Statistical Models
for Social NetworksWritten assignment
Tom A.B. Snijders
Department of Politics and Department of Statistics, Nuffield College, University of
Oxford, Oxford OX1 1NF, United Kingdom; Faculty of Behavioural and Social Sciences,
University of Groningen, 9700 AB Groningen, The Netherlands;
email: [email protected], [email protected]

Annu. Rev. Sociol. 2011. 37:131–53 First published online as a Review in Advance on April 26, 2011 The Annual Review of Sociology is online at soc.annualreviews.org This article’s doi: 10.1146/annurev.soc.012809.102709 Copyright c 2011 by Annual Reviews. All rights reserved 0360-0572/11/0811-0131$20.00

Keywords social networks, statistical modeling, inference

Abstract Statistical models for social networks as dependent variables must rep resent the typical network dependencies between tie variables such as reciprocity, homophily, transitivity, etc. This review first treats models for single (cross-sectionally observed) networks and then for network dynamics. For single networks, the older literature concentrated on con ditionally uniform models. Various types of latent space models have

been developed: for discrete, general metric, ultrametric, Euclidean,
and partially ordered spaces. Exponential random graph models were
proposed long ago but now are applied more and more thanks to the
non-Markovian social circuit specifications that were recently proposed.
Modeling network dynamics is less complicated than modeling single
network observations because dependencies are spread out in time. For
modeling network dynamics, continuous-time models are more fruitful.
Actor-oriented models here provide a model that can represent many
dependencies in a flexible way. Strong model development is now going
on to combine the features of these models and to extend them to more
complicated outcome spaces.
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INTRODUCTION
Social network analysis is a branch of social science that seems for a long time to have resisted
the integration of empirical research with
statistical modeling that has been so pervasive,
and fruitful, in other branches. This is perhaps
not surprising in view of the nature of social
networks. Networks are relational structures,
and social networks represent structures of
dyadic ties between social actors: Examples
are friendship between individuals, alliances
between firms, or trade between countries.
The nature of networks leads to dependence
between actors and also to dependence between
network ties. By contrast, statistical modeling
is normally based on assumptions of independence. The complicated nature of network
dependencies has delayed the development of
statistical models for network structures.
This article is concerned with statistical
models for networks as outcome variables,
with a focus on models relevant for social
networks. This is an area undergoing vigorous
development and intimately embedded in a
larger domain, which makes it impossible to
approximate a complete overview in a limited
number of pages. Some neighboring topics that
are not covered are models for the coevolution
of networks and individual outcomes; models
that are not probabilistic in nature, or for which
methods of statistical inference (such as estimation and testing of parameters) have not been
developed; and event networks. The last section
provides some pointers to other literature.
This review is concerned with the models,
not with the statistical methods for estimating
and testing parameters, assessing goodness of
fit, etc. The cited literature contains the details
of the statistical procedures necessary for applying these models in practice.
Notation
A social network is a structure of ties, or relational variables, between social actors. We consider mainly a fixed set {1, . . . , n} of actors and
variables Xij representing how actor i is tied to
actor j. In some cases these will be directed in
nature, so that Xij and Xji are different variables
that may assume the same or different values;
in other cases they will be nondirectional in nature, so that Xij is necessarily equal to Xji. The
most frequently employed and most strongly
developed data structure is for binary variables
X
ij, where the value 1 (or 0) represents that
there is (or there is not) a tie from i to j. Then
the mathematical object constituted by the set
{1, . . . , n} and the variables Xij is called a graph
in the nondirected case and a digraph in the directed case. The actors are called the nodes and
the ties are usually called arcs or edges, depending on whether the graph is directed or not. It
is usual to exclude the possibility of self-ties, so
that the variables Xii may be considered to be
structural zeros.
The matrix with elements Xij is called the
adjacency matrix of the graph. The adjacency
matrix as well as the graph or digraph are denoted by X. Replacing an index by a plus sign
denotes summation over that index: Thus, the
number of outgoing ties of actor i, also called
the out-degree of i, is denoted Xi+ = j Xij,
and the in-degree, which is the number of incoming ties, is X+i = j Xji.
Network Dependencies
Social networks are characterized by a number
of dependencies that have been found empirically as well as theoretically.
1. Reciprocation of directed ties is a basic
feature of social networks (found previously by Moreno 1934). This will be reflected by dependencies between Xij and
X
ji. Theoretical accounts have been made
for it from the points of view of social exchange theory (Emerson 1972) and
game theory (Axelrod 1984). Reciprocation need not be confined to pairs, but it
can circulate in larger groups (see, e.g.,
Molm et al. 2007). This then can lead to
dependence in longer cycles such as Xij,
Xjh, Xhi.
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2. Homophily, the tendency of similar actors to relate to each other, was discussed and coined by Lazarsfeld &
Merton (1954) and has been the subject of
much research (reviewed by McPherson
et al. 2001). Theoretical arguments can
be based, e.g., on opportunity, affinity,
ease of communication, reduced transaction costs and break-off risks, and organizational foci (Feld 1982) composed of
similar individuals. This leads to a higher
probability of ties being formed between
actors with similar values on relevant
covariates.
3. Transitivity of ties is expressed by the saying “friends of my friends are my friends”
and was proposed as an essential element
of networks by Rapoport (1953a,b). Davis
(1970) found large-scale empirical support for transitivity in networks. Transitivity is rooted deeply in sociology, going back to authors such as Simmel (1950
[1917]) and elaborated more recently by
Coleman (1990). If there is a tendency
toward transitivity, the existence of the
two ties Xij = Xjh = 1 will lead to an increased probability of the tie Xih = 1, the
closure of the triangle. Concatenation of
such closure events then can also lead to
the existence of larger connected groups.
Transitivity therefore has also been called
clustering (Watts 1999).
A natural measure for the transitivity in
a graph is the number of transitive triangles,
i, j,h xijxjhxih (to be divided by 6 in the case of
nondirected graphs). A natural normalization
is to divide by the number of potentially closed
triads, as proposed by Frank (1980):
i, j,h xijxjhxih
i, j,h xijxjh . 1.
In directed graphs, transitivity can have two
faces: It may point to a hierarchical ordering
or to a clustered structure. These two can
be differentiated by the aid of the number
of 3-cycles, i, j,h xijxjhxhi. A relatively high
number of 3-cycles points toward clustering, a
relatively low number toward hierarchy. Davis
(1970) empirically found that social networks
tend to contain a relatively low number
of 3-cycles, indicating the pervasiveness of
hierarchies in social networks.
4. Degree differentials, some actors being
highly connected and others having few
connections, have been studied since the
late 1940s for communication networks
by Leavitt, Bavelas, and others. This work
led to models for node centrality (reviewed by Freeman 1979). An important
theoretical account was the rich-getricher phenomenon, or Matthew effect,
elaborated in the context of bibliographic
references by de Solla Price (1976). This
effect will lead to a high dispersion
of the nodal degrees, which then may
further lead to core-periphery structures
(Borgatti & Everett 1999) or various
other types of hierarchical structures.
5. Hierarchies in directed networks, as exhibited by high transitivity and few 3-
cycles, may be local or global. A global
hierarchy will be indicated by the ordering of the in-degrees and/or out-degrees,
where the typical pattern e.g., in esteem
or advice asking, is directed from low to
high. In a statistical model for a purely
global hierarchy, such as can be seen,
e.g., in some advice networks, the degree
differentials will be sufficient to explain
the low number of 3-cycles. But local
hierarchies are possible in directed networks even when the in-degrees and outdegrees exhibit little variability.
There are many other important types of
dependencies between ties in networks that are
not mentioned here because it would require
too much space. The literature contains various ways to represent network dependencies in
statistical models. Three broad approaches may
be distinguished.
Incorporating network structure through
covariates. A first approach is to employ
a model with independent residuals and to
try and represent network dependence in
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explanatory variables. To the extent that this is
feasible, it can be done mainly in longitudinal
settings where earlier observations of the
network can be used to produce covariates (as,
for example, in Gulati & Gargiulo 1999).
Controlling for network structure. A
second approach is to control for certain
aspects of network dependencies while not
explicitly modeling them. The best-known
example of this approach is a permutational
procedure, in which nodes in the network
are permuted—one may say also that the
rows and columns in the adjacency matrix are
permuted simultaneously in such a way that the
network structure is left intact. This is called
the quadratic assignment procedure (QAP)
approach, proposed by Krackhardt (1987) and
elaborated to permutation of regression residuals [multiple-regression QAP (MRQAP)] by
Krackhardt (1988) and Dekker et al. (2007).
Lindgren (2010) recently developed another method. He used the idea of a
heteroscedasticity-consistent estimator (also
known by the affectionate term of sandwich
variance estimator) for a covariance matrix elaborated by White (1980) and which has been
very fruitful for getting asymptotically correct
standard errors for clustered data. Lindgren
(2010) applied this idea to clustering in two
dimensions—rows as well as columns of the adjacency matrix—as is seen in network data. The
assumption then is that elements Xij and Xhk in
the adjacency matrix with {i, j}∩{h, k} = ∅ are
independent. Below I make a remark about this
assumption.
A third method in the second approach is
to condition on statistics that express network
dependencies. Within the set of networks
satisfying these constraints, the distribution
is assumed to be uniform, hence the term
conditionally uniform models. In other words,
standard deviations, p-values, etc., are calculated in a reference set of networks that
obey the same network dependencies as the
observed data. This method is explained in
Conditionally Uniform Models, below.
The MRQAP and heteroscedasticityconsistent approaches are useful when research
interest focuses exclusively on the effects of
explanatory variables (predictors, covariates)
and not on modeling the network as such or
on structural dependencies. Conditionally uniform models are useful to provide a statistical
control for a few relatively simple network
dependencies while testing more complicated
structural properties.
Modeling network structure. The third
approach is to model explicitly the structural
dependencies between tie variables. In contrast
to the traditionally well-known linear and generalized linear models, which are the backbone
of statistical modeling, such models need a
potentially considerable number of parameters
to express network structure (described below).
This approach requires a change of vantage
point for researchers because hypotheses may
have to be formulated in terms not of relations
between variables, such as the regression
coefficients, correlation coefficients, or path
coefficients of the more commonly used
statistical models, but in terms of parameters
representing more complex dependencies, such
as transitive closure, which is a dependency
involving three variables at a time.
Of these three approaches, the first may
now be regarded as a relict from an earlier
period. Models expressing network structure
only through covariates while further employing independence assumptions may have had
their use in a time when more suitable methods
were not available, but they are likely to lead to
suspicious results because of misspecification
and therefore should now be avoided. The
MRQAP and heteroscedasticity-consistent approaches are useful, but they are not discussed
further here because they regard network
structure as nuisance rather than substance
and do not attempt to model network dependencies. The conditionally uniform approach
is treated briefly in Conditionally Uniform
Models, below. This article is concerned
mainly with approaches that aim to model
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networks by representing network dependency
explicitly in a stochastic model.
The Use of Probability and Statistics
to Model Networks
Many network studies are about only one single network and in that sense are N = 1 studies. Accordingly, there is a need to explain why
statistical methods are applicable at all to such
data. The use of probability models in statistical inference can be motivated generally in two
ways: (a) based on plausible model assumptions,
termed model-based inference or (b) based on
the sampling mechanism used, termed designbased inference. This distinction is explained
at length by Sterba (2009), who also shows
how the discussion about these traditions goes
back to debates in the 1920s and 1930s between two of the founding fathers of statistics,
Ronald Fisher and Jerzy Neyman, who championed model-based and design-based inference,
respectively.
In model-based inference, the researcher—
explicitly or implicitly—constructs a probability model, makes the assumption that the observed data can be regarded as the outcome
of a random draw from this model, and uses
this model to derive statistical procedures and
their properties. In social science applications,
the model is a stylized representation of social
and behavioral theories or mechanisms and the
probabilistic components express behavioral,
individual, or other differences for which the
precise values are not explicitly determined by
the model and that would assume different values in independent replications of the research.
Multiple linear regression models are an example. Molenaar (1988) gives an illuminating discussion of how to link the statistical model to
substantive and theoretical considerations and
of different types of replication. Questions that
arise include, for example, the following: Are
the new measurements of the same individuals,
or do they represent a new sample of respondents and/or other measurement instruments?
This idea of replication is linked to the idea of
inference from the data to a population. The
population is mathematically a probability distribution on the outcome space of the entire
data set and substantively (given that we are discussing statistics in the social sciences) a social
and/or behavioral process in some set of individuals in some social setting. The population of
individuals may be described in a very precise or
in more general and hypothetical terms. From
multilevel analysis (Snijders & Bosker 2011) we
have learned that a given piece of research may
be generalized simultaneously to several populations, e.g., a population of individuals and a
population of workplaces.
Probabilistic statements and properties in
design-based inference are based on the sampling mechanism, which is, in principle, under
the control of the researcher. Usually there is
some finite population of which the sample is a
subset, drawn according to a known probability
mechanism. One of the desiderata is that each
population element has a positive probability to
be included in the sample. The probability distribution of the sample as a subset of the population is known, but the values of the variables of
nonsampled population elements are unknown.
Statistical inference is from the sample to only
this finite population. If the whole population
has been observed, statistical inference has become superfluous.
In design-based inference the probability
distribution is under control of the researcher.
By contrast, in model-based inference a family
of probability distributions must be assumed,
based on plausible reasoning and basic insights
into the population and phenomena under
study, requiring that the approximate validity
be checked by diagnostic methods. Because of
the different nature of the populations to which
the data is being generalized, design-based
inference often is called descriptive, whereas
model-based inference is called analytical.
However, mixed forms do exist. Because most
social science aims at studying mechanisms
rather than describing specific populations,
statistical methods used in the social sciences
are mainly design based. However, textbooks
and approaches to teaching statistics often
found their probability models on design-based
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arguments, which leaves some researchers with
the incorrect impression that statistical inference should always be based on a probability
sample or a good approximation of such a
procedure.
The distinction between model-based and
design-based inferences applies directly to
statistical modeling of networks. This was
stressed, e.g., in Frank (2009) (see section on
probabilistic network models). Design-based
methods can be used when a sample is drawn
from a larger graph. An important class of
designs are link-tracing designs, where the
sample uncovers nodes in waves, and the ties
found for the nodes uncovered in a given wave
will in some way determine the nodes uncovered in the next wave. Examples are snowball
designs and random-walk designs (Frank
2009). Such methods are used, e.g., to try and
find members of hard-to-reach populations and
to get information about their characteristics
and the network structure of such populations.
An overview of the earlier literature is found
in Spreen (1992). Gile & Handcock (2010)
provide an overview of the recent method of
respondent-driven sampling. Model-based inference can be used in the regular case in social
network analysis where an entire network has
been observed, and it is natural or plausible to
consider that the observed network data could
also have been different akin to the query posed
by Molenaar (1988): What would happen if
you did it again? It could have been observed
at a different moment, the individuals could
have differed while the social and institutional
context remained the same, external influences
could have been different, etc. In model-based
inference it is assumed that within a population
of different networks, even though vaguely
described, the systematic patterns expressed
in the parameters of the probability model
would be the same, whereas the particular
outcome observed (in this case the outcome
would be the whole network) could have been
different. As with all model-based inferences,
there would thus be a distinction between
systematic properties of the social system and
random variability, one could say, between
signal and noise. The aim of the statistical
model is to represent the main features of the
data set—in this case, the network—by a small
number of parameter estimates and to express
the uncertainty of those estimates by standard
errors, posterior distributions, p-values, etc.,
which give an indication of how different
these estimates might be if the researcher
would “do it again.” Checking the assumptions
of the model is important to guard against
overlooking important features of the network
data set, which also could bias the features that
are being reported, and to have confidence in
the reported measures of variability.
Inference for networks is potentially even
more precarious because the traditional research design is to collect data on a single (i.e.,
N = 1) network. The inferential issue here has
an internal and an external aspect, corresponding to what Cornfield & Tukey (1956) call
the two spans of the bridge of inference. The
internal issue is that although we have only one
system, in the sense that potentially everything
depends on everything else, we also have large
numbers of actors and tie variables and we can
carry out statistical inference because reasonable assumptions of conditional independence
or exchangeability can be made. I return
to this issue in Conditional Independence
Assumptions, below. The external issue is that
from the N = 1 observed network we would
like to say something about social processes
and mechanisms more generally. Whether this
is reasonable depends on how “representative”
this network is for other networks—a dirty
word in statistics because, as Cornfield &
Tukey (1956) argued, this is a question outside
of statistics. In some cases it may be argued
that one particular network tells us a lot about
how social structure and social constraints
operate more generally. But nobody will deny
that such knowledge can be solid only if there
is a cumulation of results over replications, i.e.,
studies of broadly the same phenomenon or
process in different groups and contexts. (By
the way, doing statistical inference for N = 1
is not unusual; similar issues arise, e.g., in the
economic analysis of long time series.)
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More scientific progress can be made when
data are available for several networks that may
be regarded, in some sense, as replications of
each other: several schools, several companies,
several villages, etc. Such data sets are rare but
not exceptional. For example, Coleman (1961)
collected friendship data for 10 schools, and
more recent examples of networks collected in
larger numbers of schools are the Add Health
data set (Harris et al. 2003) and the ASSIST
study (Steglich et al. 2011). Such data sets with
multiple networks call for a multilevel study, or
meta-analysis, of social networks, enabling generalization to a population of networks. Snijders
& Baerveldt (2003) took a first step in this direction, but as yet, this area of network modeling
remains thoroughly underdeveloped.
SINGLE NETWORKS
This section discusses the main various types
of statistical models for single, i.e., crosssectionally observed, networks.
Conditionally Uniform Models
Conditionally uniform models consider a set of
statistics that the researcher wishes to control
for, and then assume that the distribution of
networks is uniform, conditional on these statistics. Thus, each network satisfying the constraints of leading to the desired statistic has
the same probability; each network not satisfying these constraints has probability 0. This
reflects the notion that the conditioning statistics contain that which is relevant in the studied
phenomena, and the rest is randomness. Conditionally uniform distributions are typically used
as straw man null hypotheses. They are used in
a strategy where network properties that the
researcher wishes to control for are put in the
conditioning statistic, and the theory that is put
to the test is expressed by a different statistic, for
which then the p-value is calculated under the
conditionally uniform distribution. This strategy has a mathematical basis in the theory of
statistical tests that are “similar” (i.e., have constant rejection probability) on the boundary
between null hypothesis and alternative hypothesis (see Lehmann & Romano 2005).
Holland & Leinhardt (1976) initiated the
study and application of this type of model,
emphasizing the uniform model for directed
graphs conditional on the dyad count, i.e., the
numbers of mutual, asymmetric, and null dyads,
denoted as the U|M, A, N distribution. They
elaborated the strategy where the test statistic is a linear function of the triad census, the
vector of counts of all possible triads (subgraphs of three nodes contained in the network). Wasserman (1977) provide a further
elaboration.
This strategy has two limitations. One is
that conditionally uniform models become very
complicated, when richer sets of conditioning
statistics are considered. For example, given
that in-degrees and out-degrees are basic
indicators of actor position, it is relevant to
condition on the in-degrees as well as the
out-degrees in the network, leading to the
so-called U |(Xi+), (X+i) distribution. This
is a distribution with difficult combinatorial
properties. Snijders (1991), Rao et al. (1996),
Roberts (2000), and Verhelst (2008) developed
ways to simulate this distribution, whereas
McDonald et al. (2007) studied ways to simulate the U |(Xi+), (X+i), M distribution, which
also conditions on the number of mutual ties in
the network. In practice, one would like to go
even further in conditioning. However, such
attempts are self-defeating given combinatorial
complexity.
Another limitation is that the rejection of the
null hypothesis does not provide a first step toward constructing a model for the phenomenon
being studied—the only conclusion is that the
observed value for the test statistic is unlikely
given the conditioning statistics if all else is random. Because of these limitations, conditionally
uniform distributions are currently not often
used.
Latent Space Models
A general strategy to represent dependencies
in data is the latent space model of Lazarsfeld
& Henry (1968). This model assumes the
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existence of latent (i.e., unobserved) variables,
such that the observed variables have a simple
probability distribution given the latent variables. The specification of the latent variables
is called the structural model, and the specification of the distribution of the observed variables conditional on the latent variables is the
measurement model. Examples are factor analysis [a model proposed long before the book
by Lazarsfeld & Henry (1968)], item response
theory, latent class analysis, and mixture models. In these examples the data are independent
across individuals, and the aim of the latent
space model is to represent parsimoniously dependencies between multiple variables within
individuals.
A number of models for social networks can
be subsumed under this category. These models all have latent variables defined for the nodes
in the graph that represent the social actors and
assume that the dyads (Xij, Xji) are conditionally independent, given these nodal variables.
In many of them, even the tie variables Xij are
assumed to be conditionally independent, given
the nodal variables.
This section reviews a number of latent
space models, defining them by the type of latent structure and the conditional distribution
of the network given the latent structure. The
latent variables are denoted by Ai for node i,
with A denoting the vector A = (A1, . . . , An).
The space of values for Ai is denoted A. In all latent structure models, this space will have some
topological structure defining the model. In all
these models, the assumption is made that the
dyads (Xij, Xji) are independent given the vector A in space A. Depending on the model,
the “locations” Ai of the nodes are regarded
as random variables, as estimable deterministic parameters, or as either. In all cases, the
choice between random and fixed is a matter of
estimation strategy.
Discrete space. In the stochastic block model
of Holland et al. (1983), Snijders & Nowicki
(1994), Nowicki & Snijders (2001), and Daudin
et al. (2008), there is a node-level categorical latent variable Ai with K possible values for some
K ≥ 2. The latent space is A = {1, . . . , K},
without any further structure. Topologically, it
is a discrete space. In the spirit of graph theory,
Nowicki & Snijders (2001) refer to these values as colors, leading to a model of a colored
graph with unobserved colors; one could also
call this a latent class model for the nodes. The
conditional distribution of the dyads (Xij, Xji) is
assumed to depend only on Ai and Aj, the colors (or classes) of i and j. Thus for each pair of
colors (c , d) ∈ A2, there is a probability vector
for the four outcomes (0,0), (0,1), (1,0), (1,1)
of the dyad (Xij, Xji). This is a stochastic version of the concept of structural equivalence of
Lorrain & White (1971). Such a model can
represent cohesive subgroups but also very different, noncohesive subgroup structures, such
as social roles. Airoldi et al. (2008) extended
this model to mixed membership models, where
each node can be a member of several classes,
thereby describing situations where the actors
may play multiple roles.
Distance models. Several variants of distance
models have been developed. A function d :
N → [0, ∞) is called a metric, or distance function, if it satisfies the following axioms:
1. d(i, i) = 0 for all i ∈ N;
2. d(i, j) = d( j, i) > 0 for all i, j ∈ N with
i = j;
3. d(i, j) ≤ d(i, k)+d( j, k) for all i, j, k ∈ N
(triangle inequality).
In the latent metric space models, it is assumed that the nodes are points in some metric space A = {1, . . . , n} with distance function
d(i, j), and the probability of a tie depends as
a decreasing function on the distance between
the points, P{Xij = 1} = π(d(i, j)). The closer
the points are, the larger the probability that
they are tied.
The general definition of metric spaces is so
wide that it does not lead to very useful models; more constraints are necessary (Hoff et al.
2002). Examples of more specific latent metric
models are defined in the following paragraphs
by posing further restrictions on the metric. It
may be noted that the symmetry of distance
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functions (Axiom 2, above) implies here that the
tie from i to j has the same probability as the tie
in the reverse direction, but the latent distance
model does not imply a special tendency to
reciprocity.
Ultrametric space. Freeman (1992) proposed
ultrametric models as a representation of group
structure in social networks. A function d: N →
[0, ∞) is called an ultrametric, or ultrametric
distance, if the triangle inequality is replaced
by the stronger requirement
d(i, j) ≤ max {d(i, k), d( j, k)}
for all i, j, k ∈ N .
This condition is called the ultrametric inequality. For ultrametric distances on finite spaces,
it is not a restriction to assume that the set of
values of the distance is a set of consecutive integers {0, 1, . . . , K} for k ≥ 1.
Ultrametric distances have the property that
for every cutoff point k, the graph with edge
set Ek = {(i, j)|0 < d(i, j) ≤ k} is a perfectly
transitive graph, meaning that it consists of
a number of mutually disconnected cliques.
Therefore, ultrametrics are useful structures for
representing the transitivity of social networks.
Schweinberger & Snijders (2003) proposed a latent ultrametric model for nondirected graphs
as a stochastic implementation of Freeman’s
(1993) idea of ultrametric spaces for representing networks and of the concept of social settings developed by Pattison & Robins (2002).
Clauset et al. (2008) later proposed almost the
same model. For each “level” k, the graph with
edge set Ek can be regarded as a “smoothed”
version of the observed graph, representing the
setting structure at level k, where k = 1 represents the most fine-grained and the maximum
value k = K the most coarse settings structure.
Euclidean space. Hoff et al. (2002) proposed
to specify the latent metric as a Euclidean
distance—in practice with a dimension K = 2,
sometimes K = 3. For K = 2, this means that
each node is represented by two real-valued coordinates (Ai1, Ai2) and the distance between
the two nodes is defined as
d(i, j) = (Ai1 – Aj1)2 + (Ai2 – Aj2)2.
This model is in line with the graphical representations of networks in two-dimensional
pictures, where the points are arranged in the
plane in such a way that nearby points are linked
more often than points separated by large distances. This model also represents transitivity,
following from the triangle inequality and further from the specific structure of Euclidean
space.
The model of Hoff et al. (2002) accommodates explanatory variables in addition to the
latent distance, letting the probability of a tie
between i and j depend on the distance and
on explanatory variables through a logistic (or
other) link function:
logit(P{Xij = 1}) = β zij – d(i, j), 2.
where zij is a vector of explanatory variables for
the pair (i, j) and β is a vector of regression
coefficients.
To represent further group structure in addition to the transitivity already implied by
the latent Euclidean distance model, Handcock
et al. (2007) complemented this model with the
assumption that the locations Ai are outcomes
of a mixture model of normal distributions.
This is a double-layer latent structure: a first
layer consisting of the locations and a second
layer consisting of the (normally distributed)
subgroups.
Sender and receiver effects. Two generations of models have been proposed representing actor differences with respect to sending and
receiving ties as well as reciprocation. Holland
& Leinhardt (1981) proposed the p1 model for
directed graphs; the name p1 was chosen because they considered this the first plausible and
viable statistical model for directed networks.
In this model, dyads (Xij, Xji) are independent;
each actor has two parameters αi, βi responsible, respectively, for the tendency of the actor to
send ties (activity, influencing the out-degrees)
and the tendency to receive ties (popularity,
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influencing the in-degrees); in addition there
are parameters influencing the total number
of ties and the tendency toward reciprocation.
The large number of parameters, two for each
actor, is a disadvantage of this model, and
Fienberg & Wasserman (1981) proposed a
model in which this number is reduced by making the parameters dependent on categorical
nodal attributes. Various other modifications
and extensions have been proposed (for a review, see Wasserman & Faust 1994).
van Duijn et al. (2004) proposed another
way to reduce the dimensionality of the parameter of this model—the p2 model—without,
however, postulating that actors with the same
attributes have identical distributions of their
incoming and outgoing ties. In this model,
the activity and popularity parameters are
regressed on nodal and/or dyadic covariates
and include random residuals, making this a
random effects model. The actor-dependent
residuals for the activity and popularity effects
are assumed to be correlated. The major
advantage of this model over the p1 model is
the possibility of including sender and receiver
covariates, which is impossible in the p1 model
because such effects are totally absorbed by the
αi and βj parameters. Hoff (2005) generalized
this model by including for the log-odds of a
tie not only random sender effects Ai, receiver
effects Bj, and reciprocity effects Cij = Cji, but
also bilinear effects DiDj, where the Di variables
also have multivariate normal distributions.
Ordered space. Hierarchical ordering is another feature that can be exhibited by networks:
a linear order, as is usual in the well-known
phenomenon of pecking orders of chickens, or
more generally a partial order. De Vries (1998)
reviewed procedures for finding linear orders,
applied to dominance relations between animals. In a partial order, all pairs of points do not
need to be ordered. The definition of a partial
order  on N is the following:
1. i  j and j  i if and only if i = j
(antisymmetry);
2. i  j and j  k implies i  k
(transitivity).
In the model proposed by Mogapi (2009), it is
assumed that the nodes i are points in a latent
partially ordered space, and probabilities of ties
depend on covariates and the ordering between
the points. Given the partial ordering and covariates zij, the probability of a tie is assumed to
be given by
logit(P{Xij = 1})

= ⎧⎪⎨ π2 + β zij if j  i π3 + β zij if i  j and j  i.

3.

⎪⎩
π1 + β zij if i  j
Exponential Random Graph Models
A different type of model represents dependencies between ties directly, rather than by conditioning on latent attributes. Frank & Strauss
(1986) introduced this line of modeling. They
defined Markov dependence for distributions
on network in analogy to distributions for
stochastic processes: Conditioning on the other
random variables, two random variables are independent unless they are tied (where a tie in
the stochastic process case would be defined as
direct sequentiality in time). For the network
case, the definition is given more precisely as
follows: The array X = (Xij) of random variables, which can also be regarded as a stochastic graph, is a Markov graph if, for each set of
four distinct nodes {i, j, h, k}, the random variables Xij and Xhk are conditionally independent,
given all the other random variables in X. This
seems a plausible kind of conditional independence suitable for social networks.
Frank & Strauss (1986) proved that this
Markov dependence for a random nondirected
graph, with the additional requirement that the
probability distribution is invariant under permutation of the nodes, is equivalent to the possibility to express the probability distribution of
the graph by
P{X = x}
=
exp θ L(x) + n k=-21 σk Sk(x) + τ T (x)
κ(θ, σ, τ) , 4.
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where L(x) = i< j xij is the edge count,
T = i< j<h xijxjhxih is the triangle count, and
Sk = i j1< j2<···< jk xi j1xi j2 . . . xi jk is the k-star
count (with S1(x) = L(x)) (see Figure 1).
The statistical parameters in this model are
θ, σ2, . . . , σn–1, and τ. Finally, κ(θ, σ, τ) is a
normalization constant to let the probabilities
sum to 1. The fact that the logarithm of the
probability is a linear combination of parameters and statistics makes this into an exponential
family of distributions (Lehmann & Romano
2005), an important class of statistical models for which many theoretical properties are
known. The statistics are the so-called sufficient
statistics, as they contain all information in the
data about the values of the parameters. The
sufficient statistics here are subgraph counts,
the frequency in the graph of small configurations: edges, k-stars, and triangles. The number
of k-stars can be expressed as
Sk(x) =
n
i=1  xki+ , 5.
which implies that the vector of k-star counts
Sk(x) for k = 1, . . . , K are a linear combination
of the first K moments
1 n
n
i=1
X k
i+(k = 1, . . . , K)
of the degree distribution. Thus, if σk = 0 for
all k larger than some value K, in a distribution
of graphs according to Equation 4, each graph
with the same moments of order up to K of the
degree distribution, and the same number of
triangles, is equiprobable. Often this model is
used while including only a few of the σk parameters for low k, so that the degree distribution
is characterized by a few low-order moments
such as the mean, variance, and skewness.
Frank (1991) and Wasserman & Pattison
(1996) generalized the Markov graph model
by proposing that the exponent in Equation 4
could contain, in principle, any statistic, thus
allowing any kind of dependence between the
tie variables:
Pθ{X = x} =
exp k θksk(x)
κ(θ) , 6.
transitive triangle 6-star
Figure 1
Examples of subgraph structures.
where the sk(x) can be any statistic depending
on the network and observed covariates. They
can be specified so as to reflect the research
questions and to obtain a good fit between
model and data. This model can theoretically
represent any distribution on the space of
graphs that gives positive probability to each
possible graph—although such a representation will not necessarily be parsimonious or
tractable. Wasserman & Pattison (1996) called
this the p∗ model; more recently it has also been
called the exponential random graph model
(ERGM). An important subclass is obtained
when the sufficient statistics sk(X) are subgraph
counts (as is the case for the Markov model).
Such models can be obtained from conditional
independence assumptions (of which, again,
Markovian dependence is one example) based
on an application of the Hammersley-Clifford
theorem (proved in Wasserman & Pattison
1996). Examples are the neighborhood models
of Pattison & Robins (2002) and the models
excluding action at a distance discussed by
Snijders (2010). An overview of the ERGM is
presented in Robins et al. (2007a) and in the
monograph by Koskinen et al. (2011).
Back to the Markov model. The two main
virtues of the Markov model are its possibility
to represent transitivity and the distribution of
degrees, as reflected by the parameters τ and
σk, and its interpretation as being equivalent
to the Markovian conditional independence
property. One difficulty is related to parameter
estimation. The normalizing factor κ in
Equation 4 cannot be easily calculated except
for uninteresting special cases, which is an
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impediment for the calculation of likelihoods
and for traditional procedures for parameter
estimation. Frank & Strauss (1986) proposed a
pseudolikelihood estimation procedure, maximizing the pseudo-log-likelihood defined as
i, j
log(Pθ,σ,τ{Xij = xij|X (–i j) = x(–i j)}),
where X (–i j) is the random graph X without
the information about the tie variable Xij. The
pseudo-log-likelihood can be seen to have the
same structure as the log-likelihood for a logistic regression model, so that standard software
can be used to compute the pseudolikelihood
estimator. The dependence between the tie
variables, however, creates problems for this
procedure. The estimates obtained under this
model and the correspondingly calculated standard errors were shown to be quite unreliable
(see, e.g., van Duijn et al. 2009). When Markov
chain Monte Carlo procedures for maximum likelihood estimation were developed
(Dahmstrom & Dahmstr ¨ om 1993, Snijders ¨
2002), however, a second difficulty came to
the surface. This relates to the probability
distributions and is not restricted to a particular
procedure for parameter estimation. Handcock
(2002) found that these distributions can
be nearly degenerate in the sense that they
concentrate the probability in one or a quite
small number of possible outcomes. Snijders
(2002) found that these distributions can have
a bimodal shape: The two modes are totally
different networks. Given that this is a model
for a single observation, such properties are
undesirable, but they will occur when the
transitivity parameter τ approaches values
required to represent the tendencies toward
transitivity observed in real-world networks,
unless the number of nodes is quite low (e.g.,
less than 30) or the average degree is small (less
than 2). These degeneracy problems can occur
even in models with τ = 0.
The conclusion of these findings is that
the Markov model is not a reasonable representation for most empirically observed
social networks when they have more than
30 nodes, average degrees more than 2, and
j i
k h
Figure 2
Creation of a 4-cycle by edges i–j and h–k.
a transitivity index (Equation 1) more than
2. Further elaborations of this problem and
relevant references can be found in Snijders
et al. (2006) and Rinaldo et al. (2009).
The social circuit model. As a less restrictive model, Snijders et al. (2006) proposed the
social circuit assumption and a set of statistics
satisfying this property. This assumption states
that, for four distinct nodes i, j, h, k, tie indicators Xij and Xhk are conditionally independent, given the rest of the graph, unless the existence of these two ties would imply a 4-cycle in
the graph (see Figure 2). An interpretation (cf.
Pattison & Robins 2002) is that in the latter
case, e.g., if Xih = X jk = 1 (as in Figure 2),
the existence of the tie i–j would imply that the
four nodes i, j, h, k are jointly included in a social setting, which would affect the conditional
probability that the tie h–k also exists.
There are many statistics satisfying the
social circuit assumption that could be chosen
to represent tendencies toward transitivity
as observed in social network data sets.
The statistics of the Markov specification
(Equation 4) are not suitable because for larger
n they do not allow probability distributions
concentrated around graphs with transitivity
indices that have any values clearly larger than
the density of the graph and smaller than 1.
Snijders et al. (2006) proposed statistics that do
satisfy this requirement and that correspond
to the social circuit assumption (although they
are by no means the only statistics obeying this
assumption). Wide experience collected since
this proposal has confirmed that these statistics
allow for a representation of a large variety
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of observed social network data sets. Two
mathematically equivalent versions have been
proposed, indicated by the epithets alternating
and geometrically weighted, respectively.
Hunter (2007) elaborated the relations between these two versions. For the geometrically
weighted versions, there are three statistics:
the geometrically weighted degree statistic
(GWD), the geometrically weighted edgewise
shared-partner statistics (GWESP), and the
geometrically weighted dyadic shared-partner
statistics (GWDSP). The GWD is a function
of the degree counts Dr = Dr(X ) defined
as the number of nodes in X with degree r.
The GWESP is a function of the edgewise
shared-partner statistics EPr defined as the
number of unordered linked pairs (i, j) that are
both connected to exactly r other nodes,
EPr =
i< j
X
ij I  k Xik X j k = r .
Here I{A} is the indicator function of the
event A, equal to 1 if A is true and 0 if it is false.
The GWDSP is a function of the dyadwise
shared-partner statistics DPr defined as the
number of unordered pairs (i, j), irrespective
of whether they are linked, that are both
connected to exactly r other nodes,
DPr =
i< j
I  k Xik X j k = r .
The edgewise and dyadwise shared-partner
statistics reflect tendencies toward transitivity.
Given a tendency toward transitive closure, for
any given pair (i, j), if there are many shared
partners, i.e., k Xik X j k is large, then the conditional probability of the edge i–j will be high.
The problem with the Markov specification
is that the conditional log-odds of this edge
increases linearly with the number of shared
partners, which is a too-strong dependence.
The exponentially weighted statistics depend
on a so-called weighting parameter, denoted
here by α, that attenuates the effects of high degrees or high numbers of shared partners. Usual
values of α are nonnegative, and higher values
mean stronger attenuation. The exponentially
weighted statistics are defined as follows:
GWD(X ) =
r
wr(α)Dr(X )
GWESP(X ) =
r
wr(α)EPr(X )
GWDSP(X ) =
r
wr(α)DPr(X ),
where wr(α) is given by
wr(α) = eα{1 – (1 – e–α)r}.
This is an increasing function of r that is
nearly linear in r if α is close to 0 and becomes
increasingly strongly concave as α gets larger.
Therefore, for α tending to 0, the model with
these statistics approximates the Markov specification, and the degeneracy problems may be
expected to become weaker asα becomes larger.
The ERGMs with these sufficient statistics
are further discussed in Snijders et al. (2006),
Hunter (2007), and Robins et al. (2007b). In
data analysis, the value of α can be set at a
predetermined value or it can be estimated, as
treated in Handcock & Hunter (2006).
There is an empirical interpretation to the
fact that ERGMs with the Markov specification cannot be fitted to realistic social network
data (assuming that the number of nodes is
more than 30, average degree more than 2, and
the transitivity coefficient clearly higher than
the density), whereas models with the alternating/geometrically weighted specification can.
The equivalence of the Markov specification
with the conditional independence assumption
implies that this assumption must be unrealistic. Thus, it is not reasonable to assume for social networks (under the stated extra conditions)
that for any four distinct nodes, i, j, k, h, the edge
indicators Xij and Xhk are conditionally independent, given the rest of the graph. By contrast, the social circuit assumption, that these
edge indicators may be conditionally independent under the additional assumption that the
existence of these two edges would not create
a 4-cycle, is a more tenable approximation to
networks observed in practice. This assumption
calls into question the robust standard error estimates of Lindgren (2010) (discussed above),
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which are based on precisely this assumption. It
is unknown whether in practice this is a restriction for the validity of these standard errors.
Conditional Independence
Assumptions
The three principles presented here for constructing network models all are based on conditionality, but in totally different ways. Conditionally uniform models condition on observed
statistics and try to assess whether these are
sufficient to represent the observed network.
Bearman et al. (2004) provide one very successful application of this principle, although
applied in a statistically informal way. These
authors investigated the dating network of
an American high school and found that to
represent the structure of the large connected
component of this network it is sufficient
to condition on the degree distribution, the
restriction to heterosexual dating, the homophilous preference for dating somebody
with similar partnership experience, and a
social taboo on 4-cycles. However, the conditionally uniform approach is successful only
in rare cases because of the combinatorial
complexity of the resulting distributions.
Latent space models postulate the existence
of a space in which the nodes occupy unobserved (latent) positions, such that the tie indicators are independent conditionally on these
positions. The estimation of these positions
yields a representation of the network that can
give a lot of insight, comparable to other visualizations, but with the extra element of a
probabilistic interpretation. These models have
a combination of rigidity and flexibility—the
assumption of a particular type of space is rigid
and limits the kinds of dependencies that can be
represented, whereas the possibility to position
the points anywhere in the space can give a lot
of flexibility. The latter flexibility is mirrored by
the multimodality of the likelihood, which may
lead to difficulties in estimation and ambiguity
of the results. Within these models, nodes can
be dropped from the observations without affecting the validity of the model assumptions
for the remaining nodes. This is convenient
for modeling, for example, to handle randomly
missing data, but may be an unlikely assumption for networks because taking out an actor
could have an impact on the relations between
the other actors.
ERGMs, when using subgraph counts as sufficient statistics, are based on conditional independence assumptions between the observed
tie variables. There is a large flexibility in specifying the sufficient statistics, and this can give
insights in dependence structures between the
ties in the network. Robins et al. (2009) provide
an example via an elaboration of this type of
model for directed networks where many different dependence structures are possible because
directions of ties can be combined in so many
ways. One of the examples treated is a network
of negative ties (difficulties in working with the
other person), where the specific dependence
structures may be of great interest.
The latent space models for categorical and
Euclidean spaces, as well as the ERGMs, have
now been applied in a variety of empirical research articles and may be regarded as part of
the advanced toolkit of the modern social scientist. To choose between these two types of
model, researchers who are interested in detailed dependence structures may profit more
from applying ERGMs, whereas those interested in positions of actors may profit more
from an appropriate latent space model. For
both these types of model, estimation may be
difficult for large networks, where “large” could
be operationalized as a number of nodes of the
order of 1000 or more. For latent space models and for ERGMs, the difficulty resides in the
multimodality of the likelihood and in achieving convergence of the algorithm, respectively.
This may change as improved computational
methods become available. However, the complexity of dependencies in networks is so great
that modeling large networks in a way that
passes the high requirements of a good statistical fit seems intrinsically difficult to achieve.
NETWORK DYNAMICS
Longitudinal social network data, also called
data about network dynamics, can be of many
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different kinds. Some examples, with their
salient restrictions, are the following. Friendship networks in a class of school children may
be recorded at a few moments in time, although
they are sure to have been changed in between.
Alliances between firms may have their starting points registered but not their termination dates. Large data sets of email or other
electronic communications have been registered but often with little additional information about the senders and receivers.
Here also, fundamental questions about dependencies arise, but now the dependencies are
spread out in time with changes in network ties
depending on structures of earlier ties in the
network. Dependencies tracked through time
are much easier to handle, however, than are
those observed at one time point.
Three basic distinctions can be made
between statistical models for network dynamics. First, the ties may have the nature of
changeable states, like friendship or enduring
collaboration, or of events, like sending a
message or spending an evening together.
Second, there is a distinction between models
where the changes are driven by the network
itself (which is meaningful only for networks
of states, not for networks of events) or by a
different, perhaps unobserved, entity. Third,
the time variable indexing the dynamic network
may be discrete or continuous.
A probabilistic model for network dynamics can be represented generally as a stochastic
process X (t)(t ∈ T ), where X(t) is the value of
the process at time t and the time domain T
may be discrete, such as an interval of consecutive integers, or continuous, such as an interval of real numbers. All proposed models for
network dynamics are based in some way on
Markov chains, which are stochastic processes
X(t) for which the earlier past can be considered forgotten in the sense that for any present
moment t0 ∈ T , the conditional probability distribution of X(t) for all future times t > t0, given
its values for the entire past t0 ≤ t, depends only
on the current value X(t0).
In some of the models proposed for network
dynamics, the network is a Markov chain. This
is applicable to networks of states, not to networks of events; for example, it would hardly
be meaningful to entertain a model where the
network of all phone calls occurring at one particular moment depends as a Markov chain on
the network of past phone calls. Other models for network dynamics can be represented as
hidden Markov models (HMM). These are defined (Cappe et al. 2005) as stochastic processes ´
X(t) for which there exists another stochastic
process A(t) that is a Markov chain and such
that for any fixed t0, the conditional probability distribution of X(t0), given A(t) for all t and
given X(t) for all t = t0, depends on only A(t0)
and nothing else.
The three distinctions mentioned above can
be related in the following ways to these definitions. First, dynamic state networks can be
represented in principle by Markov chains as
well as by HMMs, whereas dynamic event networks can be represented by HMMs. In the
latter case, the underlying network A(t) could
either be constructed by aggregating the past
observations (and hence be directly observable)
or be unobserved. Second, for Markov chains
the changes in the network are driven by the
network, as the most direct representation of
the network dynamics is a feedback process
or a self-organizing system. For HMMs, the
changes in the network are driven by the entity
A(t). Third, the time domain T may be discrete
or continuous.
Many longitudinal social network data sets
are network panel data, i.e., two or more repeated measurements on the social network existing between a fixed group of social actors
(perhaps give or take a few actors who enter
into the network or leave it during the period
of study). In this overview, attention is given
mainly to network panel data, for networks consisting of states.
Continuous-Time Models
Holland & Leinhardt (1977) had an important
insight regarding the representation of feedback occurring in network dynamics, generated by, for example, reciprocation, transitive
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closure, and degree-related processes such as
the Matthew effect: It is fruitful to employ a
continuous-time Markov process, even though
the observations are done at a few discrete time
points, and to use only the creation and termination of single ties as the basic events in
such a process, with the exclusion of simultaneous changes of more than one tie variable. As a result, complicated observations of
network change are reduced to a few basic
processes. Coleman (1964) proposed the same
principle for non-network data. Kalbfleisch &
Lawless (1985) elaborated the principle for discrete data, whereas the literature reviewed by
Singer (2008) did so for continuous data.
With continuous-time processes one can explain, for example, the change from a set of isolated points to a highly connected subgroup as a
result of the three basic processes of random tie
creation, reciprocation, and transitive closure,
operating as a feedback process according to
a Markov chain. Wasserman (1977, 1979) and
Leenders (1995, 1996) applied this approach to
models where dyads are assumed to be independent, implying that reciprocation and homophily are the only processes that can be represented. Wasserman (1980) presented a model
that represents degree-related processes.
Actor-Oriented Models
A model that allows the simultaneous representation of an arbitrary array of processes is
the actor-oriented model proposed by Snijders
& van Duijn (1997) and Snijders (2001), with
a recent tutorial presentation in Snijders et al.
(2010).
The term actor-oriented refers to the idea
of constructing the model as the result of
context-dependent choices made by the actors,
following up the suggestion by Emirbayer &
Goodwin (1994) to combine structure and
agency. Actors are thought to control their outgoing ties. In line with the principles of Holland
& Leinhardt (1977), this is a continuous-time
model in which ties are changed only one at
a time and the probabilities of changes depend
on the total current network configuration. The
frequency of tie changes is modeled by the socalled rate function λi(x; α), which indicates the
frequency per unit of time with which actor i
gets the opportunity to change an outgoing tie,
given the current network state x. The choice
of which tie variable to change is modeled using the objective function fi(x; β), which can
be interpreted as a measure of how attractive
the network state x is for actor i. α and β are
statistical parameters. To define the probability of a change, x(i j±) denotes the network that
is identical to x in all tie variables except those
for the ordered pair (i, j), and for which the tie
variable i → j in x(i j±) is the opposite of this tie
variable in x, in the sense that xij (i j±) = 1 – xij.
Furthermore, we formally define x(ii±) = x.
The model operates as follows: Suppose that
the current network is x. All actors have independent, exponentially distributed waiting
times until the next time point when they are
allowed to change one of their outgoing tie variables. Let i be the actor with the shortest waiting
time, who therefore is the one to make the next
change. Then the probability that the change is
from network x to network x(i j±) is given by
P{X changes to x(i j±)}

= nh=1 exp( fi(x(ih±); β)).

7.

exp( fi(x(i j±); β))
In Snijders (2001), this formula is motivated
on the basis of the myopic stochastic optimization of the objective function, as is often used in
game-theoretical models of network formation
(e.g., Bala & Goyal 2000). When this change
has been made (if there was a change, which in
this model has probability less than 1), the process starts all over again but now from the new
state.
Model specification. The choice of the rate
function λi(x; α) and the objective function
fi(x; β) will reflect the research questions, underlying theory, and substantive knowledge.
The rate function often is constant or dependent on measures of the amount of activity
and resources put by actor i in determining
or optimizing her network position. The objective function is usually specified as a linear
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combination
fi(x; β) =
k
βks ki(x), 8.
where the s ki(x), called effects, are functions of
the personal network of i. Examples are
• s ki(x) = j xij (out-degree), reflecting
average degrees;
• s ki(x) = j xijxji (number of reciprocated dyads of actor i ), reflecting reciprocation;
• s ki(x) = j,h xij xjh xih (number of transitive triplet of actor i ), reflecting transitivity; and
• s ki(x) = j xij x+j (sum of in-degrees of
actor i’s network contacts), reflecting the
Matthew effect.
Snijders (2001) and Snijders et al. (2010) provide an extensive list of possible effects.
Use of continuous-time models to represent network panel data. Given an initial network for, say, time t1 = 0, the process described
above defines a continuous-time Markov chain
with time parameter {t ≥ 0}. When network
panel data have been observed at time points
t1 = 0 < t2 < · · · < tM , for some M ≥ 2,
the dynamics of the process between consecutive time points tm and tm+1 is unobserved,
which can be accounted for in the estimation procedure by simulating this dynamic (cf.
Snijders 2001). The distribution of X(t) does
not need to be stationary, but the transition
probability distribution is assumed to be stationary except for any time-changing parameters that may be incorporated in the parameter
vectors α and β. Usually, in the rate function,
a multiplicative parameter that depends on the
time interval (tm, tm+1) is included to reflect the
total amount of change observed when going
from observation x(tm) to observation x(tm+1).
Dynamic Exponential Random
Graph Models
Discrete-time extensions of the ERGM for
observations x(t1), x(t2), . . . , x(tM ) can be
formulated by the model
Pθ{X (t1) = x(t1), . . . , X (tM ) = x(tM )} =
exp k θ1ks1k (x(t1)) + mM=-11 k θ2ks2k(x(tm), x(tm+1))
κ(θ) ,
where the effects s 1k(x(t1)) and parameters θ 1k
are used to represent the distribution of the
network at X(t1), while s 2k(x(tm), x(tm+1)) and
θ 2k represent the conditional distribution of
X (tm+1) given X (tm). Here again, the distribution of X(t) is not necessarily stationary in t, but
the conditional distribution of X (tm+1) given
X (tm) is stationary, unless some of the components in s2k depend also on m.
Robins & Pattison (2001) proposed this
model, which was further elaborated by
Hanneke et al. (2010). It has in principle the
same generality and the same difficulties as
the ERGM for single observed networks. It
does not have the parsimonious approach of
the actor-oriented model, which, owing to its
definition in continuous time, represents network change in terms of its most simple building block: simple tie changes. To obtain a good
fit, this will lead to greater model complexity
and, hence, more complicated interpretation,
unless the successive networks x(tm) and x(tm+1)
are very close to each other.
Hidden Markov Models
Several kinds of discrete-time HMMs have
been proposed where the underlying variables
A(tm) are Markov chains for which the marginal
distributions as well as the conditional distributions of A(tm+1) given A(tm) are multivariate
normal. Xing et al. (2010) proposed one such
model, which is a dynamic version of the mixed
membership model of Airoldi et al. (2008).
Their model has two sets of latent variables:
probabilities of class membership and probabilities of ties between various classes, both of
which may change over time. These probabilities assume multivariate normal distributions,
which are transformed to the required domain
of probability vectors.
Sarkar & Moore (2005) generalized the latent Euclidean distance model of Hoff et al.
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(2002) to longitudinal network data. They used
a random-walk model for the changes in the latent locations of the nodes. Westveld & Hoff
(2011) proposed another HMM, although they
did not use this term. They extended the random effects model with sender, receiver, and
reciprocity effects of van Duijn et al. (2004) and
Hoff (2005) to a dynamic model tracked over
time. For the random effects, an autoregressive normal model is assumed—because this is a
Markov chain, the resulting model is an HMM.
REVIEW AND FORWARD LOOK
During the past 10 years, tremendous developments have taken place in network modeling in general, including statistical inference.
Most of the models reviewed here have been applied fruitfully in diverse areas of social science.
The current challenge of dealing with complex
network dependencies in statistical inference is
being met by recently developed models and
methods (these models, but not these methods,
are here reviewed).
This article focuses on two of the basic types
of network data: single and longitudinal observations of graphs (directed or nondirected, but
this is not a major distinction), interpretable as
states. Such an approach, though limited, covers
a large domain, allows us to illustrate important
issues in the representation of network dependencies, and has many applications. In this last
section, some connections to other models and
additional literature are mentioned.
Other Network Models
A closely related stream of network models
has been developed by researchers with a
background in statistical physics and computer science. Widely known models are the
Watts-Strogatz small-world model (Watts
1999), which deals with large networks and
combines the features of transitivity, limited
degrees, and limited path lengths (geodesics),
as well as the scale-free network model (de Solla
Price 1976, Barabasi & Albert 1999), which ´
yields networks where the degree distribution
has a power law distribution, implying that
some nodes will have very large degrees—the
“hubs” in the network. Newman et al. (2002),
Watts (2004), and, more recently, Toivonen
et al. (2009) review this stream of literature. All
these models may be regarded as micro-macro
models in the sense that they are built on simple
rules for the formation of ties and aim to investigate the network-level structures that are
generated. The resulting insights have percolated into the literature on statistical modeling
of social networks: Robins et al. (2005) show
how models with small-world properties can be
obtained from ERGMs, whereas Snijders et al.
(2010) demonstrate the importance attached
to degree-related effects in stochastic actororiented models. Kolaczyk (2009) presents an
in-depth treatment of models from physics as
well as statistical backgrounds.
Network modeling in economics has
focused on optimal network structures where
actors have relatively simple utility functions,
e.g., with a cost on links and a benefit for
reaching other nodes indirectly. Jackson (2005)
provides a review of network formation models
in economics. The extensive work in this area
has led to three recent books: Goyal (2007),
Vega-Redondo (2007), and Jackson (2008).
The book by Vega-Redondo (2007) also
contains much material on the techniques from
statistical mechanics that are used extensively
in the physics literature on networks.
A particular feature of many published models for network dynamics assumes that nodes are
added sequentially, making some ties to previously created nodes, and that, once created, ties
remain forever. This helps tractability for deriving mathematical properties but makes them
unsuitable for modeling dynamics of networks
involving tie creation as well as tie deletion on
a given, fixed node set.
There exist many more statistical models for
data with a network structure. Part of this literature is labeled machine learning, which is the
term used by computer scientists when referring to inferential problems. Kolaczyk (2009)
and Goldenberg et al. (2009) provide extensive
reviews.
148 Snijders
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Further Work
The field reviewed here is in a state of vigorous development, and the models treated are
being extended in various ways. One such extension is for other types of network structure:
valued graphs, signed graphs, bipartite graphs,
etc. For bipartite networks, for example,
ERGMs and actor-oriented models were developed by Wang et al. (2009) and Koskinen
& Edling (2010), respectively. Koehly &
Pattison (2005) discussed multivariate ERGMs,
whereas T.A.B Snijders, A. Lomi & V. Torlo`
(manuscript under review) proposed actororiented models for multivariate networks.
Network structure can also be combined with
other structures as dependent variables; see
Steglich et al. (2010), which describes the actororiented model for the coevolution of networks
and behavior.
Another extension is the combination of several of the principles reviewed here. An example is the combination by Krivitsky et al. (2009)
of latent Euclidean distances, latent sender and
receiver effects, and covariate effects, which are
different elements in the framework of latent
space models. Latent space elements could also
be combined with ERGMs or actor-oriented
models.
DISCLOSURE STATEMENT
The author is not aware of any affiliations, memberships, funding, or financial holding that might
be perceived as affecting the objectivity of this review.
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Annual Review
of Sociology
Volume 37, 2011
Contents
Prefatory Chapters
Reflections on a Sociological Career that Integrates Social Science
with Social Policy
William Julius Wilson ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 1
Emotional Life on the Market Frontier
Arlie Hochschild ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣21
Theory and Methods
Foucault and Sociology
Michael Power ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣35
How to Conduct a Mixed Methods Study: Recent Trends in a Rapidly
Growing Literature
Mario Luis Small ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣57
Social Theory and Public Opinion
Andrew J. Perrin and Katherine McFarland ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣87
The Sociology of Storytelling
Francesca Polletta, Pang Ching Bobby Chen, Beth Gharrity Gardner,
and Alice Motes ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 109
Statistical Models for Social Networks
Tom A.B. Snijders ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 131
The Neo-Marxist Legacy in American Sociology
Jeff Manza and Michael A. McCarthy ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 155
Social Processes
Societal Reactions to Deviance
Ryken Grattet ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 185
v
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Formal Organizations
U.S. Health-Care Organizations: Complexity, Turbulence,
and Multilevel Change
Mary L. Fennell and Crystal M. Adams ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 205
Political and Economic Sociology
Political Economy of the Environment
Thomas K. Rudel, J. Timmons Roberts, and JoAnn Carmin ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 221
The Sociology of Finance
Bruce G. Carruthers and Jeong-Chul Kim ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 239
Political Repression: Iron Fists, Velvet Gloves, and Diffuse Control
Jennifer Earl ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 261
Emotions and Social Movements: Twenty Years of Theory
and Research
James M. Jasper ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 285
Employment Stability in the U.S. Labor Market:
Rhetoric versus Reality
Matissa Hollister ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 305
The Contemporary American Conservative Movement
Neil Gross, Thomas Medvetz, and Rupert Russell ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 325
Differentiation and Stratification
A World of Difference: International Trends in Women’s
Economic Status
Maria Charles ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 355
The Evolution of the New Black Middle Class
Bart Landry and Kris Marsh ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 373
The Integration Imperative: The Children of Low-Status Immigrants
in the Schools of Wealthy Societies
Richard Alba, Jennifer Sloan, and Jessica Sperling ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 395
Gender in the Middle East: Islam, State, Agency
Mounira M. Charrad ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 417
Individual and Society
Research on Adolescence in the Twenty-First Century
Robert Crosnoe and Monica Kirkpatrick Johnson ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 439
vi Contents
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Diversity, Social Capital, and Cohesion
Alejandro Portes and Erik Vickstrom ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 461
Transition to Adulthood in Europe
Marlis C. Buchmann and Irene Kriesi ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 481
The Sociology of Suicide
Matt Wray, Cynthia Colen, and Bernice Pescosolido ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 505
Demography
What We Know About Unauthorized Migration
Katharine M. Donato and Amada Armenta ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 529
Relations Between the Generations in Immigrant Families
Nancy Foner and Joanna Dreby ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 545
Urban and Rural Community Sociology
Rural America in an Urban Society: Changing Spatial
and Social Boundaries
Daniel T. Lichter and David L. Brown ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 565
Policy
Family Changes and Public Policies in Latin America [Translation]
Br´ ıgida Garc´ıa and Orlandina de Oliveira ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 593
Cambios Familiares y Pol´ ıticas Publicas en Am ´ erica Latina [Original, ´
available online at http://arjournals.annualreviews.org/doi/abs/
10.1146/annurev-soc-033111-130034]
Br´ ıgida Garc´ıa and Orlandina de Oliveira ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 613
Indexes
Cumulative Index of Contributing Authors, Volumes 28–37 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 635
Cumulative Index of Chapter Titles, Volumes 28–37 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 639
Errata
An online log of corrections to Annual Review of Sociology articles may be found at
http://soc.annualreviews.org/errata.shtml
Contents vii
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