields of sociology. Both research lines
have been featured in leading sociological and methodological journals
(e.g., Sociological Methodology, Sociological Methods & Research, Social
Networks, Social Forces). To make the methodological work accessible to
the applied social networks community, Christoph Stadtfeld develops and
contributes to scientific software packages.
RELATED ESSAYS
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AIDS and Social Networks (Sociology), Alexander Weinreb et al.
Domestic Institutions and International Conflict (Political Science), Giacomo
Chiozza
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
Architecture of Markets (Sociology), Neil Fligstein and Ryan Calder
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Migrant Networks (Sociology), Filiz Garip and Asad L. Asad
Interdependence, Development, and Interstate Conflict (Political Science),
Erik Gartzke
Herd Behavior (Psychology), Tatsuya Kameda and Reid Hastie
How Networks Form: Homophily, Opportunity, and Balance (Sociology),
Kevin Lewis
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
Culture, Diffusion, and Networks in Social Animals (Anthropology), Janet
Mann and Lisa Singh
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
How Do Labor Market Networks Work? (Sociology), Brian Rubineau and
Roberto M. Fernandez
The Role of Social Mechanisms in the Formation of Social Inequalities
(Sociology),Martin Diewald
Social Network Analysis in the Study of Ethnic Inequalities (Psychology),
Frank Kalter
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Diffusion: From Facebook to (Management) Fashion (Sociology), David
Strang and Kelly Patterson
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The Micro–Macro Link in Social
Networks
Christoph Stadtfeld
Abstract
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Important questions in the social sciences are concerned with the link between
micro-level behavior and aggregate macro-level outcomes. This essay proposes
that studies of the micro–macro link in social systems can utilize conceptual
representations and analytical strategies from the field of social networks. In
particular, statistical network models and research strategies from agent-based
network modeling can be combined to investigate dynamics and the emergence
of structure. An empirical case study illustrates how stochastic actor-oriented
models can be applied as empirically calibrated agent-based simulations. The
fruitfulness of this approach is demonstrated by a Schelling-inspired case study on
the emergence of segregation in social networks. It is shown that even individuals
without homophilous preferences may find themselves in segregated structures due
to the complex interaction of different network mechanisms. The example thereby
illustrates how social networks can serve as a conceptual and analytical framework
to study the micro–macro link in dynamic, interdependent, and multi-mechanistic
social systems.
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INTRODUCTION
The observation that societies are more than the sum of their parts, and
that complex phenomena can emerge from the actions of many interdependent social actors is a central motif of Sociology, ranging from Émile
Durkheim’s work (Sawyer, 2002) to recent developments in the field of
analytical sociology (Hedström & Bearman, 2009). Many aspects of complex
social systems can be well represented by social networks in which the
dynamically changing relations between social units give rise to processes
and structural outcomes that cannot merely be described by an aggregation
of the units’ desires, beliefs, and actions. Complex dependence between
individual and relational observations as found in social networks (Lusher,
Johan, & Garry, 2013; Robins, 2015) can, in fact, constrain the set of action
Emerging Trends in the Social and Behavioral Sciences.
Robert A. Scott and Marlis Buchmann (General Editors) with Stephen Kosslyn (Consulting Editor).
© 2018 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.
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opportunities of social actors, and social networks can affect their desires
and beliefs, for example, through processes of social influence or knowledge
diffusion (Friedkin, 1998). Social networks have been used to represent
societally relevant macro-level outcomes such as sub-groups, social distances, network segregation along attributes, or hierarchies (Robins, 2015).
At the same time, the behavior of individual actors in a network can be well
described with sociological, economic, and social-psychological micro-level
theories (Kadushin, 2012; Robins, 2015). I propose that the link between
those two levels, the micro-level of individual desires, beliefs, and actions
and the macro-level of complex network phenomena, can be explored by
applying newly developed statistical and computational techniques from
the field of social network analysis.
Section titled “A Micro–Macro Framework for Social Networks” explains
a conceptual and analytical framework for the study of micro–macro links
in social networks. It starts with Coleman’s micro–macro model and links
it to the dynamic, interdependent, and multi-mechanistic nature of social
networks. The discussed analytical strategy builds upon the seminal work
of Snijders and Steglich (2015) who propose the study of micro–macro
links in social networks through empirically calibrated simulation models.
Section titled “Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration” exemplarily shows how the full Coleman cycle can be explored with a combination of novel statistical and
computational network analysis techniques. I will illustrate how individuals without preferences for homophily may find themselves embedded
in homogeneous clusters of similar nodes. This case study is related to
Schelling’s model on neighborhood segregation (Schelling, 1978), but
uses the formulation of a multi-mechanistic network process in which the
dynamic relation between three interdependent mechanisms—homophily,
reciprocity, and transitivity—explains the emergent macro-level outcome of
network segregation. Section titled “Network Representations and Outlook”
broadens the scope of the paper by discussing how advanced network
representations could be utilized to address more complex micro–macro
problems and concludes with a discussion of promising future research
directions.
A MICRO–MACRO FRAMEWORK FOR SOCIAL NETWORKS
SOCIAL NETWORK DYNAMICS IN COLEMAN’S MODEL
Coleman’s (1990) model can serve as a starting point in an attempt to
develop a conceptual framework for the study of the micro–macro link
in social networks (Figure 1). The analytical goal is to explain how social
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Emergence
Dependence
constraints
Aggregation
Individual
Desires
opportunities
beliefs
Actions
Figure 1 A social network perspective on Coleman’s micro–macro model.
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networks change through time and how macro-level phenomena emerge
(the dashed arrow). This is achieved by investigating the underlying
individual micro-level dynamics (the three solid arrows). First, by understanding how macro-level structures affect individuals’ desires, beliefs,
and opportunities (Hedström, 2005) and, in particular, how dependence
and opportunity constraints can limit and structure the set of possible
behavioral actions. Second, by understanding how individuals’ actions
are based on their desires, beliefs, and opportunities, for example, when
they decide to create or modify interpersonal relations, or when their
behavior is related to their current network position. Third, by understanding how the multi-mechanistic and interdependent actions of many
individuals aggregate and thereby shape social networks on the macro
level.
While Coleman’s framework is conceptually appealing, it is hard to study
on the whole. Empirical network studies, behavioral research studies in
small settings, and studies of theoretically motivated models are exemplary
approaches that have been utilized to examine parts of it. In observational
network studies, it is possible to examine how individuals’ actions—like
the creation and dissolution of network ties (the lower-right box)—are
associated with their structural network position (the upper-left box). The
desires, opportunities, and beliefs of individuals (the lower-left box) are
important in constructing meaningful theoretical frameworks but they often
remain unobserved. Detailed behavioral studies, for example, situated in
experimental lab settings, can aim at understanding the causal link between
individuals’ desires, opportunities, and constraints and their relational
and individual actions (the lower solid arrow). Owing to the typically
limited scale of experimental studies it is, however, hard to represent
network dependence and network constraints. Theoretical studies can, for
example, by using mathematical formulations or simulation frameworks,
aim at establishing the link between the actions of many interdependent
individuals and aggregate macro-level observations. However, due to the
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complexity of network processes is difficult to calibrate the theoretical
empirically.
THE DYNAMIC, INTERDEPENDENT, AND MULTI-MECHANISTIC NATURE OF SOCIAL NETWORKS
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Social networks are dynamic systems that are constructed and changed by the
social actions of a potentially large number of actors. They formally consist
of two types of entities. First, one or more sets of nodes, representing social
actors (e.g., individuals or organizations) or nonsocial entities (e.g., social
foci, social settings, affiliations, or the internal state of social actors). Second,
one or more sets of relations that connect pairs of nodes (e.g., kinship between
individuals, collaboration between organizations, or individuals’ affiliation
with a specific social setting). Social networks are flexible in how they represent social actors and can express their behavior, desires, or beliefs either as
node attributes or as network affiliations.
Social networks are a way to describe dependence between individuals’ relation and actions explicitly and to investigate how dependence between social
actors can enable emergent phenomena. The study of dependence is indeed
at the core of social network analysis (Robins, 2015). Within a dyad (a pair of
nodes) dependence may arise because of mutual exchange processes (Emerson, 1976). Dependence between more than two nodes can originate from
transitive processes in which two nodes are more likely to connect if they link
to the same third node. Transitivity can be explained, for example, by cognitive balance mechanisms (Heider, 1958). Dependence that involves more
than three nodes can, for example, be related to degree popularity mechanisms (Merton, 1968) or to in-group cohesion and intergroup conflict (Tajfel
& Turner, 1979). Social network researchers have discussed the translations
of a variety of theoretically motivated micro-level mechanisms into mathematical representations of network dependence (Lusher et al., 2013; Robins,
2015; Snijders, 2017).
Importantly, none of these dependence mechanisms in social networks
operates in isolation, but typically, multiple mechanisms operate simultaneously within one network and jointly affect the actions of social
actors. This multi-mechanistic nature of social network processes is one
possible explanation of complex emergent phenomena (as illustrated in
section titled Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration). State-of-the-art statistical network models have sophisticated ways of expressing dependence in
multi-mechanistic systems (Robins, 2015). Some of these models have
been developed to be fit to dynamic data explicitly (Snijders, 2017; Stadtfeld, Hollway, & Block, 2017), however, also network models that are
applied to cross-sectional data typically assume that the static network
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observations emerge from dynamic network processes (Lusher et al.,
2013).
INTEGRATING TWO APPROACHES
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Social networks research so far mostly takes one of two approaches to studying the link between the micro and the macro level. The first approach aims
at inferring micro-level models from empirically observed (macro-level)
network data. The most prominent methods in the field are exponential random graph models and stochastic actor-oriented models (SAOMs) (Lusher
et al., 2013; Snijders, 2017). Block, Stadtfeld, and Snijders (2016) discuss their
similarities and differences. The second approach aims at understanding
how theoretically grounded micro-level mechanism that describe actors’ (or
agents’) behaviors, strategies, and actions lead to the emergent macro-level
phenomena through agent-based simulation models (Bianchi & Squazzoni,
2015). A related research area is work by computational scientists and
physicists who explain widely observed network level structures with
straightforward micro-level models (e.g., small world patterns, Watts &
Strogatz, 1998).
Statistical network models tend to be more complex than agent-based
models in terms of the number of parameters and mechanisms that are
considered simultaneously. At the same time, they often have more rigorous assumptions than agent-based models. For example, micro-level
interpretations of exponential random graph models and SAOMs tend to
think of agents as myopic and assume that individuals with equivalent
attributes and network position will follow the same behavioral patterns.
Agent-based network models, in comparison, often aim at expressing
strategic (forward-looking) behavior and actor heterogeneity. A promising
approach to integrating the two research traditions is to develop computational network simulation models that are empirically calibrated (as
discussed by Hedström & Åberg, 2005) while acknowledging the dynamic,
interdependent, and multi-mechanistic nature of social networks. Such
models could capitalize on solutions to problems discovered in the statistical
network literature that relate to near-degeneracy of simulation models
(Snijders, Pattison, Robins, & Handcock, 2006), and extend findings on the
multi-mechanistic nature of social networks explored in a variety of empirical studies. At the same time, empirically calibrated models can be extended
to express, for example, strategic considerations and actor heterogeneity.
The proposed combination of statistical and computational models is in line
with paradigms discussed in the field analytical sociology (Coleman, 1990;
Hedström & Bearman, 2009). The application of such models could open
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new insights into the emergence of macro-level phenomena in complex
social systems.
SCHELLING’S MODEL OF SEGREGATION AS A MULTI-MECHANISTIC
NETWORK MODEL – AN ILLUSTRATION
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I illustrate how to investigate the micro–macro link with a straightforward
example that is inspired by Thomas Schelling’s foundational model of residential segregation (Schelling, 1978). In one formulation of Schelling’s model,
agents (imagined as coins of two colors) are placed on the 64 squares of a
checkerboard, but move their position if they are unsatisfied with the composition of their neighborhood. The neighborhood of an agent i is defined by
the coins positioned on the (max. eight) squares that are adjacent to its current square. Coins are satisfied with their position if the ratio of neighbors
of the other color is below a threshold ?. If nodes collectively act according to these rules, they will typically find themselves in neighborhoods in
which the ratio of differently colored nodes is eventually much lower than
?—the segregation that emerges on the macro-level thus that cannot merely
be explained with the individuals’ preferences. I translate Schelling’s model
into a multi-mechanistic network model in which the size and the structure
of each agents’ neighborhood is modeled with endogenous network mechanisms. Other than in Schelling’s original formulation I allow that some of
the actors have no preference for homophily at all—their emerging neighborhoods will be of specific interest.
In a first analytical step, I fit a SAOM (Snijders, 2017) to data of friendship
relations collected among school children. The resulting micro-level model
is cross-sectional rather than longitudinal, thus describes the underlying actor-oriented processes under the assumption that the empirically
observed network is drawn from the model’s stationary distribution
(Snijders & Steglich, 2015).
In a second analytical step, I use the micro-level model as an agent-based
simulation model (Snijders & Steglich, 2015) to investigate the emergence
of network segregation. In the following, I use the term homophily only for
the micro-level preferences of individual actors, while network segregation
refers to the (macro-level) ratio of network relations that connect nodes
of the same attribute. By randomly varying the preferences of actor types
within the agent-based simulation, I illustrate how individuals’ position in
a network depends on the preferences of others. In particular, I show that
even actors who have no preference for homophily will find themselves
positioned in networks where the majority of their network neighbors is of
their type. This outcome will be observed because actors are assumed to have
additional preferences—they prefer maintaining ties that are embedded
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intransitive and reciprocal structures. This illustrates the importance of
considering the multi-mechanistic nature of social networks.
INFERRING MICRO-MECHANISMS FROM EMPIRICAL DATA
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The empirical starting point is a single friendship network collected in the
“Wired into Each Other” study by Károly Takács and colleagues (Pál, Stadtfeld, Grow, & Takács, 2016; Vörös & Snijders, 2017). It is shown in Figure 2.
The data are fit to a cross-sectional SAOM. These models conceive of
network structure as drawn from a stationary distribution of an underlying
network process in which individuals (the “actors”) create and dissolve
network ties according to a set of shared preferences. The model that I fit
is specified with four parameters—a relatively small number compared to
many empirical network studies. The first parameter is concerned with the
number of times that individuals have to others. It indicates that individuals
with many friendship ties will be more likely to drop one rather than
creating an additional tie. This “outdegree” effect can be linked to the fact
that network ties are often costly to maintain. It is important to include
in SAOM specifications and is often understood as the model intercept
(Snijders, 2017). The second parameter expresses that individuals will be
more likely to create and maintain mutual network ties rather than one-sided
relationships. The “reciprocity” mechanism is a fundamental mechanism
in social networks (Emerson, 1976). The third parameter is concerned with
Figure 2 An empirically observed friendship network with a high proportion of
in-group (same-gender) relations.
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individuals’ preference to be embedded intransitive structures, meaning that
if individuals A–B are friends and B–C are friends then A and C will have a
higher tendency to be friends as well. This “transitivity” mechanism in social
networks is again considered important in many contexts and can be linked
to some of the most prominent sociological and social-psychological theories within social networks research (Heider, 1958; Feld, 1981; Granovetter,
1973). Transitivity is operationalized in terms of “geometrically weighted
edge-wise shared partners” (Snijders et al., 2006), indicating that individuals
have a preference to maintain ties in transitive structures, but the number
of shared friends (the individuals B in the example above) has a sub-linear
effect on the tendency of A and C to connect. The fourth parameter models
the preference to form ties to individuals with the same attribute—gender in
the empirical example. “Homophily” is considered an important social force
in many social networks as well (McPherson, Smith-Lovin, & Cook, 2001). A
“rate parameter” (Snijders, 2017) is not estimated in cross-sectional SAOMs
as the model is a stationary distribution and the rate is infinite in theory
(and set to a very high value in practice). Modern specifications of SAOMs
typically include a number of additional effects that, for example, also
consider additional attributes, degree-related effects, different triadic effects,
or interactions between different mechanisms (Block, 2018; Snijders, 2017).
Estimation results are shown in Figure 3. The outdegree parameter is
negative, indicating that the overall network density is low, while reciprocity, transitivity, and gender homophily have a positive effect on the
creation and maintenance of friendship ties. The fit of the model is good
in terms of the structures that are explicitly included—the micro model
generates networks with the correct density, reciprocity, transitivity, and
level of gender segregation. The fit in terms of other macro-level statistics,
such as degree distributions, or specific types of triads will be unsatisfying,
but can be corrected by considering additional network mechanisms. The
straightforward model specification is, however, purposeful in this illustration, as it is comparable to Schelling’s model of residential segregation. Both
Outdegree (1)
Reciprocity (2)
Transitivity (GWESP) (3)
Homophily (gender) (4)
−2
0
2
Figure 3 Results from the stochastic actor-oriented model with a straightforward
four-parameter specification. Point estimates and 95% confidence intervals are
shown.
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models express preferences for similarity in neighborhoods and personal
networks, respectively. The network models explicitly consider individuals’
preferences to maintain a positive number of ties (i.e., having neighbors
at all in Schelling’s view), that are reciprocal and embedded in transitive
structures. The three mechanisms together define and structure individuals’
personal networks—their neighborhoods. These endogenous network
mechanisms serve a similar purpose to the exogenous space constraints on
Schelling’s checkerboard that enforce the formation of neighborhoods in the
first place.
STUDYING MACRO-STRUCTURES FROM MICRO-MODELS
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In the next step, the empirically estimated model is utilized as an agent-based
simulation model. However, I vary the percentage of nodes with homophily
preferences—this extension deviates from the assumption of the SAOM
that nodes with the same attributes and network positions will express the
same behavior. To be able to generate different preference compositions
with meaningful subsets, the model is applied to a larger network of 200
nodes (100 males, 100 females), rather than simulating networks as small
as the empirical network in Figure 2 (33 nodes). However, in principle, the
findings can be replicated with a network of the original size. The estimated
parameters from Figure 3 remain unchanged. Only for the nodes without
homophily preference, the homophily parameter is set to zero so that
only outdegree, reciprocity, and transitivity matter for these individuals’
relational actions.
Figure 4 shows prototypical networks simulated from the empirical model.
The percentage of nodes with a preference for gender homophily is 10% (20
(a) 10%
(b) 50%
(c) 90%
Figure 4 Networks simulated from the empirically calibrated micro-model on a
set of 200 nodes. The percentage of nodes with preference for gender homophily
is 10% (20 nodes), 50% (100 nodes), and 90% (180 nodes).
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Percentage of ties to similar nodes...
0.9
0.8
...Of nodes with homophily preference
...Of nodes without homophily preference
0.7
0.6
0.5
0.25
0.50
0.75
Proportion of nodes with homophily preference in the network
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Figure 5 The more nodes with a homophily preference in the network (x-axis),
the larger is the share of ties to nodes of the same type (y-axis). This holds both for
the nodes with a homophily preference (blue line) and the nodes without
homophily preference (red line). Values were estimated from 200 simulations each.
nodes), 50% (100 nodes), and 90% (180 nodes) in the three panels. Nodes with
homophily preference are indicated as circles, those without as squares. It is
evident that the more nodes with homophily preference are in the network,
the higher is the level of network segregation. The last of the three networks is
indeed highly segregated and it appears that also nodes without homophily
preferences (the squares) are mostly linked to nodes of their own color.
I now investigate the level of homogeneity in the personal networks of
homophilous and non-homophilous nodes. Results are shown in Figure 5.
The more homophilous nodes are in the network, the more homogeneous
will their personal networks be—this is the case both for homophilous (blue
line) and non-homophilous nodes (red line). The reason for this phenomenon
of unintended individual consequences lies in the multi-mechanistic social
network process. All individuals strive for reciprocal and transitive relations.
For a small minority of non-homophilous nodes, these needs will be easiest
to satisfy when they “accept” reciprocal connections to others of the same
color or embed themselves in dense clusters with homophilous nodes of the
same color. The level of homogeneity in individuals’ personal networks thus
does not only depend on their own preferences, but also on the preference of
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others in the network. This effect also operates in the reverse direction: The
more non-homophilous nodes are in the networks, the more diverse will the
networks of homophilous nodes be.
The study illustrates how the level of network segregation is partly
explained by the amplifying effect of transitivity on homophily (Stadtfeld &
Pentland, 2015). The complex interaction of these fundamental micro-level
network processes has been investigated in empirical network studies
(Block, 2018; Stadtfeld & Pentland, 2015; Goodreau, Kitts, & Morris, 2009),
but to my knowledge not in agent-based simulation studies.
NETWORK REPRESENTATIONS AND OUTLOOK
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I believe that the combination of statistical and computational network analysis techniques can open new insights into the dynamic, interdependent, and
multi-mechanistic nature of social networks. Many societal phenomena can
be expressed with social networks, and recently proposed network representations have the potential to importantly extend the scope of micro–macro
network studies in the social sciences.
Recent publications have emphasized how multivariate, two-mode,
and weighted network representations can be linked to detailed network
mechanisms. Multivariate and weighted networks can be used to express
the co-evolution between positive and negative relationships (Labianca &
Brass, 2006; Pál et al., 2016), different strength of network ties (Elmer, Boda,
& Stadtfeld, 2017), or the associations between networks of different types
(Boda, 2018; Lazega & Pattison, 1999). Two-mode network can describe
the affiliations of social actors to nonsocial entities. Such representations
are very powerful, as they allow to explicitly model actors’ preferences,
beliefs, activities, social foci, internal structures, or affiliation with social
settings (Snijders, Lomi, & Torlo, 2013; Stadtfeld, Mascia, Pallotti, & Lomi,
2016). Statistical network models have further been developed to express
the coevolution of individual outcomes and network change (Niezink &
Snijders, 2017; Steglich, Snijders, & Michael, 2010). Thereby, they can connect
to agent-based simulation studies that explore processes of polarization or
social influence (Mäs, Flache, Takács, & Jehn, 2013).
In this essay, I discussed how the conceptual micro–macro model of Coleman can be utilized in the study of social networks that are characterized by
their dynamic, interdependent, and multi-mechanistic nature. I proposed
to integrate recent advances in statistical network models and agent-based
simulations. The application of this approach was illustrated in a straightforward case study on the emergence of segregation in social networks.
Other macro-level outcomes can be explored similarly. For example, how
individual behavior relates to the emergence of sub-groups, hierarchical
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status systems, social distances, or to the polarization of political opinions.
This essay provides a conceptual framework to approach such questions on
the micro–macro links in social networks.
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Stadtfeld, C., Hollway, J., & Block, P. (2017). Dynamic network actor models: Investigating coordination ties through time. Sociological Methodology, 47, 1–40.
Stadtfeld, C., Mascia, D., Pallotti, F., & Lomi, A. (2016). Assimilation and differentiation: A multilevel perspective on organizational and network change. Social
Networks, 44, 363–374.
Stadtfeld, C., & Pentland, A. (2015). Partnership ties shape friendship networks: A
dynamic social network study. Social Forces, 94, 453–477.
Steglich, C., Snijders, T. A. B., & Pearson, M. (2010). Dynamic networks and behavior:
Separating selection from influence. Sociological Methodology, 40, 329–393.
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Christoph Stadtfeld is an assistant professor of Social Networks at ETH
Zürich, Switzerland. He develops methods for the statistical analysis of
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dynamic social network data and publishes theoretical-empirical work on
network dynamics in different subfields of sociology. Both research lines
have been featured in leading sociological and methodological journals
(e.g., Sociological Methodology, Sociological Methods & Research, Social
Networks, Social Forces). To make the methodological work accessible to
the applied social networks community, Christoph Stadtfeld develops and
contributes to scientific software packages.
RELATED ESSAYS
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AIDS and Social Networks (Sociology), Alexander Weinreb et al.
Domestic Institutions and International Conflict (Political Science), Giacomo
Chiozza
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
Architecture of Markets (Sociology), Neil Fligstein and Ryan Calder
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Migrant Networks (Sociology), Filiz Garip and Asad L. Asad
Interdependence, Development, and Interstate Conflict (Political Science),
Erik Gartzke
Herd Behavior (Psychology), Tatsuya Kameda and Reid Hastie
How Networks Form: Homophily, Opportunity, and Balance (Sociology),
Kevin Lewis
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
Culture, Diffusion, and Networks in Social Animals (Anthropology), Janet
Mann and Lisa Singh
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
How Do Labor Market Networks Work? (Sociology), Brian Rubineau and
Roberto M. Fernandez
The Role of Social Mechanisms in the Formation of Social Inequalities
(Sociology),Martin Diewald
Social Network Analysis in the Study of Ethnic Inequalities (Psychology),
Frank Kalter
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Diffusion: From Facebook to (Management) Fashion (Sociology), David
Strang and Kelly Patterson
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The Micro–Macro Link in Social
Networks
Christoph Stadtfeld
Abstract
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Important questions in the social sciences are concerned with the link between
micro-level behavior and aggregate macro-level outcomes. This essay proposes
that studies of the micro–macro link in social systems can utilize conceptual
representations and analytical strategies from the field of social networks. In
particular, statistical network models and research strategies from agent-based
network modeling can be combined to investigate dynamics and the emergence
of structure. An empirical case study illustrates how stochastic actor-oriented
models can be applied as empirically calibrated agent-based simulations. The
fruitfulness of this approach is demonstrated by a Schelling-inspired case study on
the emergence of segregation in social networks. It is shown that even individuals
without homophilous preferences may find themselves in segregated structures due
to the complex interaction of different network mechanisms. The example thereby
illustrates how social networks can serve as a conceptual and analytical framework
to study the micro–macro link in dynamic, interdependent, and multi-mechanistic
social systems.
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INTRODUCTION
The observation that societies are more than the sum of their parts, and
that complex phenomena can emerge from the actions of many interdependent social actors is a central motif of Sociology, ranging from Émile
Durkheim’s work (Sawyer, 2002) to recent developments in the field of
analytical sociology (Hedström & Bearman, 2009). Many aspects of complex
social systems can be well represented by social networks in which the
dynamically changing relations between social units give rise to processes
and structural outcomes that cannot merely be described by an aggregation
of the units’ desires, beliefs, and actions. Complex dependence between
individual and relational observations as found in social networks (Lusher,
Johan, & Garry, 2013; Robins, 2015) can, in fact, constrain the set of action
Emerging Trends in the Social and Behavioral Sciences.
Robert A. Scott and Marlis Buchmann (General Editors) with Stephen Kosslyn (Consulting Editor).
© 2018 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.
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opportunities of social actors, and social networks can affect their desires
and beliefs, for example, through processes of social influence or knowledge
diffusion (Friedkin, 1998). Social networks have been used to represent
societally relevant macro-level outcomes such as sub-groups, social distances, network segregation along attributes, or hierarchies (Robins, 2015).
At the same time, the behavior of individual actors in a network can be well
described with sociological, economic, and social-psychological micro-level
theories (Kadushin, 2012; Robins, 2015). I propose that the link between
those two levels, the micro-level of individual desires, beliefs, and actions
and the macro-level of complex network phenomena, can be explored by
applying newly developed statistical and computational techniques from
the field of social network analysis.
Section titled “A Micro–Macro Framework for Social Networks” explains
a conceptual and analytical framework for the study of micro–macro links
in social networks. It starts with Coleman’s micro–macro model and links
it to the dynamic, interdependent, and multi-mechanistic nature of social
networks. The discussed analytical strategy builds upon the seminal work
of Snijders and Steglich (2015) who propose the study of micro–macro
links in social networks through empirically calibrated simulation models.
Section titled “Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration” exemplarily shows how the full Coleman cycle can be explored with a combination of novel statistical and
computational network analysis techniques. I will illustrate how individuals without preferences for homophily may find themselves embedded
in homogeneous clusters of similar nodes. This case study is related to
Schelling’s model on neighborhood segregation (Schelling, 1978), but
uses the formulation of a multi-mechanistic network process in which the
dynamic relation between three interdependent mechanisms—homophily,
reciprocity, and transitivity—explains the emergent macro-level outcome of
network segregation. Section titled “Network Representations and Outlook”
broadens the scope of the paper by discussing how advanced network
representations could be utilized to address more complex micro–macro
problems and concludes with a discussion of promising future research
directions.
A MICRO–MACRO FRAMEWORK FOR SOCIAL NETWORKS
SOCIAL NETWORK DYNAMICS IN COLEMAN’S MODEL
Coleman’s (1990) model can serve as a starting point in an attempt to
develop a conceptual framework for the study of the micro–macro link
in social networks (Figure 1). The analytical goal is to explain how social
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Emergence
Dependence
constraints
Aggregation
Individual
Desires
opportunities
beliefs
Actions
Figure 1 A social network perspective on Coleman’s micro–macro model.
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networks change through time and how macro-level phenomena emerge
(the dashed arrow). This is achieved by investigating the underlying
individual micro-level dynamics (the three solid arrows). First, by understanding how macro-level structures affect individuals’ desires, beliefs,
and opportunities (Hedström, 2005) and, in particular, how dependence
and opportunity constraints can limit and structure the set of possible
behavioral actions. Second, by understanding how individuals’ actions
are based on their desires, beliefs, and opportunities, for example, when
they decide to create or modify interpersonal relations, or when their
behavior is related to their current network position. Third, by understanding how the multi-mechanistic and interdependent actions of many
individuals aggregate and thereby shape social networks on the macro
level.
While Coleman’s framework is conceptually appealing, it is hard to study
on the whole. Empirical network studies, behavioral research studies in
small settings, and studies of theoretically motivated models are exemplary
approaches that have been utilized to examine parts of it. In observational
network studies, it is possible to examine how individuals’ actions—like
the creation and dissolution of network ties (the lower-right box)—are
associated with their structural network position (the upper-left box). The
desires, opportunities, and beliefs of individuals (the lower-left box) are
important in constructing meaningful theoretical frameworks but they often
remain unobserved. Detailed behavioral studies, for example, situated in
experimental lab settings, can aim at understanding the causal link between
individuals’ desires, opportunities, and constraints and their relational
and individual actions (the lower solid arrow). Owing to the typically
limited scale of experimental studies it is, however, hard to represent
network dependence and network constraints. Theoretical studies can, for
example, by using mathematical formulations or simulation frameworks,
aim at establishing the link between the actions of many interdependent
individuals and aggregate macro-level observations. However, due to the
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complexity of network processes is difficult to calibrate the theoretical
empirically.
THE DYNAMIC, INTERDEPENDENT, AND MULTI-MECHANISTIC NATURE OF SOCIAL NETWORKS
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Social networks are dynamic systems that are constructed and changed by the
social actions of a potentially large number of actors. They formally consist
of two types of entities. First, one or more sets of nodes, representing social
actors (e.g., individuals or organizations) or nonsocial entities (e.g., social
foci, social settings, affiliations, or the internal state of social actors). Second,
one or more sets of relations that connect pairs of nodes (e.g., kinship between
individuals, collaboration between organizations, or individuals’ affiliation
with a specific social setting). Social networks are flexible in how they represent social actors and can express their behavior, desires, or beliefs either as
node attributes or as network affiliations.
Social networks are a way to describe dependence between individuals’ relation and actions explicitly and to investigate how dependence between social
actors can enable emergent phenomena. The study of dependence is indeed
at the core of social network analysis (Robins, 2015). Within a dyad (a pair of
nodes) dependence may arise because of mutual exchange processes (Emerson, 1976). Dependence between more than two nodes can originate from
transitive processes in which two nodes are more likely to connect if they link
to the same third node. Transitivity can be explained, for example, by cognitive balance mechanisms (Heider, 1958). Dependence that involves more
than three nodes can, for example, be related to degree popularity mechanisms (Merton, 1968) or to in-group cohesion and intergroup conflict (Tajfel
& Turner, 1979). Social network researchers have discussed the translations
of a variety of theoretically motivated micro-level mechanisms into mathematical representations of network dependence (Lusher et al., 2013; Robins,
2015; Snijders, 2017).
Importantly, none of these dependence mechanisms in social networks
operates in isolation, but typically, multiple mechanisms operate simultaneously within one network and jointly affect the actions of social
actors. This multi-mechanistic nature of social network processes is one
possible explanation of complex emergent phenomena (as illustrated in
section titled Schelling’s Model of Segregation as A Multi-Mechanistic
Network Model – An Illustration). State-of-the-art statistical network models have sophisticated ways of expressing dependence in
multi-mechanistic systems (Robins, 2015). Some of these models have
been developed to be fit to dynamic data explicitly (Snijders, 2017; Stadtfeld, Hollway, & Block, 2017), however, also network models that are
applied to cross-sectional data typically assume that the static network
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observations emerge from dynamic network processes (Lusher et al.,
2013).
INTEGRATING TWO APPROACHES
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Social networks research so far mostly takes one of two approaches to studying the link between the micro and the macro level. The first approach aims
at inferring micro-level models from empirically observed (macro-level)
network data. The most prominent methods in the field are exponential random graph models and stochastic actor-oriented models (SAOMs) (Lusher
et al., 2013; Snijders, 2017). Block, Stadtfeld, and Snijders (2016) discuss their
similarities and differences. The second approach aims at understanding
how theoretically grounded micro-level mechanism that describe actors’ (or
agents’) behaviors, strategies, and actions lead to the emergent macro-level
phenomena through agent-based simulation models (Bianchi & Squazzoni,
2015). A related research area is work by computational scientists and
physicists who explain widely observed network level structures with
straightforward micro-level models (e.g., small world patterns, Watts &
Strogatz, 1998).
Statistical network models tend to be more complex than agent-based
models in terms of the number of parameters and mechanisms that are
considered simultaneously. At the same time, they often have more rigorous assumptions than agent-based models. For example, micro-level
interpretations of exponential random graph models and SAOMs tend to
think of agents as myopic and assume that individuals with equivalent
attributes and network position will follow the same behavioral patterns.
Agent-based network models, in comparison, often aim at expressing
strategic (forward-looking) behavior and actor heterogeneity. A promising
approach to integrating the two research traditions is to develop computational network simulation models that are empirically calibrated (as
discussed by Hedström & Åberg, 2005) while acknowledging the dynamic,
interdependent, and multi-mechanistic nature of social networks. Such
models could capitalize on solutions to problems discovered in the statistical
network literature that relate to near-degeneracy of simulation models
(Snijders, Pattison, Robins, & Handcock, 2006), and extend findings on the
multi-mechanistic nature of social networks explored in a variety of empirical studies. At the same time, empirically calibrated models can be extended
to express, for example, strategic considerations and actor heterogeneity.
The proposed combination of statistical and computational models is in line
with paradigms discussed in the field analytical sociology (Coleman, 1990;
Hedström & Bearman, 2009). The application of such models could open
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new insights into the emergence of macro-level phenomena in complex
social systems.
SCHELLING’S MODEL OF SEGREGATION AS A MULTI-MECHANISTIC
NETWORK MODEL – AN ILLUSTRATION
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I illustrate how to investigate the micro–macro link with a straightforward
example that is inspired by Thomas Schelling’s foundational model of residential segregation (Schelling, 1978). In one formulation of Schelling’s model,
agents (imagined as coins of two colors) are placed on the 64 squares of a
checkerboard, but move their position if they are unsatisfied with the composition of their neighborhood. The neighborhood of an agent i is defined by
the coins positioned on the (max. eight) squares that are adjacent to its current square. Coins are satisfied with their position if the ratio of neighbors
of the other color is below a threshold 𝜃. If nodes collectively act according to these rules, they will typically find themselves in neighborhoods in
which the ratio of differently colored nodes is eventually much lower than
𝜃—the segregation that emerges on the macro-level thus that cannot merely
be explained with the individuals’ preferences. I translate Schelling’s model
into a multi-mechanistic network model in which the size and the structure
of each agents’ neighborhood is modeled with endogenous network mechanisms. Other than in Schelling’s original formulation I allow that some of
the actors have no preference for homophily at all—their emerging neighborhoods will be of specific interest.
In a first analytical step, I fit a SAOM (Snijders, 2017) to data of friendship
relations collected among school children. The resulting micro-level model
is cross-sectional rather than longitudinal, thus describes the underlying actor-oriented processes under the assumption that the empirically
observed network is drawn from the model’s stationary distribution
(Snijders & Steglich, 2015).
In a second analytical step, I use the micro-level model as an agent-based
simulation model (Snijders & Steglich, 2015) to investigate the emergence
of network segregation. In the following, I use the term homophily only for
the micro-level preferences of individual actors, while network segregation
refers to the (macro-level) ratio of network relations that connect nodes
of the same attribute. By randomly varying the preferences of actor types
within the agent-based simulation, I illustrate how individuals’ position in
a network depends on the preferences of others. In particular, I show that
even actors who have no preference for homophily will find themselves
positioned in networks where the majority of their network neighbors is of
their type. This outcome will be observed because actors are assumed to have
additional preferences—they prefer maintaining ties that are embedded
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intransitive and reciprocal structures. This illustrates the importance of
considering the multi-mechanistic nature of social networks.
INFERRING MICRO-MECHANISMS FROM EMPIRICAL DATA
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The empirical starting point is a single friendship network collected in the
“Wired into Each Other” study by Károly Takács and colleagues (Pál, Stadtfeld, Grow, & Takács, 2016; Vörös & Snijders, 2017). It is shown in Figure 2.
The data are fit to a cross-sectional SAOM. These models conceive of
network structure as drawn from a stationary distribution of an underlying
network process in which individuals (the “actors”) create and dissolve
network ties according to a set of shared preferences. The model that I fit
is specified with four parameters—a relatively small number compared to
many empirical network studies. The first parameter is concerned with the
number of times that individuals have to others. It indicates that individuals
with many friendship ties will be more likely to drop one rather than
creating an additional tie. This “outdegree” effect can be linked to the fact
that network ties are often costly to maintain. It is important to include
in SAOM specifications and is often understood as the model intercept
(Snijders, 2017). The second parameter expresses that individuals will be
more likely to create and maintain mutual network ties rather than one-sided
relationships. The “reciprocity” mechanism is a fundamental mechanism
in social networks (Emerson, 1976). The third parameter is concerned with
Figure 2 An empirically observed friendship network with a high proportion of
in-group (same-gender) relations.
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individuals’ preference to be embedded intransitive structures, meaning that
if individuals A–B are friends and B–C are friends then A and C will have a
higher tendency to be friends as well. This “transitivity” mechanism in social
networks is again considered important in many contexts and can be linked
to some of the most prominent sociological and social-psychological theories within social networks research (Heider, 1958; Feld, 1981; Granovetter,
1973). Transitivity is operationalized in terms of “geometrically weighted
edge-wise shared partners” (Snijders et al., 2006), indicating that individuals
have a preference to maintain ties in transitive structures, but the number
of shared friends (the individuals B in the example above) has a sub-linear
effect on the tendency of A and C to connect. The fourth parameter models
the preference to form ties to individuals with the same attribute—gender in
the empirical example. “Homophily” is considered an important social force
in many social networks as well (McPherson, Smith-Lovin, & Cook, 2001). A
“rate parameter” (Snijders, 2017) is not estimated in cross-sectional SAOMs
as the model is a stationary distribution and the rate is infinite in theory
(and set to a very high value in practice). Modern specifications of SAOMs
typically include a number of additional effects that, for example, also
consider additional attributes, degree-related effects, different triadic effects,
or interactions between different mechanisms (Block, 2018; Snijders, 2017).
Estimation results are shown in Figure 3. The outdegree parameter is
negative, indicating that the overall network density is low, while reciprocity, transitivity, and gender homophily have a positive effect on the
creation and maintenance of friendship ties. The fit of the model is good
in terms of the structures that are explicitly included—the micro model
generates networks with the correct density, reciprocity, transitivity, and
level of gender segregation. The fit in terms of other macro-level statistics,
such as degree distributions, or specific types of triads will be unsatisfying,
but can be corrected by considering additional network mechanisms. The
straightforward model specification is, however, purposeful in this illustration, as it is comparable to Schelling’s model of residential segregation. Both
Outdegree (1)
Reciprocity (2)
Transitivity (GWESP) (3)
Homophily (gender) (4)
−2
0
2
Figure 3 Results from the stochastic actor-oriented model with a straightforward
four-parameter specification. Point estimates and 95% confidence intervals are
shown.
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models express preferences for similarity in neighborhoods and personal
networks, respectively. The network models explicitly consider individuals’
preferences to maintain a positive number of ties (i.e., having neighbors
at all in Schelling’s view), that are reciprocal and embedded in transitive
structures. The three mechanisms together define and structure individuals’
personal networks—their neighborhoods. These endogenous network
mechanisms serve a similar purpose to the exogenous space constraints on
Schelling’s checkerboard that enforce the formation of neighborhoods in the
first place.
STUDYING MACRO-STRUCTURES FROM MICRO-MODELS
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In the next step, the empirically estimated model is utilized as an agent-based
simulation model. However, I vary the percentage of nodes with homophily
preferences—this extension deviates from the assumption of the SAOM
that nodes with the same attributes and network positions will express the
same behavior. To be able to generate different preference compositions
with meaningful subsets, the model is applied to a larger network of 200
nodes (100 males, 100 females), rather than simulating networks as small
as the empirical network in Figure 2 (33 nodes). However, in principle, the
findings can be replicated with a network of the original size. The estimated
parameters from Figure 3 remain unchanged. Only for the nodes without
homophily preference, the homophily parameter is set to zero so that
only outdegree, reciprocity, and transitivity matter for these individuals’
relational actions.
Figure 4 shows prototypical networks simulated from the empirical model.
The percentage of nodes with a preference for gender homophily is 10% (20
(a) 10%
(b) 50%
(c) 90%
Figure 4 Networks simulated from the empirically calibrated micro-model on a
set of 200 nodes. The percentage of nodes with preference for gender homophily
is 10% (20 nodes), 50% (100 nodes), and 90% (180 nodes).
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Percentage of ties to similar nodes...
0.9
0.8
...Of nodes with homophily preference
...Of nodes without homophily preference
0.7
0.6
0.5
0.25
0.50
0.75
Proportion of nodes with homophily preference in the network
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Figure 5 The more nodes with a homophily preference in the network (x-axis),
the larger is the share of ties to nodes of the same type (y-axis). This holds both for
the nodes with a homophily preference (blue line) and the nodes without
homophily preference (red line). Values were estimated from 200 simulations each.
nodes), 50% (100 nodes), and 90% (180 nodes) in the three panels. Nodes with
homophily preference are indicated as circles, those without as squares. It is
evident that the more nodes with homophily preference are in the network,
the higher is the level of network segregation. The last of the three networks is
indeed highly segregated and it appears that also nodes without homophily
preferences (the squares) are mostly linked to nodes of their own color.
I now investigate the level of homogeneity in the personal networks of
homophilous and non-homophilous nodes. Results are shown in Figure 5.
The more homophilous nodes are in the network, the more homogeneous
will their personal networks be—this is the case both for homophilous (blue
line) and non-homophilous nodes (red line). The reason for this phenomenon
of unintended individual consequences lies in the multi-mechanistic social
network process. All individuals strive for reciprocal and transitive relations.
For a small minority of non-homophilous nodes, these needs will be easiest
to satisfy when they “accept” reciprocal connections to others of the same
color or embed themselves in dense clusters with homophilous nodes of the
same color. The level of homogeneity in individuals’ personal networks thus
does not only depend on their own preferences, but also on the preference of
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others in the network. This effect also operates in the reverse direction: The
more non-homophilous nodes are in the networks, the more diverse will the
networks of homophilous nodes be.
The study illustrates how the level of network segregation is partly
explained by the amplifying effect of transitivity on homophily (Stadtfeld &
Pentland, 2015). The complex interaction of these fundamental micro-level
network processes has been investigated in empirical network studies
(Block, 2018; Stadtfeld & Pentland, 2015; Goodreau, Kitts, & Morris, 2009),
but to my knowledge not in agent-based simulation studies.
NETWORK REPRESENTATIONS AND OUTLOOK
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I believe that the combination of statistical and computational network analysis techniques can open new insights into the dynamic, interdependent, and
multi-mechanistic nature of social networks. Many societal phenomena can
be expressed with social networks, and recently proposed network representations have the potential to importantly extend the scope of micro–macro
network studies in the social sciences.
Recent publications have emphasized how multivariate, two-mode,
and weighted network representations can be linked to detailed network
mechanisms. Multivariate and weighted networks can be used to express
the co-evolution between positive and negative relationships (Labianca &
Brass, 2006; Pál et al., 2016), different strength of network ties (Elmer, Boda,
& Stadtfeld, 2017), or the associations between networks of different types
(Boda, 2018; Lazega & Pattison, 1999). Two-mode network can describe
the affiliations of social actors to nonsocial entities. Such representations
are very powerful, as they allow to explicitly model actors’ preferences,
beliefs, activities, social foci, internal structures, or affiliation with social
settings (Snijders, Lomi, & Torlo, 2013; Stadtfeld, Mascia, Pallotti, & Lomi,
2016). Statistical network models have further been developed to express
the coevolution of individual outcomes and network change (Niezink &
Snijders, 2017; Steglich, Snijders, & Michael, 2010). Thereby, they can connect
to agent-based simulation studies that explore processes of polarization or
social influence (Mäs, Flache, Takács, & Jehn, 2013).
In this essay, I discussed how the conceptual micro–macro model of Coleman can be utilized in the study of social networks that are characterized by
their dynamic, interdependent, and multi-mechanistic nature. I proposed
to integrate recent advances in statistical network models and agent-based
simulations. The application of this approach was illustrated in a straightforward case study on the emergence of segregation in social networks.
Other macro-level outcomes can be explored similarly. For example, how
individual behavior relates to the emergence of sub-groups, hierarchical
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status systems, social distances, or to the polarization of political opinions.
This essay provides a conceptual framework to approach such questions on
the micro–macro links in social networks.
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Christoph Stadtfeld is an assistant professor of Social Networks at ETH
Zürich, Switzerland. He develops methods for the statistical analysis of
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dynamic social network data and publishes theoretical-empirical work on
network dynamics in different subfields of sociology. Both research lines
have been featured in leading sociological and methodological journals
(e.g., Sociological Methodology, Sociological Methods & Research, Social
Networks, Social Forces). To make the methodological work accessible to
the applied social networks community, Christoph Stadtfeld develops and
contributes to scientific software packages.
RELATED ESSAYS
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AIDS and Social Networks (Sociology), Alexander Weinreb et al.
Domestic Institutions and International Conflict (Political Science), Giacomo
Chiozza
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
Architecture of Markets (Sociology), Neil Fligstein and Ryan Calder
Emerging Trends in Social Network Analysis of Terrorism and Counterterrorism (Sociology), David Knoke
Problems Attract Problems: A Network Perspective on Mental Disorders
(Psychology), Angélique Cramer and Denny Borsboom
Migrant Networks (Sociology), Filiz Garip and Asad L. Asad
Interdependence, Development, and Interstate Conflict (Political Science),
Erik Gartzke
Herd Behavior (Psychology), Tatsuya Kameda and Reid Hastie
How Networks Form: Homophily, Opportunity, and Balance (Sociology),
Kevin Lewis
Network Research Experiments (Methods), Allen L. Linton and Betsy Sinclair
Culture, Diffusion, and Networks in Social Animals (Anthropology), Janet
Mann and Lisa Singh
The Role of School-Related Peers and Social Networks in Human Development (Political Science), Chandra Muller
How Do Labor Market Networks Work? (Sociology), Brian Rubineau and
Roberto M. Fernandez
The Role of Social Mechanisms in the Formation of Social Inequalities
(Sociology),Martin Diewald
Social Network Analysis in the Study of Ethnic Inequalities (Psychology),
Frank Kalter
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Diffusion: From Facebook to (Management) Fashion (Sociology), David
Strang and Kelly Patterson
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