Abstract
This paper addresses the formation of social norms of cooperation through interaction
in repeated Public Goods Games, using novel multilevel techniques.
Cooperation has traditionally been understood as the interplay of
static factors such as shared social identity and pre-existing norms.
This study investigates the dynamic emergence of cooperative norms in
the presence or absence of social categorization.
A small effect of categorization was found: Categorization helps initiate and maintain
higher levels of cooperation. However, the differences in emergent cooperation
between small groups were much stronger than the differences between the
Categorization and Non-Categorization conditions.
Using explorative analyses, three distinct classes of groups were found.
Within groups, group members follow nearly identical rules for
their choice of cooperative behavior.
We argue that individual behavior converged because of the social
interactions within these groups. Overall, the development of cooperation
is best predicted by the process of norm formation that occurs when social identities emerge.
Abstract
This paper proposes that common measures for network transitivity,
based on the enumeration of transitive triples, do not reflect the
theoretical statements about transitivity they aim to describe.
These statements are often formulated as comparative conditional probabilities,
but these are not directly reflected by simple functions of enumerations.
We think that a better approach is obtained by considering the
probability of a tie between two randomly drawn nodes,
conditional on selected features of the network.
Two measures of transitivity based on correlation coefficients between
the existence of a tie and the existence, or the number,
of two-paths between the nodes are developed, and called “Transitivity Phi”
and “Transitivity Correlation.”
Some desirable properties for these measures are studied and compared
to existing clustering coefficients, in both random (Erdös–Renyi)
and in stylized networks (windmills). Furthermore, it is shown that in a directed graph,
under the condition of zero Transitivity Correlation,
the total number of transitive triples is determined by four underlying features:
size, density, reciprocity, and the covariance between in- and outdegrees.
Also, it is demonstrated that plotting conditional probability of ties,
given the number of two-paths, provides valuable insights into empirical
regularities and irregularities of transitivity patterns.
Abstract
Using data on 32 actors and ties among them drawn from available court
files, we combine analytical sociology with statistical
models for networks in order to analyse a case of a counterfeit
alcohol distribution network from the Czech Republic. We formulate
a theory of action and identify relational mechanisms which
could explain how the structure of the network emerged and
describe. We use the exponential random graph model to test
these mechanisms. The analysis reveals that the two actors capable of
manufacturing the poisonous mixture were considerably
though not optimally proximate to others enabling fast distribution
of the mixture. Our model results that the structure was
formed by mechanisms of triadic closure, negative tendency to
concentrate ties, and tie translation of pre-existing ties into operational ties.
We conclude with the discussion of the implications our
approach for the study of criminal networks.
Abstract
Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data,
collected by observing a network and a behaviour in a panel design.
The parameters of SAOMs are usually estimated by the method of moments (MoM)
implemented by a stochastic approximation algorithm, where statistics defining
the moment conditions correspond in a natural way to the parameters.
Here, we propose to apply the generalized method of moments (GMoM),
using more statistics than parameters. We concentrate on statistics
depending jointly on the network and the behaviour, because of the
importance of their interdependence, and propose to add contemporaneous
statistics to the usual cross-lagged statistics. We describe the stochastic
algorithm developed to approximate the GMoM solution. A small simulation study
supports the greater statistical efficiency of the GMoM estimator compared to the MoM.
Abstract
Next to homophily, three other mechanisms are discussed that may play
a role in how numerical actor attributes affect directed social networks:
aspiration, conformity, and sociability. A four-parameter specification
is proposed for the effects of numerical actor attributes on directed
social networks.
Abstract
In some panel data studies, respondents are nested in social contexts,
like classrooms or organizations. The social relations, such as friendship
or collaboration, between the individuals in such contexts can be represented
by social networks. Like individual outcomes (e.g., behavior or performance),
the social relations between individuals are not static.
Social networks and individual outcomes change over time and can mutually affect each other.
In this chapter, we present a statistical model to study the interdependent
dynamics (or coevolution) of network structure and individual outcomes.
We assume panel observations of networks and individual attributes to be
the discrete-time realizations of an underlying continuous-time process,
which is modeled by a stochastic differential equation for the attribute
dynamics and a Markov chain model for the network dynamics.
We illustrate the proposed method by a study of the coevolution of
friendship ties and mathematics grades among 1160 students in 39 classrooms.
Abstract
The present study is an analysis of a Czech political corruption network known as
the Rath affair reconstructed with publicly available data. We argue that for the study of
criminal networks it is fruitful to follow a multiplex approach, i.e., to distinguish several
interdependent network dimensions and study how they are interrelated. Relational elements
in corruption are identified, and we propose three dimensions that are essential for understanding
the Rath network: pre-existing ties (e.g., marriage or co-membership of the same
party), resource transfer (e.g., bribing), and collaboration (e.g., communication).
The aim of the study was threefold. We aimed to examine if the network exhibits
a core/periphery structure,
to investigate the multiplex structure of the network by assessing
the overlap of the main dimensions of the network,
and to determine the central and multiplex actors while
considering the differentiation of centrality according to the three network dimensions. The
core/periphery model appears to have a perfect fit to the aggregated network, leading to a
four-block adjacency matrix. Studying the frequency of ties in these blocks shows that
collaboration ties are present in all the blocks, while resource transfer ties are mainly
locatedbetween the core and periphery, and pre-existing ties are rare generally. We also
identify central actors,none of which are strategically positioned, occupying more visible
positions instead. The majority of actors display strong multiplexity in the composition of
their own ties. In the conclusion the potential usefulness of multiplex descriptive measures
and of mixed methods approaches, implications of our results for trust in criminal networks,
and potential merits of analytical sociology approach are discussed.
Abstract
The aim of the present study was to develop and validate a safe sex norm
questionnaire as an appropriate instrument which would be adaptable to the female sex
worker (FSW) population.
A list of 34 items was finalized, with subscales for actual behavioral norms and for
perceived norms.
Non-parametric Mokken scale analysis was used for assessing
scalability and unidimensionality of the questionnaire in a sample of 170 FSWs in Tehran,
and showed that the two subscales were distinct constructs, and all
items are good indicators for the constructs.
Our findings support that the safe sex norm questionnaire is a valid and reliable
measure that would be useful to harm reduction programs and help effective HIV prevention
among female sex workers.
Abstract
This paper compares several imputation methods for
missing data in network analysis on a diverse set of simulated
networks under several missing data mechanisms. Previous work
has highlighted the biases in descriptive statistics of networks
introduced by missing data. The results of the current study
indicate that the default methods (analysis of available cases
and null-tie imputation) do not perform well with moderate
or large amounts of missing data. The results further indicate
that multiple imputation using sophisticated imputation models
based on exponential random graph models (ERGMs) lead to
acceptable biases even under large amounts of missing data.
Abstract
Missing data on network ties are a fundamental problem for network analysis.
In this paper, we present a new method with two variants to handle missing data
due to actor non-response in the framework of stochastic actor-oriented models (SAOMs).
The proposed method imputes missing tie variables in the first wave either by
using a Bayesian exponential random graph model (BERGMs) or a stationary SAOM
and imputes missing tie variables in later waves utilizing a SAOM.
The proposed method is compared to the standard SAOM missing data treatment
as well as recently proposed methods.
The multiple imputation procedure provided more reliable point estimates than the default treatment.
Abstract
Longitudinal social network studies can easily suffer from insufficient statistical
power. Studies that simultaneously investigate change of network ties
and change of nodal attributes (selection and influence studies) are particularly
at risk because the number of nodal observations is typically much
lower than the number of observed tie variables. This article presents a
simulation-based procedure to evaluate statistical power of longitudinal
social network studies in which stochastic actor-oriented models are to be
applied. Two detailed case studies illustrate how statistical power is strongly
affected by network size, number of data collection waves, effect sizes,
missing data, and participant turnover. These issues should thus be explored
in the design phase of longitudinal social network studies.
Abstract
We examined social mechanisms that account for similarity
in the social context of school-sponsored extracurricular sports activities
among friends. We distinguish two social mechanisms: "shared sports activities that
lead to friendship," whereby friendship formation and maintenance are conditioned
by joint sports participation, and "friendship that leads to shared sports activities,"
a form of social influence whereby adolescents join or maintain certain
sports activities based on their friends' choices.
Using a longitudinal sample of 1,776 10th graders at five high schools in
Southern California, we employed a stochastic actor-oriented multivariate
dynamic model to model the dynamic interplay between the two-mode affiliation
network of adolescents' participation in sports activities and the one-mode
friendship network. As a corresponding descriptive method,
we propose a quantitative measure for the relative strength of the
two-mentioned mechanisms as explanations of the association between
the one-mode and the two-mode network.
Further, we introduce two specifications that represent homophily effects
in the two-mode network and apply them to test gender homophily
in sports participation. The results provide strong evidence for both mechanisms,
with friendship leading to shared sport activities as stronger than shared
sports activities leading to friendships in explaining adolescents'
friendships with co-participants.
Abstract
Social networks and the attributes of the actors in these networks are not
static; they may develop interdependently over time. The stochastic
actor-oriented model allows for statistical inference on the mechanisms driving
this co-evolution process. In earlier versions of this model, dynamic actor
attributes are assumed to be measured on an ordinal categorical scale. We
present an extension of the stochastic actor-oriented model that does away
with this restriction using a stochastic differential equation to model the
evolution of continuous actor attributes. We estimate the parameters by a
procedure based on the method of moments. The proposed method is applied to
study the dynamics of a friendship network among the students at an
Australian high school. In particular, we model the relationship between
friendship and obesity, focusing on body mass index as a continuous co-evolving
attribute.
Abstract
This is a comment on the paper "Dynamic Network Actor Models:
Investigating Coordination Ties through Time"
by Christoph Stadtfeld, James Hollway, and Per Block, published in this volume of
Sociological Methodology.
First the position of their proposed model in relation to other publications for statistical dynamic
network models is described. This is followed by a discussion of the model for
coordination between the two actors at either side of the tie.
Next, it is argued that the rate function may be modeled more in detail in event models,
compared to models for panel data with a small number of waves.
Finally, a remark is made about how to deal with unknown time orderings in event data.
Abstract
Two approaches for the statistical analysis of social network generation
are widely used; the tie-oriented exponential random graph model
(ERGM) and the stochastic actor-oriented model (SAOM) or Siena
model. While the choice for either model by empirical researchers often
seems arbitrary, there are important differences between these models
that current literature tends to miss. First, the ERGM is defined on the
graph level, while the SAOM is defined on the transition level. This allows
the SAOM to model asymmetric or one-sided tie transition dependence.
Second, network statistics in the ERGM are defined globally but are
nested in actors in the SAOM. Consequently, dependence assumptions in
the SAOM are generally stronger than in the ERGM. Resulting from both,
meso- and macro-level properties of networks that can be represented
by either model differ substantively and analyzing the same network
employing ERGMs and SAOMs can lead to distinct results. Guidelines for
theoretically founded model choice are suggested.
Abstract
Social relations are multiplex by nature: actors in a group are tied together
by various types of relationships. To understand and explain group processes
it is, therefore, important to study multiple social networks simultaneously
in a given group. However, with multiplexity the complexity of data also
increases. Although some multivariate network methods (e.g. Exponential Random
Graph Models, Stochastic Actor-oriented Models) allow to jointly analyze
multiple networks, modeling becomes complicated when it focuses on more than
a few (2–4) network dimensions. In such cases, dimension reduction methods
are called for to obtain a manageable set of variables.
Drawing on existing statistical methods and measures, we propose a procedure
to reduce the dimensions of multiplex network data measured in multiple groups.
We achieve this by clustering the networks using their pairwise similarities,
and constructing composite network measures as combinations of the networks
in each resulting cluster. The procedure is demonstrated on a dataset of 21
interpersonal network dimensions in 18 Hungarian high-school classrooms.
The results indicate that the network items organize into three
well-interpretable clusters: positive, negative, and social role
attributions. We show that the composite networks defined on these three
relationship groups overlap but do not fully coincide with the network measures
most often used in adolescent research, such as friendship and dislike.
Abstract
We examined the dynamics of the perception of 'dislike' ties (reputational dislike)
among adolescents within the contexts of friendship, perceived popularity,
substance use, and Facebook use. Survey data were collected from a
longitudinal sample of 238 adolescents from the 11th and 12th grades
in one California high school. We estimated stochastic actor-based network
dynamic models, using reports of reputational dislike, friendships,
and perceived popularity, to identify factors associated with the maintenance
and generation reputational dislike ties. The results showed that high-status
adolescents and more frequent Facebook users tended to become perceived as or
stay disliked by their peers over time. There was a tendency for friendships
to promote the creation and maintenance of reputational disliking but not
vice versa. Adolescents tended to perceive others as disliked when their friends
also perceived them as disliked. There was no evidence that either cigarette
smoking or drinking alcohol affected reputational dislike dynamics.
This study highlights the important role that the hierarchical peer system,
online peer context, and friendships play in driving information diffusion
of negative peer relations among adolescents.
Abstract
This chapter gives a general introduction to Stochastic Actor-Oriented
Models for Network Dynamics from a statistical point of view.
An e-print link to this article is here at the Siena website.
Abstract
Stochastic Actor Oriented Models for Network Dynamics are used for the statistical
analysis of longitudinal network data collected as a panel. The probability model defines
an unobserved stochastic process of tie changes, where social actors add new ties or drop
existing ties in response to the current network structure; the panel observations are
snapshots of the resulting changing network. The statistical analysis is based on
computer simulations of this process, which provides a great deal of flexibility in
representing data constraints and dependence structures.
In this Chapter we begin by defining the basic model.
We then explicate a new model for nondirected ties, including
several options for the specification of how pairs of actors coordinate tie changes. Next,
we describe coevolution models. These can be used to model the dynamics of several
interdependent sets of variables, such as the analysis of panel data on a network and the
behavior of the actors in the network, or panel data on two or more networks. We finish
by discussing the differences between Stochastic Actor Oriented Models and some other
longitudinal network models. A major distinguishing feature is the treatment of time,
which allows straightforward application of the model to panel data with different time
lags between waves. We provide a variety of applications in political science throughout.
Emmanuel Lazega
and Tom A.B. Snijders (eds).
Multilevel Network Analysis for the Social Sciences.
ISBN 978-3-319-24518-8 |
Abstract
First, an overview is given of the current state of multilevel modeling and of
statistical modeling of social networks, the latter with some extra attention
for network models with a multilevel flavour.
It is argued that both of these modeling approaches can be seen as responses
to Coleman's (1959) plea for a methodology that is oriented
to social organization rather than treating society as a collection
of atomized individuals.
Second, the combination of these approaches is discussed.
'Multilevel' and 'network analysis' can be combined in several ways.
'Multilevel Network Analysis' is defined as the combined network analysis
of several 'parallel' groups; a two-step approach was taken by
Snijders and Baerveldt (2003), integrated approaches were taken for
the p2 model bij Zijlstra, van Duijn, and Snijders (2006),
for the latent space model by Sweet, Thomas, and Junker (2013).
The 'Analysis of Multilevel Networks', on the other hand, is defined
– following Wang, Robins, Pattison, and Lazega (2013) – as
the analysis of a network structure with nodes of different types,
where the meaning of ties depends on the types of nodes they connect.
The implementation of such models is presented for Exponential Random Graph Models
and for Stochastic Actor-Oriented Models.
Other chapters in this volume contain examples; some of Multilevel Network
Analysis, others of the Analysis of Multilevel Networks.
Abstract
The stochastic actor-oriented model (Snijders, Sociological Methodology, 2001)
models the evolution of
networks over time, given panel data in a fixed group of actors,
where at each panel wave the network between these
actors (a digraph structure) as well as attribute variables for
these actors are observed. The parameters of this model
usually are estimated by a stochastic approximation version of
the method of moments. Statistics that correspond to
the parameters in a natural way are used for fitting the model.
Here we present an estimator based on the generalized
method of moments, i.e., using more statistics than parameters,
for minimizing the distance between observed statistics
and their expected values. Again, the resulting equation is solved
by stochastic approximation. Several algorithmic
issues arise that have to be solved in order to obtain a stable procedure.
For some examples we study the resulting gain
in statistical efficiency.
Abstract
This paper examines the collaboration structures and dynamics of the coauthorship
network of all Slovenian researchers. Its goal is to identify the key factors
driving collaboration and the main differences in collaboration behavior across scientific
fields and disciplines.
Two approaches to modelling network dynamics are combined in
this paper: the small-world model and the mechanism of preferential attachment, also
known as the process of cumulative advantage.
Stochastic-actor-based modelling of coauthorship
network dynamics uses data for the complete longitudinal co-authorship networks
for the entire Slovenian scientific community from 1996 to 2010. We confirmed the
presence of clustering in all fields and disciplines. Preferential attachment is far more
complex than a single global mechanism. There were two clear distinctions regarding
collaboration within scientific fields and disciplines. One was that some fields had an
internal national saturation inhibiting further collaboration. The second concerned the
differential impact of collaboration with scientists from abroad on domestic collaboration.
In the natural, technical, medical, and biotechnical sciences, this promotes collaboration
within the Slovenian scientific community while in the social sciences and humanities this
inhibits internal collaboration.
Abstract
Stochastic actor-based models for network dynamics have the primary aim
of statistical inference about processes of network change, but
may be regarded as a kind of agent-based models. Similar to many other
agent-based models, they are based on local rules for actor behavior.
Different from many other agent-based models, by including elements of generalized
linear statistical models they aim to be realistic detailed representations
of network dynamics in empirical data sets. Statistical parallels to
micro-macro considerations can be found in the estimation of parameters
determining local actor behavior from empirical data, and the
assessment of goodness of fit from the correspondence with network-level
descriptives.
This article studies several network-level consequences of
dynamic actor-based models applied to represent cross-sectional network data.
Two examples illustrate how network-level characteristics can be obtained
as emergent features implied by micro-specifications of actor-based models.
Abstract
The complex interplay between bullying/victimization and defending was
examined using a longitudinal social network approach (stochastic actor-based
models). The (co)evolution of these relations within three elementary schools
(Grades 2–5 at Time 1, ages 8–11, N = 354 children) was investigated across
three time points within a year. Most bullies and defenders were in the
same grade as the victims, although a substantial number of bullies and defenders
were in other grades (most often one grade higher). Defenders were usually
of the same gender as the victims, whereas most bullies were boys, with boys
bullying both boys and girls. In line with goal-framing theory, multiplex
network analyses provided evidence for the social support hypothesis (victims
with the same bullies defended each other over time) as well as the
retaliation hypothesis (defenders run the risk of becoming victimized by
the bullies of the
victims they defend). In addition, the analysis revealed that bullies with
the same victims defended each other over time and that defenders of bullies
initiated harassment of those bullies' victims. This study can be seen as a
starting point in unraveling the relationship dynamics among bullying, victimization,
and defending networks in schools.
Abstract
Antisocial behavior typically increases during early adolescence,
but the possibility of seasonal variation has not been examined.
In this study, trajectories of antisocial behavior were estimated
for early adolescent boys and girls. Data were obtained from a
3-year longitudinal study of 11 middle schools in the western
United States (n = 5,742), with assessments completed four times
per academic year. Antisocial behavior increased steadily throughout
6th grade, but beginning in 7th grade for boys and 8th grade for
girls it declined during the school year.
Significant increases between Grades 6–7 and 7–8 were
found for both genders. Trajectories varied by contextual
and individual-level social victimization and gender.
Implications for theoretical development and future studies are discussed.
Abstract
A complete survey of a network in a large population may be prohibitively difficult and costly. So it is
important to estimate models for networks using data from various network sampling designs, such
as link-tracing designs. We focus here on snowball sampling designs, designs in which the members
of an initial sample of network members are asked to nominate their network partners, their network
partners are then traced and asked to nominate their network partners, and so on. We assume an
exponential random graph model (ERGM) of a particular parametric form and outline a conditional
maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure
is intended to complement the likelihood approach developed by Handcock and Gile (2010) by providing
a practical means of estimation when the size of the complete network is unknown and/or the complete
network is very large. We report the outcome of a simulation study with a known model designed to
assess the impact of initial sample size, population size, and number of sampling waves on properties of
the estimates. We conclude with a discussion of the potential applications and further developments of
the approach.
Abstract
A novel version the stochastic actor-based modeling (SABM) framework
is applied to model the diffusion of
first alcohol use through middle school–wide longitudinal networks of early
adolescents, aged approximately
11-14 years. Models couple a standard SABM for friendship network evolution
with a proportional hazard model for
first alcohol use. Meta-analysis of individual models for 12 schools found
significant effects for friendship selection
based on the same alcohol use status and for an increased rate of onset to
first use based on exposure to already-onset
peers. Onset rate was greater at higher grades and among participants who
spent more unsupervised time with
friends. Neither selection nor exposure effects interacted with grade,
adult supervision, or gender.
Abstract
Longitudinal social networks are increasingly given by event data,
i.e., data coding the time and type of interaction between social actors.
Examples include networks stemming from computer-mediated communication,
open collaboration in wikis, phone call data, and interaction among political actors.
In this paper we propose a general model for networks of dyadic, typed events.
We decompose the probability of events into two components:
the first modeling the frequency of interaction and the second modeling the
conditional event type, i.e., the quality of interaction, given that interaction takes place.
While our main contribution is methodological, for illustration
we apply our model to data about political cooperation and conflicts collected
with the Kansas Event Data System. Special emphasis is given to the fact that
some explanatory variables
affect the frequency of interaction while others rather determine the
level of cooperativeness vs. hostility, if interaction takes place.
Furthermore, we analyze if and how model components controlling for
network dependencies affect findings on the effects of more traditional
predictors such as geographic proximity or joint alliance membership.
We argue that modeling the conditional event type is a valuable - and in some cases superior -
alternative to previously proposed models for networks of typed events.
Abstract
This chapter considers networks as relational structures
in a given set of social actors, and provides an overview of
models and some empirical results for dynamics of social networks,
considered in a setting of social actors optimizing a utility
function that is based, among other things, on their network
embeddedness (excluding purely rule-based models).
Some attention is also paid to network equilibrium,
this being relevant for network dynamics as a potential final state.
The chapter starts with discussing some basic empirical
regularities for social networks,
and how they can be understood from a rational actor perspective.
This is a background to the rest of the chapter,
focusing on models for representing networks.
It then turns to a discussion of game-theoretic models for network
equilibrium and network dynamics. The game-theoretic approach has
difficulties in becoming well aligned to empirical reality;
the latter is the purpose of the statistical models treated next.
Network dynamics is important especially in studies where not only
the network but also actor properties are endogenized.
Therefore, the second part of the chapter discusses models for
the joint dynamics of networks and actor characteristics,
both in a game-theoretic and in a statistical framework.
Abstract
We propose a new stochastic actor-oriented model
for the co-evolution of two-mode and one-mode networks.
The model posits that activities of a set of actors, represented in the
two-mode network, co-evolve with exchanges and interactions between the actors,
as represented in the one-mode network.
The model assumes that the actors, not the activities, have agency.
The empirical value of the model is demonstrated by examining
how employment preferences co-evolve
with friendship and advice relations in a group of seventy-five MBA students.
The analysis shows that activity in the two-mode network,
as expressed by number of employment preferences,
is related to activity in the friendship network,
as expressed by outdegrees.
Further, advice ties between students lead to
agreement with respect to employment preferences.
In addition, considering the multiplexity of
advice and friendship ties yields a better understanding of the
dynamics of the advice relation:
tendencies to reciprocation and homophily
in advice relations are mediated to an important extent
by friendship relations.
The discussion pays attention to the implications of this
study in the broader context of current efforts to model the
co-evolutionary dynamics of social networks and individual behavior.
Abstract
Three relations between elementary school children were investigated: networks of general dislike and
bullying were related to networks of general like. These were modeled using multivariate cross-sectional
(statistical) network models. Exponential random graph models for a sample of 18 classrooms, numbering
393 students, were summarized using meta-analyses. Results showed (balanced) network structures
with positive ties between those who were structurally equivalent in the negative network. Moreover,
essential structural parameters for the univariate network structure of positive (general like) and negative
(general dislike and bullying) tie networks were identified. Different structures emerged in positive and
negative networks. The results provide a starting point for further theoretical and (multiplex) empirical
research about negative ties and their interplay with positive ties.
Abstract
Geographic proximity is a determinant factor of friendship. Friendship datasets that include detailed geographic information are scarce, and when this information is available, the dependence of friendship on distance is often modelled by pre-specified parametric functions or derived from theory without further empirical assessment. This paper aims to give a detailed representation of the association between distance and the likelihood of friendship existence and friendship dynamics, and how this is modified by a few basic social and individual factors. The data employed is a three-wave network of 336 adolescents living in a small Swedish town, for whom information has been collected on their household locations.
The analysis is a three-step process that combines (1) nonparametric logistic regressions to unravel the overall functional form of the dependence of friendship on distance, without assuming it has a particular strength or shape; (2) parametric logistic regressions to construct suitable transformations of distance that can be employed in (3) stochastic models for longitudinal network data, to assess how distance, individual covariates, and network structure shape adolescent friendship dynamics. It was found that the log-odds of friendship existence and friendship dynamics decrease smoothly with the logarithm of distance.
For adolescents in different schools the dependence is linear, and stronger than for adolescents in the same school. Living nearby accounts, in this dataset, for an aspect of friendship dynamics that is not explicitly modelled by network structure or by individual covariates. In particular, the estimated distance effect is not correlated with reciprocity or transitivity effects.
The second edition of an extensive textbook on multilevel analysis.
Material about this book is available
at a separate web page.
Chapters
Abstract.
The issue of the influence of norms on behavior is as old as sociology itself. This paper explores the
effect of normative homophily (i.e. "sharing the same normative choices") on the evolution of the advice
network among lay judges in a courthouse. Blau's (1955, 1964) social exchange theory suggests that
members select advisors based on the status of the advisor. Additional research shows that members of an
organization use similarities with others in ascribed, achieved or inherited characteristics, as well as other
kinds of ties, to mitigate the potentially negative effects of this strong status rule. We elaborate and test
these theories using data on advisor choice in the Commercial Court of Paris.Weuse a jurisprudential case
about unfair competition (material and "moral" damages), a case thatwesubmitted to all the judges of this
court, to test the effect of normative homophily on the selection of advisors, controlling for status effects.
Normative homophily is measured by the extent to which two judges are equally "punitive" in awarding
damages to plaintiffs. Statistical analyses combine longitudinal advice network data collected among
the judges with their normative dispositions. Contrary to what could be expected from conventional
sociological theories, we find no pure effect of normative homophily on the choice of advisors. In this
case, therefore, sharing the same norms and values does not have, by itself, a mitigating effect and does
not contribute to the evolution of the network. We argue that status effects, conformity and alignments on
positions of opinion leaders in controversies still provide the best insights into the relationship between
norms, structure and behavior.
Key words:Advice networks, Longitudinal analysis, Homophily, Norms, Social selection, Status, Learning.
Abstract.
Studies of peer effects in educational settings confront two main problems.
The first is the presence of endogenous sorting which confounds the effects
of social influence and social selection on individual attainment.
The second is how to account for the local network dependencies through
which peer effects influence individual behavior. We empirically address
these problems using longitudinal data on academic performance, friendship,
and advice seeking relations among students in a full-time graduate academic
program. We specify stochastic agent-based models that permit estimation of
the interdependent contribution of social selection and social influence
to individual performance. We report evidence of peer effects.
Students tend to assimilate the average performance of their friends
and of their advisors. At the same time, students attaining similar
levels of academic performance are more likely to develop friendship and advice ties.
Together, these results imply that processes of social influence and
social selection are sub-components of a more general a co-evolutionary process
linking network structure and individual behavior.
We discuss possible points of contact between our findings and
current research in the economics and sociology of education.
Key words: Peer effects; Stochastic actor-oriented models; Social networks; Network dynamics; Education
Abstract.
Statistical models for social networks as dependent variables must represent
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 conditionally
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.
Key words: Social networks, Statistical modeling, Inference
Abstract.
This article examines the dynamics of peer relationships across the first
2 grades of Dutch junior high schools (average age
13 - 14). Specifically, we studied how gender and compositional
changes in classrooms structured the changes in peer
relationships between the 2 grades.
Expectations were derived from past research, and we tested whether these held when
methods for data analysis were applied that control appropriately for
the dependence structure of the data (more specifically,
multilevel analysis and a multilevel application of actor-oriented models
for network evolution). Analyses revealed
that the stability of peer acceptance was relatively low and that it
was affected neither by the level of classroom
stability nor by gender. Dyadic relationships were moderately stable.
Tendencies toward reciprocity, network closure, and
gender similarity shaped the changes in networks of peer relationships
within classes. Contrary to past findings, female
newcomers in classrooms were equally as well accepted as male newcomers
or established class members.
Key words: Social networks, Statistical modeling, Inference
Abstract.
This article examines the use of various research designs in the social sciences as
well as the choices that are made when a quasi-experimental design is used. A content analysis
was carried out on articles published in 18 social science journals with various impact
factors. The presence of quasi-experimental studies was investigated as well as choices in the
design and analysis stage. It was found that quasi-experimental designs are not very often
used in the inspected journals, and when they are applied they are not very well designed
and analyzed. These findings suggest that the literature on how to deal with selection bias
has not yet found its way to the practice of the applied researcher.
Key words: Quasi-experiments, Social science, Selection bias, Research designs, Content analysis.
DOI: http://dx.doi.org/10.1080/0022250X.2010.485707.
Abstract.
For exponential random graph models, under quite general conditions, it is
proved that induced subgraphs on node sets disconnected from the other
nodes still have distributions from an exponential random graph model.
This can help in the theoretical interpretation of such models. An
application is that for saturated snowball samples from a potentially larger
graph which is a realization of an exponential random graph model, it is
possible to do the analysis of the observed snowball sample within the
framework of exponential random graph models without any knowledge of
the larger graph.
Key words: Connected component, network delineation, network boundary, random graphs, snowball sample.
DOI: http://dx.doi.org/10.1214/09-AOAS313.
Abstract.
A model for network panel data is discussed, based on the assumption
that the observed data are discrete observations of a
continuous-time Markov process on the space of all directed graphs on a given
node set, in which changes in tie variables are independent
conditional on the current graph. The model for tie changes is
parametric
and designed for applications to social network analysis, where the
network dynamics can be interpreted as being generated by choices
made by the social actors represented by the nodes of the graph. An
algorithm for calculating the Maximum Likelihood estimator is
presented, based on data augmentation and stochastic approximation.
An application to an evolving friendship network is given and a small
simulation study is presented which suggests that for small data sets
the Maximum Likelihood estimator is more efficient than the earlier
proposed Method of Moments estimator.
Key words: Graphs, Longitudinal data, Method of moments, Stochastic approximation, Robbins-Monro algorithm.
Abstract.
A recurrent problem in the analysis of behavioral dynamics, given a simultaneously evolving
social network, is the difficulty of separating effects of partner selection from effects of social
influence. Because misattribution of selection effects to social influence, or vice versa, suggests
wrong conclusions about the social mechanisms underlying the observed dynamics, special
diligence in data analysis is advisable. While a dependable and valid method would benefit
several research areas, according to the best of our knowledge, it has been lacking in the extant
literature. In this paper, we present a recently developed family of statistical models that enables
researchers to separate the two effects in a statistically adequate manner. To illustrate our
method, we investigate the roles of homophile selection and peer influence mechanisms in the
joint dynamics of friendship formation and substance use among adolescents. Making use of a
three-wave panel measured in the years 1995-97 at a school in Scotland, we are able to assess the
strength of selection and influence mechanisms and quantify the relative contributions of
homophile selection, assimilation to peers, and control mechanisms to observed similarity of
substance use among friends.
Key words: statistical modeling, social networks, graphs, longitudinal, network dynamics, smoking, alcohol consumption.
The methods proposed in this paper are implemented in the SIENA program .
Software implementation.
This method is implemented in visone
(release of September 5, 2012).
There are some tutorials in the visone wiki on how to use it:
an example-based illustration using Wikipedia edit data can be found on
http://visone.info/wiki/index.php/Wikipedia_edit_networks_(tutorial)
and an example with the political event data is on
http://visone.info/wiki/index.php/Event_networks_(tutorial).
The implementation can deal with general event types and weights and with general date/time formatting
Key words: statistical modeling, longitudinal, Markov chain, agent-based model, peer selection, peer influence.
Key words: Smoking; Adolescents; Selection; Influence; Friends; Reciprocity; Siena
Key words: Residuals, Hausman test, empirical Bayes, spline functions, deletion residuals, influence diagnostics, non-linear transformations, mixed models, Hierarchical Linear Model.
Abstract.
Multiple regression quadratic assignment procedures (MRQAP)
tests are permutation tests for multiple
linear regression model coefficients for data organized in
square matrices of relatedness among n
objects. Such a data structure is typical in social network
studies, where variables indicate some type
of relation between a given set of actors.
We present a new permutation method (called "double semipartialing",
or DSP) that complements the family of extant approaches to MRQAP tests.
We assess the
statistical bias (type I error rate) and statistical power of the set
of five methods, including DSP, across a
variety of conditions of network autocorrelation,
of spuriousness (size of confounder effect),
and of skewness in the data.
These conditions are explored across three assumed
data distributions: normal, gamma,
and negative binomial.
We find that the Freedman-Lane method and the DSP method are the most robust
against a wide array of these conditions.
We also find that all five methods perform better if the test
statistic is pivotal.
Finally, we find limitations of usefulness for MRQAP tests:
All tests degrade under
simultaneous conditions of extreme skewness and
high spuriousness for gamma and negative binomial
distributions.
Key words:MRQAP, Mantel tests, permutation tests, social networks, network autocorrelation, collinearity, dyadic data.
DOI: http://dx.doi.org/10.1016/j.jspi.2007.04.011.
Abstract.
We consider a continuous-time model for the evolution of social networks.
A social network is here conceived as a (di-)graph
on a set of vertices, representing actors, and the changes of interest are
creation and disappearance over time of (arcs) edges in the graph.
Hence we model a collection of random edge indicators that are not,
in general, independent. We explicitly model the
interdependencies between edge indicators that arise from interaction between social entities.
A Markov chain is defined in terms of an embedded chain with holding times and
transition probabilities. Data are observed at fixed points in time and hence we are
not able to observe the embedded chain directly.
Introducing a prior distribution for the parameters we may implement an MCMC
algorithm for exploring the posterior distribution of the
parameters by simulating the evolution of the embedded process between observations.
Key words: Longitudinal social networks; Data augmentation; Bayesian inference; Random graphs.
Key words: Social networks; ERGM; Dependence structure
Key words: delinquency; friendship networks; interdependence; SIENA
Abstract.
This article reviews new specifications for exponential random graph models proposed by
Snijders et
al. (2006) and demonstrates their improvement over homogeneous
Markov random graph models in fitting empirical network data. Not only do the new specifications show
improvements in goodness of fit for various data sets, but they also help to avoid the problem of neardegeneracy
that often afflicts the fitting of Markov random graph models in practice, particularly to network
data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows
estimation of parameters of exponential graph models for many (but not all) cases where it is impossible
to estimate parameters of homogeneous Markov graph models. The new specifications were used to model
a large number of classical small-scale network data sets and showed a dramatically better performance
than Markov graph models. We also review three current programs for obtaining maximum likelihood
estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with
less accurate pseudo-likelihood estimates. Finally, we discuss whether homogeneous Markov random graph
models may be superseded by the new specifications, and how additional elaborations may further improve
model performance.
Key words: Exponential random graph models; p* models; Statistical models for social networks
Abstract.
A parametric, continuous-time Markov model for digraph panel data is considered.
The parameter is estimated by the method of moments.
A convenient method for estimating the variance-covariance matrix of the moment estimator relies on the delta method, requiring the Jacobian matrix - that is, the matrix of partial derivatives - of the estimating function.
The Jacobian matrix was estimated hitherto by Monte Carlo methods based on finite differences.
Three new Monte Carlo estimators of the Jacobian matrix are proposed,
which are related to the likelihood ratio / score function method of derivative estimation
and have theoretical and practical advantages compared to the finite differences method.
Some light is shed on the practical performance of the methods
by applying them
in a situation where the true Jacobian matrix is known and in a situation where the true Jacobian matrix is unknown.
Key words: digraphs, continuous-time Markov processes, gradient estimation, likelihood ratio / score function method, variance reduction, control variates.
The methods proposed in this paper are implemented in the SIENA program .
Abstract.
A deeper understanding of the relation
between individual behavior and individual actions on one hand and
the embeddedness of individuals in social structures on the other
hand can be obtained by empirically studying the dynamics of
individual outcomes and network structure, and how these mutually
affect each other. In methodological terms, this means that behavior
of individuals -- indicators of performance and success, attitudes
and other cognitions, behavioral tendencies -- and the ties between
them are studied as a social process evolving over time, where
behavior and network ties mutually influence each other. We propose
a statistical methodology for this type of investigation and
illustrate it by an example.
Key words: statistical modeling, social networks, graphs, longitudinal, network dynamics.
The methods proposed in this paper are implemented in the SIENA program .
Abstract.
We analyse the co-evolution of social networks and substance use behaviour of adolescents
and address the problem of separating the effects of homophily and assimilation. Adolescents
who prefer friends with the same substance-use behaviour exhibit the homophily principle.
Adolescents who adapt their substance use behaviour to match that of their friends display
the assimilation principle. We use the Siena software to illustrate the co-evolution of
friendship networks, smoking, cannabis use and drinking among sport-active teenagers.
Results indicate strong network selection effects occurring with a preference for same sex
reciprocated relationships in closed networks. Assimilation occurs among cannabis and
alcohol but not tobacco users. Homophily prevails among tobacco and alcohol users.
Cannabis use influences smoking behavior positively (i.e., increasing cannabis increases
smoking). Weaker effects include drinkers smoking more and cannabis users drinking more.
Homophily and assimilation are not significant mechanisms with regard to sporting activity
for any substance. There is, however, a significant reduction of sporting activity among
smokers. Also, girls engaged in less sport than boys. Some recommendations for health
promotion programmes are made.
Key words: statistical modeling, social networks, graphs, longitudinal, network dynamics.
The methods proposed in this paper are implemented in the SIENA program .
Abstract.
Social networks can be defined as the patterns of ties between social
actors.
This paper gives a review of recently developed statistical models
and estimation methods for the analysis of social network
panel data.
To represent the feedback processes inherent in network dynamics,
it is helpful to regard such panel data as momentary observations
on a continuous-time process on the space of directed graphs.
Tie-oriented and actor-oriented stochastic models are presented,
which can reflect endogenous network dynamics
as well as effects of exogenous variables.
These models do not allow explicit calculations, but they can be implemented
as computer simulation models.
Stochastic approximation methods can be used to estimate the parameters.
An example is given where the models are applied to an early precursor
of email communication.
Abstract.
Abstract
The purpose of this study is to examine whether peer relations within classrooms were related
to students' academic progress, and if so, whether this can be explained by students' relatedness
and engagement, in line with Connell and Wellborn's self-system model. We analyzed data of
18,735 students in 796 school classes in Dutch junior high schools, using multilevel analysis.
Academic progress, conceptualized as regular promotion to the next year versus grade retention,
moving upward, and moving downward in the track system, was measured at the time of
transition between Grades 1 and 2 (equivalent to US Grades 7 and 8). The results indicated that
students who were accepted by their peers had lower probabilities to retain a grade or to move
downward in the track system. Although peer acceptance was associated with relatedness and
engagement, these variables did not explain why peer acceptance was associated to academic progress. Furthermore, peer acceptance and relatedness were more strongly related in classes with
more negative class climates.
Abstract.
The chapter in this volume
by Dronkers and Hox presents an interesting multilevel event history analysis of divorce risks.
The sibling design gives excellent opportunities for studying the similarity between
brothers and sisters in the risks of divorce. Various discussion points are raised,
all of which bear in some way upon the choice of predictor variables in the multilevel
logistic regression. Questions are posed about
the level of detail of modeling time trends; about the fact that sampling weights are a
function of number of siblings; and about the inclusion in the fixed part of the model
of the fraction of previously divorced siblings, which is correlated with the family-level
random intercept.
Abstract.
We give a nontechnical introduction into recently developed methods for analyzing the coevolution of social networks and behavior(s) of the network actors. This coevolution is crucial for a variety of research topics that currently receive a lot of attention, such as the role of peer groups in adolescent development. A family of dynamic actor-driven models for the coevolution process is sketched, and it is shown how to use the SIENA software for estimating these models. We illustrate the method by analyzing the coevolution of friendship networks, taste in music, and alcohol consumption of teenagers.
Key words: network dynamics, longitudinal, social networks, stochastic modeling.
The methods proposed in this paper are implemented in the SIENA program .
DOI: http://dx.doi.org/10.1111/j.1467-9531.2006.00176.x
Abstract.
The most
promising class of statistical models for expressing structural
properties of social networks observed at one moment in time, is the
class of Exponential Random Graph Models (ERGMs), also known as p*
models. The strong point of these models is that they can represent
a variety of structural tendencies, such as transitivity, that
define complicated dependence patterns not easily modeled by more
basic probability models. Recently, MCMC algorithms have been
developed which produce approximate Maximum Likelihood estimators.
Applying these models in their traditional specification to observed
network data often has led to problems, however, which can be traced
back to the fact that important parts of the parameter space
correspond to nearly degenerate distributions, which may lead to
convergence problems of estimation algorithms, and a poor fit to
empirical data.
This paper proposes new specifications of
Exponential Random Graph Models. These specifications represent
structural properties such as transitivity and heterogeneity of
degrees by more complicated graph statistics than the traditional
star and triangle counts. Three kinds of statistic are proposed:
geometrically weighted degree distributions, alternating
k-triangles, and alternating independent two-paths. Examples are
presented both of modeling graphs and digraphs, in which the new
specifications lead to much better results than the earlier existing
specifications of the ERGM. It is concluded that the new
specifications increase the range and applicability of the ERGM as a
tool for the statistical analysis of social networks.
Key words: statistical modeling, social networks, graphs, transitivity, clustering, maximum likelihood, MCMC, p* model.
Also see Snijders (2002).
The methods
proposed in this paper are implemented in the SIENA program , part of the StOCNET package.
Abstract.
This paper proposes a multilevel extension to the p2 model for the analysis of social networks. In the p2 model dichotomous tie observations between actors in a given set can be regressed on explanatory variables. The multilevel p2 model is a model for social networks with a multilevel data structure, e.g., networks observed in multiple schools. It defines an identical model for the independent observations of the same type of social network, where the parameters can be allowed to vary across the social networks using random effects. For the multilevel p2 model a Bayesian MCMC algorithm has been developed, which is briefly described here. The model is applied to investigate reported received practical support among Dutch high school pupils of different ethnic backgrounds.
The methods proposed in this paper are implemented in the StOCNET package.
Abstract.
This chapter treats statistical methods for network evolution. It
is argued that it is most fruitful to consider models where network
evolution is represented as the result of many (usually non-observed)
small changes occurring between the consecutively observed networks.
Accordingly, the focus is on models where a continuous-time network
evolution is assumed although the observations are made at discrete
time points (two or more).
Three models are considered in detail, all based on the assumption
that the observed networks are outcomes of a Markov process
evolving in continuous time. The independent arcs model is a trivial
baseline model. The reciprocity model expresses effects of reciprocity,
but lacks other structural effects. The actor-oriented model is based
on a model of actors changing their outgoing ties as a consequence of
myopic stochastic optimization of an objective function. This framework
offers the flexibility to represent a variety of network effects. An
estimation algorithm is treated, based on a Markov chain Monte Carlo
implementation of the method of moments.
Key words: network evolution, Markov process, stochastic actor-oriented network model.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA program.
Abstract.
In research on the social capital of individuals,
there has been little standardisation of measurement instruments.
In this paper we propose two innovations. First, a new
measurement method: the Resource Generator; an instrument with concretely
worded items covering `general' social capital in a population, that
combines advantages of earlier techniques. Construction, use, and first
empirical findings are discussed for a representative sample (N = 1,004)
for the Dutch population in 1999-2000. Second, we propose to
investigate social capital by latent trait analysis, and we
identify separately accessed portions of social capital: prestige and
education related social capital, entrepreneurial social capital, skills
social capital, and personal support social capital. This underlines that
social capital measurement needs multiple measures, and cannot be reduced
to one total measure of indirectly `owned' resources. Constructing a theory-based
Resource Generator can be a challenge for different
contexts of use, but also retrieve meaningful
information for investigating the productivity and goal specificity of social capital.
This paper is part of the
Ph.D. research by Martin van der Gaag on measurement of social capital.
Abstract.
A measure for explained variation is proposed for stochastic actor-driven models
for data on social networks. The measure is based on the entropy of the distribution of the choices
made by the actors during the network evolution process. This measure can be helpful in the
specification and interpretation of statistical models for longitudinal network data.
Key words: Complete network, Longitudinal study, Dynamics, Explained variation, Coefficient of Determination, Entropy.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA program .
Abstract.
A random effects model is proposed for the analysis of binary dyadic data that
represent a social network or directed graph, using nodal and/or dyadic attributes
as covariates. The network structure is reflected by modeling the dependence
between the relations to and from the same actor or node. Parameter estimates
are proposed that are based on an iterated generalized least squares procedure.
An application is presented to a data set on friendship relations between American
lawyers.
The methods proposed in this paper are implemented in the StOCNET package.
Abstract.
A class of statistical models is proposed which aims to recover latent
settings structures in social networks. Settings may be regarded as
clusters of vertices. The measurement model builds on two assumptions.
The observed network is assumed to be generated by hierarchically
nested latent transitive structures, expressed by ultrametrics.
It is assumed that expected tie strength decreases with ultrametric
distance. The approach could be described as model-based clustering
with an ultrametric space as the underlying metric to capture the dependence
in the observations. Maximum likelihood methods as well
as Bayesian methods are applied for statistical inference. Both approaches
are implemented using Markov chain Monte Carlo methods.
The methods proposed in this paper are implemented in the StOCNET package.
Abstract. Markov chains can be used for the modeling of complex longitudinal network data. One class of probability models to model the evolution of social networks are stochastic actor-oriented models for network change, proposed by Snijders (1996, 2001). These models are continuous-time Markov chain models that are implemented as simulation models. In this paper an extension of the simulation algorithm of stochastic actor-oriented models is proposed to include networks of changing composition. In empirical research, the composition of networks may change due to actors joining or leaving the network at some points in time. The composition changes are modeled as exogenous events that occur at given time points and are implemented in the simulation algorithm. The estimation of the network effects and the effects of actor and dyadic attributes that influence the evolution of the network, is based on the simulation of Markov chains.
Key words: network evolution, Markov process, stochastic actor-oriented network model, changing composition.
Also see Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA. program.
Abstract.
This is a chapter in the volume on
the 1999 SCALE conference on social capital (Amsterdam, december 9-11, 1999).
The chapter presents a conceptual approach to the measurement of social
capital as defined on the level of individuals, with the aim to develop a
yardstick for social capital that can be used in prospective studies
investigating its productivity and goal specificity. It discusses several
theoretical choices that should be made before starting
measurements, and introduces an empirical approach to the
construction of domain specific social capital measures.
This paper is part of the
Ph.D. research by Martin van der Gaag on measurement of social capital.
Key words: MANOVA, incomplete data, missing at random, hierarchical linear model, Hotelling's test, Wald test, Lawley - Hotelling trace criterion, trend tests, compound symmetry model.
Abstract.
Degrees (the number of links attached to a given node) play a particular
and important role in empirical network analysis because of their obvious
importance for expressing the position of nodes.
It is argued here that there is no general straightforward relation
between the degree distribution on one hand and structural aspects on
the other hand, as this relation depends on further characteristics of
the presumed model for the network. Therefore empirical inference from
observed network characteristics to the processes that could be responsible
for network genesis and dynamics cannot be based only, or mainly, on the
observed degree distribution.
As an elaboration and practical implementation of this point,
a statistical model for the dynamics of networks, expressed as digraphs
with a fixed vertex set,
is proposed in which the outdegree distribution is governed by parameters
that are not connected to the parameters for the structural dynamics.
The use of such an approach in statistical modeling
minimizes the influence of the observed degrees on the conclusions
about the structural aspects of the network dynamics.
The model is a stochastic actor-oriented model, and deals
with the degrees in a manner resembling Tversky's
Elimination by Aspects approach.
A statistical procedure for parameter estimation in
this model is proposed, and an example is given.
Also see
Snijders (2001).
The methods proposed in this paper are implemented
in the SIENA. program.
Abstract.
A multilevel approach is proposed to the study of the evolution
of multiple networks. In this approach, the basic evolution process
is assumed to be the same, while parameter values may differ
between different networks.
For the network evolution process,
stochastic actor-oriented models are used, of which the parameters
are estimated by Markov chain Monte Carlo methods.
This is applied to the study of effects of delinquent behavior
on friendship formation, a question of long standing in criminology.
The evolution of friendship is studied empirically in 19 school classes.
It is concluded that there is evidence for an effect of
similarity in delinquent behavior on friendship evolution.
Similarity of the degree of
delinquent behavior has a positive effect on tie formation
but also on tie dissolution.
The last result seems to contradict criminological theories, and deserves
further study.
Key words: actor-oriented model; longitudinal data; social networks; criminology; adolescents.
Also see Snijders (2001).
Abstract.
Markov graphs and exponential random graph models are an important family
of probability distributions for graphs and digraphs because they allow
the kind of dependency that is often considered in social network
analysis, e.g., transitivity of choice. To estimate parameters in these
statistical models, pseudo-likelihood methods have been proposed, but they
are of doubtful value. Maximum likelihood (ML) estimates would be better
but are hard to calculate.
These can be approximated, however, by MCMC methods that solve the moment
equation. The use of MCMC methods in these models often is hampered by
convergence problems, of which the cause can be traced to steepness of the
moments as functions of the parameters;
moreover, in the region where this steepness
occurs, the distribution can have a bimodal shape, which in itself already
leads to serious convergence problems.
A possible way out of these problems is to model the degrees more
carefully. On one hand, precisely modeling the degrees may confine the
algorithm to a region in the parameter space where the moment function is
well-behaved and where the distribution has a unimodal shape. On the other
hand, modeling the degrees may lead to a better fitting model, which also
can lead to a better-behaving algorithm.
Three types of specification of exponential random digraph models are
considered: (1) conditional on the number of ties; (2) conditional on all
in- and out-degrees; (3) conditional on the number of ties, and icluding
incidental vertex parameters. In some examples, it is investigated
how well it is possible to achieve convergence in the MCMC parameter
estimation, and how the parameter estimates differ between these
specifications.
Also see Snijders (JoSS, 2002).
The methods proposed in this paper are implemented
in
the StOCNET package.
Abstract.
In this study we try to estimate the size of the homeless population
in Budapest by using
two "non-standard" sampling methods: snowball sampling and the
capture-recapture method. Using
two methods and three different data sets we are able to compare
the methods as well as the results,
and we also suggest some further applications. Apart from the
practical purpose of our study there
is a methodological one as well: to use two relatively unknown
methods for the estimations of this
very peculiar kind of population.
Key words: snowball sampling, capture-recapture, hidden population, homeless.
Abstract.
A number of estimation methods of the variance components
in Wing & Kristofferson's model for inter-response times are examined
and compared by means of a simulation study.
The estimation methods studied are the method of moments,
maximum likelihood, and an alternative approach in which
the WK-model is recognized as a moving average model.
Key words: discrete motor responses, moving average model, EM, maximum likelihood, method of moments.
By clicking here you can run the JAVA applet that is used in this paper to demonstrate proprties of the treated probability model.
The estimation procedure in this publication is available in the program SIENA.
Abstract.
This paper is about estimating the parameters of the
exponential random graph model, also known as the p* model,
using frequentist Markov chain Monte Carlo (MCMC) methods.
The exponential random graph model is simulated using Gibbs
or Metropolis-Hastings sampling.
The estimation procedures considered are based on
the Robbins-Monro algorithm for approximating
a solution to the likelihood equation.
A major problem with exponential random
graph models resides in the fact that such models
can have, for certain parameter values, bimodal
(or multimodal) distributions
for the sufficient statistics such as the number of ties.
The bimodality of the exponential graph distribution
for certain parameter values seems a severe limitation
to its practical usefulness.
The possibility of bi- or multimodality is reflected in the possibility that the
outcome space is divided into two (or more) regions
such that the more usual type of MCMC algorithms,
updating only single relations, dyads, or triplets,
have extremely long sojourn times
within such regions, and a negligible probability to
move from one region to another.
In such situations, convergence to the target distribution
is extremely slow.
To be useful, MCMC algorithms must be able to make transitions
from a given graph to a very different graph.
It is proposed to include transitions to the graph complement
as updating steps
to improve the speed of convergence to the target distribution.
Estimation procedures implementing these ideas work satisfactorily for some
data sets and model specifications, but not for all.
Key words: p* model; Markov graph; digraphs; exponential family; maximum likelihood; method of moments; Robbins-Monro algorithm; Gibbs sampling; Metropolis-Hastings algorithm.
Also see Snijders, Pattison, Robins, and Handcock (2006).
The methods proposed in this paper are implemented in
the SIENA program in
the StOCNET package,
and also in the standalone pnet program and the R package statnet.
Abstract.
Available variance component tests are reviewed and three new score tests are presented.
In the first score test, the asymptotic normal distribution of the test statistic
is used as a reference distribution.
In the other two score tests, a Satterthwaite approximation is used
for the null distribution of the test statistic.
We evaluate the performance of the score tests and other available tests
by means of a Monte Carlo study.
The new tests are computationally relatively cheap and have
good power properties.
Key words: multilevel models; variance components; random coefficients; score tests; Monte Carlo study.
DOI: http://dx.doi.org/10.1111/0081-1750.00099.
Abstract.
A class of statistical models is proposed for longitudinal network data.
The dependent variable is the changing (or evolving) relation network,
represented by two or more observations of a directed graph
with a fixed set of nodes.
The nodes are modeled as actors whose choices determine the network.
Individual and dyadic exogenous variables can be used as covariates.
The change in the network is modeled as the stochastic result of
network effects (reciprocity, transitivity, etc.) and these covariates.
The existing network structure is a dynamic constraint for the
evolution of the structure itself.
The models are continuous time Markov chain models that
can be implemented as simulation models.
The network evolution is modeled as the consequence of the actors
making new choices, or withdrawing existing choices, on the basis
of functions, with fixed and random components, that the actors
try to maximize.
The models parameters must be estimated from observed data.
For estimating and testing these models,
statistical procedures are proposed which are based on the method of moments.
The statistical procedures are implemented
using a stochastic approximation algorithm based on
computer simulations of the network evolution process.
Key words: actor-oriented model; longitudinal data; continuous-time Markov process; Robbins-Monro algorithm; simulation models; method of moments; stochastic approximation; simulated moments; random utility; Markov chain Monte Carlo.
This paper is related to various other papers;
these can be found by searching in this publication list for the key word SIENA.
The methods proposed in this paper are implemented
in the program SIENA.
Abstract.
A statistical approach to a posteriori
blockmodeling for digraphs and
valued digraphs is proposed.
The probability model assumes that the vertices
of the digraph are partitioned
into several unobserved (latent) classes and that the
probability of a relationship between two vertices
depends only on the classes to which they belong.
A Bayesian estimator, based on Gibbs sampling, is proposed.
The basic model is not identified, because class labels are arbitrary.
The resulting identifiability problems are solved by restricting inference to
the posterior distributions of invariant functions of the
parameters and the vertex class membership.
In addition, models are considered where class labels are identified by
prior distributions for the class membership of some of the vertices.
The model is illustrated by an example from the social networks literature
(Kapferer's tailor shop).
Key words: Colored graph; Gibbs sampling; latent class model; social network; cluster analysis; mixture model.
This paper continues earlier work published as
Nowicki and Snijders (1997).
The methods proposed in this paper are implemented
in
the StOCNET package.
Key words: item response theory, person fit, asymptotic approximations.
Abstract.
The relation between multilevel analysis and multistage sampling
is discussed.
After this, much attention is paid to the determination of sample sizes
in multilevel analysis.
Key words: loneliness, item response theory, Rasch model, dimensionality, aging.
Abstract.
This paper considers a design where the
objects to be scaled are the higher level units; nested within each
object are lower level units, called `subjects';
and a set of dichotomous items is administered to each subject.
The subjects are regarded as strictly parallel tests
for the objects.
Examples are the scaling of teachers on the basis of their pupils' responses,
or of neighborhoods on the basis of responses by inhabitants.
A two-level version is elaborated of the non-parametric
scaling method first proposed by Mokken (1971).
The probabilities of positive responses to the items are assumed to be
increasing functions of the value on a latent trait.
The latent trait value for each subject is
composed of an object-dependent value and a
subject-dependent deviation from this value .
The consistency of responses within, but also between objects
is expressed by two-level versions of Loevinger's H coefficients.
The availability of parallel tests is used to calculate
a reliability coefficient.
Key words: Multi-level models, item response theory, reliability, parallel tests, ecometrics.
An extensive textbook on multilevel analysis.
Material about this book is available
at a separate web page.
Chapters
Abstract.
This paper is about social capital as a second-order resource of
individuals.
In spite of its growing popularity, social capital has mostly been
measured in ad hoc fashions.
This paper discusses possible approaches that could be taken to
measure the social capital of individuals.
What kinds of questions should be posed to the individual,
and how should these questions be integrated to a measure
of his or her social capital?
Several domains of well-being should be distinguished,
and social capital should be measured for these domains separately.
It is argued that aggregation over alters is not additive,
because the main distinction is between having no alter, or
at least one alter who could provide a given resource.
Aggregation over resources is necessary but debatable;
it can be based on either a common valuation, or on statistical
asociations, or on substitutability in the production of the individual's
well-being.
For studying the statistical association between second-order resources
available to a given individual, a distinction is proposed between,
on one hand, within-alter associations, and on the other,
within-ego associations.
The elaboration of these ideas into a questionnaire and a concrete
measurement instrument is being carried out in the
SCALE research programme and its 1999 survey of the
'social networks of the Dutch'.
Key words: social resources, aggregation.
This is further elaborated in the Ph.D. research by Martin van der Gaag on measurement of social capital.
Abstract.
Multilevel models are proposed to study relational or dyadic
data from multiple persons in families or other groups.
The variable under study is assumed to refer to a
dyadic relation between individuals in the groups.
The proposed models are elaborations of the Social Relations Model.
The different roles of father, mother, and child
are emphasized in these models.
Multilevel models provide researchers with a
method to estimate the variances and correlations of the
Social Relations Model, as well as to incorporate the effects of
covariates and to test specialized models, even for possibly incomplete data.
MLn/MLwiN macros for fitting these models can be obtained from my macro page.
Key words: network dynamics, longitudinal social network data, continuous-time Markov chain.
The methods used in this paper are implemented in the SIENA. program.
Key words: rational choice, friendship, Markov processes, random utility models, simulation, empirical test.
Key words: Multilevel analysis, network analysis, longitudinal models, mathematical modeling, gossip.
Abstract.
Actor-oriented models are proposed for the statistical analysis of
longitudinal social network data. These models are implemented as
simulation models, and the statistical evaluation
is based on the method of moments and the Robbins-Monro process
applied to computer simulation outcomes.
In this approach,
the calculations that are required for statistical inference are too
complex to be carried out analytically, and therefore they are replaced
by computer simulation.
The statistical models are continuous-time Markov chains.
It is shown how the reciprocity model of Wasserman
and Leenders can be formulated as a special case of the actor-oriented model.
Key words: Social networks, statistical modeling, actor-oriented model, continuous-time Markov chain, Robbins-Monro process.
Also see Snijders (2001) and the SIENA program.
Key words: Colored graph, EM algorithm, Gibbs sampling, latent class model, social network.
Also see Nowicki and Snijders (2001)
and the associated computer program
BLOCKS.
Key words. Dynamic access model, policy networks, computer simulation, method of moments, Robbins Monro process.
Abstract.
A class of models is proposed for longitudinal network data. These
models are along the lines of methodological individualism: actors use
heuristics to try to achieve their individual goals, subject to constraints.
The current network structure is among these constraints. The models
are continuous time Markov chain models that can be implemented as
simulation models. They incorporate random change in addition to the
purposeful change that follows from the actors' pursuit of their goals,
and include parameters that must be estimated from observed data.
Statistical methods are proposed for estimating and testing these
models. These methods can also be used for parameter estimation for
other simulation models. The statistical procedures are based on the
method of moments, and use computer simulation to estimate the
theoretical moments. The Robbins-Monro process is used to deal with
the stochastic nature of the estimated theoretical moments. An example
is given for Newcomb's fraternity data, using a model that expresses
reciprocity and balance.
Key words: methodological individualism; Markov process; Newcomb data; balance; Robbins-Monro process; simulation models; method of moments; simulated moments; random utility.
Key words: multilevel analysis, hierarchical linear model, random coefficients.
Key words: Personal network, snowball sample, multilevel analysis,hierarchical linear model, random effects, cocaine.
Also see van Duijn, van Busschbach and Snijders (1999).
Key words: R-squared, explained variance, coefficient of determination, multilevel analysis, misspecification.
Key words: Network sampling; random graphs; link-tracing designs.
Key words: Unfolding, item response theory, unimodal response models, total positivity, unidimensional scaling, measurement theory.
Key words: hierarchical linear model, multilevel research, sample design.
Key words: Snowball Sampling, Weighting, Parameter Estimations, Social Networks.
Key words: lognormal prior, Dirichlet prior, gamma prior, posterior mode, Rasch's multiplicative Poisson model, empirical Bayes estimation.
Abstract.
Data in the form of zero-one matrices where conditioning on the
marginals is relevant arise in diverse fields such as social networks and
ecology; directed graphs constitute an important special case. An
algorithm is given for the complete enumeration of the family of all
zero-one matrices with given marginals and with a prespecified set of
cells with structural zero entries. Complete enumeration is
computationally feasible only for relatively small matrices. Therefore, a
more useable Monte Carlo simulation method for the uniform
distribution over this family is given, based on unequal probability
sampling and ratio estimation. This method is applied to testing
reciprocity of choices in social networks.
Key words: adjacency matrices, random digraphs, networks, ecology, Monte Carlo methods, unequal probability sampling, reciprocity.
Key words: Consensus, Dice coefficient, Jaccard coefficient, Simple Matching coefficient, Multivariate binary data, Observer agreement, Similarity coefficients, Beta distribution.
Key words: Counts; personal networks; reliability; reliability of change; binomial distribution; random effects; empirical Bayes; regression.
Abstract.
A method is presented for testing change of digraphs (representing
some binary relation) observed at two points in time, labeled I and II.
The test is conditional on the entire digraph at time I, the numbers of
new arcs to and from each actor, and the numbers of disappeared arcs
to and from each actor. A new arc is defined as an arc existing at time
II but not at time I; a disappeared arc is an arc existing at time I but not
at time II. In particular, tests are conditional simultaneously on in-
degrees and out-degrees at times I and II. The elements of the dyad
transition matrix, indicating the numbers of dyads of some particular
type (mutual, asymmetric, of null) at time I, and of some (same or
other) type at time II, are proposed as possible test statistics.
Also see
Snijders (Psychometrika, 1991).
Key words: Subsistence agriculture, risk, early warning.
Key words: bipartite graphs, conditionally uniform distribution.
Key words: combination of tests, equality of correlated proportions, incomplete data, asymptotically most powerful test, Monte Carlo study, antithetic variates, power comparison.
Key words: antithetic variates, Monte Carlo, variance reduction, change-point test, Wilcoxon test.
Key words: graph heterogeneity, graph centrality, random graphs, degree variance.
Key words: Nonparametric tests, bivariate symmetry and asymmetry, locally most powerful tests, asymptotic normality.
Key words: communality, internal consistency, Heywood case, positive definite.
Key words: graph heterogeneity, graph centrality, random graphs, degree variance.
Key words: Empirical Bayes classification, complete class, monotone procedures.
Key words: Markov chain, ethology, transition analysis.