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Selected publications of Tom A.B. Snijders

A complete publication list is available by clicking here.

The publications for which a link is provided, are presented strictly for personal use only.

Tom A.B. Snijders, Malick Faye, and Julien Brailly (2020). Network dynamics with a nested node set: Sociability in seven villages in Senegal.
Statistica Neerlandica .

We propose two complementary ways to deal with a nesting structure in the node set of a network; such a structure may be called a multilevel network, with a node set consisting of several groups. First, within-group ties are distinguished from between-group ties by considering them as two distinct but interrelated networks. Second, effects of nodal variables are differentiated according to the levels of the nesting structure, to prevent ecological fallacies. This is elaborated in a study of two repeated observations of a sociability network in seven villages in Senegal, analyzed using the Stochastic Actor-oriented Model.

Josh A. Lospinoso and Tom A. B. Snijders (2019). Goodness of fit for Stochastic Actor-Oriented Models.
Methodological Innovations, 12, 2059799119884282.

We propose a Mahalanobis distance–based Monte Carlo goodness of fit testing procedure for the family of stochastic actor-oriented models for social network evolution. A modified model distance estimator is proposed to help the researcher identify model extensions that will remediate poor fit.

Nynke M. D. Niezink, Tom A. B. Snijders, and Marijtje van Duijn (2019). No longer discrete: Modeling the dynamics of social networks and continuous behavior.
Sociological Methodology, 49, 295-340.

The dynamics of individual behavior are related to the dynamics of the social structures in which individuals are embedded. This implies that to study social mechanisms such as social selection or peer influence, we need to model the evolution of social networks and the attributes of network actors as interdependent processes. The stochastic actor-oriented model is a statistical approach to study network-attribute coevolution based on longitudinal data. In its standard specification, the co-evolving actor attributes are assumed to be measured on an ordinal categorical scale. Continuous variables first need to be discretized to fit into such a modeling framework. This article presents an extension of the stochastic actor-oriented model that does away with this restriction by using a stochastic differential equation to model the evolution of a continuous attribute. A measure for explained variance is proposed, and an interpretation of parameter sizes is given. The proposed method is illustrated by a study of the relationship between friendship, alcohol consumption and self-esteem among adolescents.

Kim Titlestad, T.A.B. Snijders, Kevin Durrheim, Michael Quayle, and Tom Postmes (2019). The dynamic emergence of cooperative norms in a social dilemma.
Journal of Experimental Social Psychology, 84, 103799.

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.

David Dekker, David Krackhardt, and Tom A. B. Snijders (2019). Transitivity correlation: A descriptive measure of network transitivity.
Network Science , 7, 353-375.

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.

Tomáš Diviák, Jan Kornelis Dijkstra, and Tom A.B. Snijders (2019). Poisonous connections: a case study on a Czech counterfeit alcohol distribution network.
Global Crime, DOI: 10.1080/17440572.2019.1645653.

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.

Amati, Viviana, Schönenberger, Felix, and Snijders, Tom A.B. (2019). Contemporaneous Statistics for Estimation in Stochastic Actor-Oriented Co-Evolution Models .
Psychometrika, 84, 1068-1096.

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.

Tom A.B. Snijders and Alessandro Lomi (2019). Beyond Homophily: Incorporating Actor Variables in Statistical Network Models.
Network Science, 7, 1-19.

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.

Nynke M.D. Niezink and Tom A.B. Snijders (2018). Continuous-Time Modeling of Panel Data with Network Structure. In: Kees van Montfort, Johan H.L. Oud, and Manuel C. Voelkle (eds), Continuous Time Modeling in the Behavioral and Related Sciences, pp. 111-134.

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.

Tomáš Diviák, Jan Kornelis Dijkstra, and Tom A. B. Snijders (2018). Structure, multiplexity, and centrality in a corruption network: the Czech Rath affair.
Trends in Organized Crime , 22, 274–297.

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.

Zahra Jorjoran Shushtari, Seyed Ali Hosseini, Homeira Sajjadi, Yahya Salimi, A. Shahesmaeili, and Tom Snijders (2018). Safe sex norm questionnaire for female sex workers: development and validation study in Iran.
Public Health, 164, 82-90.

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.

Robert W. Krause, Mark Huisman, Christian E.G. Steglich, and Tom A.B. Snijders (2018). Missing network data: A comparison of different imputation methods.
Proceedings of the 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining.

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.

Robert W. Krause, Mark Huisman, and Tom A.B. Snijders (2018). Multiple imputation for longitudinal network data.
Italian Journal of Applied Statistics, 30, 33-57.

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.

Christoph Stadtfeld, Tom A. B. Snijders, Christian Steglich, and Marijtje van Duijn (2018). Statistical Power in Longitudinal Network Studies.
Sociological Methods and Research , in press.

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.

Kayo Fujimoto, Tom A.B. Snijders and Thomas W. Valente (2018). Multivariate dynamics of one-mode and two-mode networks: Explaining similarity in sports participation among friends.
Network Science, 6, 370-395.

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.

Nynke M. D. Niezink and Tom A.B. Snijders (2017). Co-evolution of Social Networks and Continuous Actor Attributes.
The Annals of Applied Statistics, 11, 1948-1973.

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.

Tom A.B. Snijders (2017). Comment: Modeling of Coordination, Rate Functions, and Missing Ordering Information.
Sociological Methodology, 47, 41-47.

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.

Per Block, Christoph Stadtfeld, and Tom A. B. Snijders (2019). Forms of dependence: Comparing SAOMs and ERGMs from basic principles.
Sociological Methods and Research,48, 202-239.

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.

András Vörös and Tom A.B. Snijders (2017). Cluster analysis of multiplex networks: Defining composite network measures.
Social Networks , 49, 93-112.

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.

Kayo Fujimoto, Tom A.B. Snijders, and Thomas W. Valente (2017). Popularity breeds contempt: The evolution of reputational dislike relations and friendships in high school.
Social Networks, 48, 100-109.

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.

Snijders, Tom A.B. (2017). Stochastic Actor-Oriented Models for Network Dynamics.
Annual Review of Statistics and its Application, Volume 4 (2017), 343-363.
DOI: 10.1146/annurev-statistics-060116-054035

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.

Snijders, Tom A.B. and Pickup, Mark (2017). Stochastic Actor-Oriented Models for Network Dynamics.
Pp. 221-247 in Oxford Handbook of Political Networks, edited by Jennifer Nicoll Victor, Alexander H. Montgomery, and Mark Lubell. Oxford: Oxford University Press.
Also available in OUP Handbooks-online.


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.
Cham: Springer, 2016. Methodos Series: Methodological Prospects in the Social Sciences.

ISBN 978-3-319-24518-8
ISBN 978-3-319-24520-1 (eBook)
DOI: 10.1007/978-3-319-24520-1

At least two methodologies have helped social scientists of the past two generations in overcoming the traditional divide between individualistic and holistic approaches in the social sciences: multilevel analysis and social network analysis. The purpose of this book is to provide an exploration of the diverse ways in which these two methodologies can be brought together in statistical approaches to multilevel network analysis, specifically their combination in the development of three areas: theory, techniques, and empirical applications in the social sciences.

Table of contents

  1. Emmanuel Lazega and Tom A.B. Snijders: Introduction
  2. Tom A.B. Snijders : The Multiple Flavours of Multilevel Issues for Networks
  3. Emmanuel Lazega : Synchronization Costs in the Organizational Society: Intermediary Relational Infrastructures in the Dynamics of Multilevel Networks
  4. Filip Agneessens and Johan Koskinen : Modeling Individual Outcomes Using a Multilevel Social Influence (MSI) Model: Individual Versus Team Effects of Trust on Job Satisfaction in an Organisational Context
  5. Mark Tranmer and Emmanuel Lazega: Multilevel Models for Multilevel Network Dependencies
  6. Peng Wang, Garry Robins, and Petr Matous: Multilevel Network Analysis Using ERGM and Its Extension
  7. Mengxiao Zhu, Valentina Kuskova, Stanley Wasserman, and Noshir Contractor: Correspondence Analysis of Multirelational Multilevel Networks
  8. Aleš Žiberna and Emmanuel Lazega: Role Sets and Division of Work at Two Levels of Collective Agency: The Case of Blockmodeling a Multilevel (Inter-individual and Inter-organizational) Network
  9. Elisa Bellotti, Luigi Guadalupi, and Guido Conaldi: Comparing Fields of Sciences: Multilevel Networks of Research Collaborations in Italian Academia
  10. Julien Brailly, Guillaume Favre, Josiane Chatellet, and Emmanuel Lazega: Market as a Multilevel System
  11. Julia Brennecke and Olaf N. Rank: Knowledge Networks in High-Tech Clusters: A Multilevel Perspective on Interpersonal and Inter-organizational Collaboration
  12. Guillaume Favre, Julien Brailly, Josiane Chatellet, and Emmanuel Lazega: Inter-organizational Network Influence on Long-Term and Short-Term Inter-individual Relationships: The Case of a Trade Fair for TV Programs Distribution in Sub-Saharan Africa
  13. James Hollway and Johan Koskinen: Multilevel Bilateralism and Multilateralism: States’ Bilateral and Multilateral Fisheries Treaties and Their Secretariats
  14. Paola Zappa and Alessandro Lomi: Knowledge Sharing in Organizations: A Multilevel Network Analysis
  15. Emmanuel Lazega and Tom A.B. Snijders: General Conclusion

Tom A.B. Snijders (2016). The Multiple Flavours of Multilevel Issues for Networks.
Chapter 2 (p. 15-46) in Emmanuel Lazega and Tom A.B. Snijders (eds.), Multilevel Network Analysis for the Social Sciences, Cham: Springer.

ISBN 978-3-319-24518-8
ISBN 978-3-319-24520-1 (eBook)
DOI: 10.1007/978-3-319-24520-1

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.

Viviana Amati, Felix Schönenberger, and Tom A.B. Snijders (2015). Estimation of stochastic actor-oriented models for the evolution of networks by generalized method of moments.
Journal de la Société Française de Statistique, 156, 140-165.

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.

Anuška Ferligoj, Luka Kronegger, Franc Mali, Tom A.B. Snijders, and Patrick Doreian (2015). Scientific collaboration dynamics in a national scientific system.
Scientometrics, 104, 985-1012.

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.

Tom A.B. Snijders and Christian E.G. Steglich (2015). Representing Micro-Macro Linkages by Actor-Based Dynamic Network Models.
Sociological Methods & Research, 44, 222-271.


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.

Gijs Huitsing, Tom A.B. Snijders, Marijtje A.J. van Duijn, and René Veenstra (2014). Victims, bullies, and their defenders: a longitudinal study of the coevolution of positive and negative networks.
Development and Psychopathology, 26, 645-659.

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.

Tom A.B. Snijders (2014). Siena: Statistical Modeling of Longitudinal Network Data.
Pp. 1718-1725 in Reda Alhajj and Jon Rokne (eds.) Encyclopedia of Social Network Analysis and Mining. New York: Springer.


John M. Light, Julie C. Rusby, Kimberley M. Nies, and Tom A.B. Snijders (2014). Antisocial behavior trajectories and social victimization within and between school years in early adolescence.
Journal of Research on Adolescence, 24, 332-336.


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.

Philippa E. Pattison, Garry L. Robins, Tom A. B. Snijders, and Peng Wang (2013). Conditional estimation of exponential random graph models from snowball sampling designs.
Journal of Mathematical Psychology 57, 284-296.


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.

John M. Light, Charlotte C. Greenan, Julie C. Rushby, Kimberly M. Nies, and Tom A.B. Snijders (2013). Onset to First Alcohol Use in Early Adolescence: A Network Diffusion Model.
Journal of Research on Adolescence, 23, 487-499.


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.

Jürgen Lerner, Margit Bussmann, Tom A. B. Snijders, and Ulrik Brandes (2013). Modeling frequency and type of interaction in event networks.
Corvinus Journal of Sociology and Social Policy vol.4 (1), 3-32.

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.

Rafael Wittek, Tom A.B. Snijders, and Victor Nee (eds.), The Handbook of Rational Choice Social Research. Stanford, CA: Stanford University Press (2013).

Tom A.B. Snijders, "Network Dynamics", pp. 252-279 in Rafael Wittek, Tom A.B. Snijders, and Victor Nee (eds.), The Handbook of Rational Choice Social Research. Stanford, CA: Stanford University Press (2013).

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.

D. Lusher, J. Koskinen, and G. Robins,
Exponential Random Graph Models for Social Networks, Cambridge University Press, 2013.

To this book I contributed three chapters, with co-authors:

Tom A.B. Snijders, Alessandro Lomi, and Vanina Jasmine Torló (2013). A model for the multiplex dynamics of two-mode and one-mode networks, with an application to employment preference, friendship, and advice.
Social Networks, 35, 265-276.


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.

Gijs Huitsing, Marijtje A.J. van Duijn, Tom A.B. Snijders, Peng Wang, Miia Sainiod, Christina Salmivalli, René Veenstra (2012). Univariate and multivariate models of positive and negative networks: Liking, disliking, and bully-victim relationships.
Social Networks, 34, 645-657.


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.

Paulina Preciado, Tom A.B. Snijders, William J. Burk, Håkan Stattin, and Margaret Kerr (2012). Does proximity matter? Distance dependence of adolescent friendships. Social Networks, 34, 18-31.


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.

Snijders, Tom A.B., and Bosker, Roel J.
Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling, second edition.
London etc.: Sage Publishers, 2012

ISBN 9781849202008 (hardcover), ISBN 9781849202015 (pbk). xii + 368 p.

The second edition of an extensive textbook on multilevel analysis.
Material about this book is available at a separate web page.


  1. Introduction
  2. Multilevel theories, multi-stage sampling, and multilevel models
  3. Statistical treatment of clustered data
  4. The random intercept model
  5. The hierarchical linear model
  6. Testing and model specification
  7. How much does the model explain?
  8. Heteroscedasticity
  9. Missing data
  10. Assumptions of the hierarchical linear model
  11. Designing multilevel studies
  12. Other methods and models
  13. Imperfect hierarchies
  14. Survey weights
  15. Longitudinal data
  16. Multivariate multilevel models
  17. Discrete dependent variables
  18. Software

Lazega, E., Mounier, L., Snijders, T.A.B., and Tubaro, P. (2012). Norms, status and the dynamics of advice networks: A case study . Social Networks, 34, 323-332.

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.

Alessandro Lomi, Tom A.B. Snijders, Christian E.G. Steglich, and Vanina Jasmine Torló (2011). Why Are Some More Peer Than Others? Evidence from a Longitudinal Study of Social Networks and Individual Academic Performance. Social Science Research, 40 (2011), 1506-1520.

DOI: .

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

Tom A.B. Snijders. (2011). Statistical Models for Social Networks . Annual Review of Sociology, 37, 129-151.


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

Miranda J. Lubbers, Tom A.B. Snijders, and Margaretha P.C. van der Werf (2011). Dynamics of Peer Relationships Across the First Two Years of Junior High as a Function of Gender and Changes in Classroom Composition. Journal of Research on Adolescence, 21, 488-504.
DOI: .

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

Tom A.B. Snijders. (2011). Network Dynamics. Chapter 33 (pp. 501-513) in John Scott and Peter J. Carrington (eds.), The SAGE Handbook of Social Network Analysis. London: Sage, 2011.

Marie-Claire E. Aussems, Anne Boomsma, and Tom A.B. Snijders. (2011). The use of quasi-experiments in the social sciences: a content analysis. Quality and Quantity, 45, 21-42.

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.

Snijders, T.A.B. (2010). Conditional Marginalization for Exponential Random Graph Models . Journal of Mathematical Sociology, 34, 239-252.


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.

Snijders, Tom A.B., Koskinen, Johan, and Schweinberger, Michael (2010). Maximum Likelihood Estimation for Social Network Dynamics . Annals of Applied Statistics, 4, 567-588.


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.

Christian E.G. Steglich, Tom A.B. Snijders, and Michael Pearson (2010). Dynamic Networks and Behavior: Separating Selection from Influence. Sociological Methodology, 40, 329-392.
DOI: .

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 .

Ulrik Brandes, Jürgen Lerner, and Tom A. B. Snijders: Networks Evolving Step by Step: Statistical Analysis of Dyadic Event Data. Proc. 2009 Intl. Conf. Advances in Social Network Analysis and Mining (ASONAM 2009), pp.200-205. IEEE Computer Society, 2009.

With few exceptions, statistical analysis of social networks is currently focused on cross-sectional or panel data. On the other hand, automated collection of network-data often produces event data, i. e., data encoding the exact time of interaction between social actors. In this paper we propose models and methods to analyze such networks of dyadic events and to determine the factors that influence the frequency and quality of interaction. We apply our methods to empirical datasets about political conflicts and test several hypotheses concerning reciprocity and structural balance theory.

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 and an example with the political event data is on
The implementation can deal with general event types and weights and with general date/time formatting

Snijders, T.A.B., Doreian, P. (2010). Introduction to the special issue on network dynamics. Social Networks, 32, 1-3.

This journal issue contains the first of two connected special issues on Dynamics of Social Networks. This second special issue will appear later this year. For a rather long time, attention to dynamic aspects in Social Network Analysis took the form of descriptive studies. However, over the last fifteen years model-based approaches to studying network change have been flowering. Landmarks were three special issues on Network Evolution of the Journal of Mathematical Sociology, edited by Frans Stokman and Patrick Doreian, in 1996 (with a book version: Doreian and Stokman, 1997), 2001, and 2003. These three special issues demonstrated how formal and statistical modeling and empirical analysis were coming together. The 2001 and 2003 special issues were focused on joining of theoretical developments with the analysis of empirical data using advanced modeling. This special issue presents a continuation of jointly using theories and modeling to understand social network phenomena.

Snijders, T.A.B., Steglich, C.E.G., and van de Bunt, G.G. (2010). Introduction to actor-based models for network dynamics . Social Networks, 32, 44-60.

Stochastic actor-based models are models for network dynamics that can represent a wide variety of influences on network change, and allow to estimate parameters expressing such influences, and test corresponding hypotheses. The nodes in the network represent social actors, and the collection of ties represents a social relation. The assumptions posit that the network evolves as a stochastic process "driven by the actors', i.e., the model lends itself especially for representing theories about how actors change their outgoing ties. The probabilities of tie changes are in part endogenously determined, i.e., as a function of the current network structure itself, and in part exogenously, as a function of characteristics of the nodes ('actor covariates') and of characteristics of pairs of nodes ('dyadic covariates'). In an extended form, stochastic actor-based models can be used to analyze longitudinal data on social networks jointly with changing attributes of the actors: dynamics of networks and behavior.
This paper gives an introduction to stochastic actor-based models for dynamics of directed networks, using only a minimum of mathematics. The focus is on understanding the basic principles of the model, understanding the results, and on sensible rules for model selection.

Key words: statistical modeling, longitudinal, Markov chain, agent-based model, peer selection, peer influence.

L. Mercken, T.A.B. Snijders, C. Steglich, E. Vartiainen, H. de Vries (2010). Dynamics of adolescent friendship networks and smoking behavior . Social Networks, 32, 72-81.

The mutual influence of smoking behavior and friendships in adolescence is studied. It is attempted to disentangle influence and selection processes in reciprocal and non-reciprocal friendships. An actor-based model is described for the co-evolution of friendship networks and smoking behavior. This model considers alternative selection and influence mechanisms, and models continuous-time changes in network and behavior. The data consists of a longitudinal sample of 1326 Finnish adolescents in 11 high schools. Findings suggest that selection as well as influence processes play an important role in adolescent smoking behavior. Selection had a relatively stronger role than influence, in particular when selecting non-reciprocal friends. The strength of both influence and selection processes decreased over time.

Key words: Smoking; Adolescents; Selection; Influence; Friends; Reciprocity; Siena

Emmanuel Lazega, Lise Mounier, Tom Snijders (guest editors). Revue Française de Sociologie, Special issue on Dynamics of Networks (Numéro Spécial sur la Dynamique des Réseaux), vol. 49 (2008) no. 3.
Emmanuel Lazega, Lise Mounier, Tom Snijders. Presentation of the special issue, pp. 463-465.
Emmanuel Lazega, Lise Mounier, Tom Snijders, and Paola Tubaro. Réseaux, normes et controverses. pp. 467-498.

Martin Van der Gaag, Tom A. B. Snijders, and Henk Flap. Position Generator Measures and their Relationship to other Social Capital Measures.
Chapter 2 in Nan Lin & Bonnie Erickson (eds.), Social Capital: Advances in Research. New York: Aldine de Gruyter (2008).

Snijders, Tom A.B., and Berkhof, Johannes, Diagnostic checks for multilevel models. Chapter 3 (pp. 141-175) in Jan de Leeuw & Erik Meijer (eds.), Handbook of Multilevel Analysis, Springer (2008).

By clicking here you can view or download a preprint of this chapter in .pdf format.
This chapter is about diagnostics for the two-level Hierarchical Linear Model (HLM). It treats various types of residuals and influence diagnostics. The choice between fixed and random effects (which some authors like to base on the Hausman test) is discussed. Methods to estimate and test non-linear fixed effects of explanatory variables are discussed extensively.

Key words: Residuals, Hausman test, empirical Bayes, spline functions, deletion residuals, influence diagnostics, non-linear transformations, mixed models, Hierarchical Linear Model.

David Dekker, David Krackhardt, and Tom A.B. Snijders. Sensitivity of MRQAP Tests to Collinearity and Autocorrelation Conditions .
Psychometrika, 72 (2007), 563-581.

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.

Koskinen, Johan H., and Snijders, Tom A.B., (2007). Bayesian inference for dynamic social network data. Journal of Statistical Planning and Inference, 137 (2007), 3930-3938.


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.

Lubbers, Miranda J., and Snijders, Tom A.B. (2007), A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes. Social Networks, 29, 489-507.

This paper describes an empirical comparison of four specifications of the exponential family of random graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood, Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes. At the level of all classes combined, earlier substantive conclusions were supported by all specifications.However, the different specifications led to different conclusions for individual classes. PL produced unreliable estimates (whenMLis regarded as the standard) and had more convergence problems than ML. Furthermore, the estimates of covariate effects were affected considerably by controlling for network structure, although the precise specification of the structural part (Markov or partial conditional dependence) mattered less.

Key words: Social networks; ERGM; Dependence structure

Burk, William J., Steglich, Christian E.G., and Snijders, Tom A.B. (2007). Beyond dyadic interdependence: Actor-oriented models for co-evolving social networks and individual behaviors. International Journal of Behavioral Development, 31, 397-404.

Actor-oriented models are described as a longitudinal strategy for examining the co-evolution of social networks and individual behaviors.We argue that these models provide advantages over conventional approaches due to their ability to account for inherent dependencies between individuals embedded in a social network (i.e., reciprocity, transitivity) and model interdependencies between network and behavioral dynamics. We provide a brief explanation of actor-oriented processes, followed by a description of parameter estimates, model specification, and selection procedures used by the Simulation Investigation for Empirical Network Analyses (SIENA) software program (Snijders, Steglich, Schweinberger, & Huisman, 2006).To illustrate the applicability of these models, we provide an empirical example investigating the co-evolution of friendship networks and delinquent behaviors in a longitudinal sample of Swedish adolescents with the goal of simultaneously assessing selection and influence processes. Findings suggest both processes play a substantive role in the observed dynamics of delinquent behaviors, with influence having a relatively stronger role than selection (especially in reciprocated friendships).

Key words: delinquency; friendship networks; interdependence; SIENA

Robins, Garry L., Tom A.B. Snijders, Peng Wang, Mark Handcock, and Philippa Pattison. Recent developments in exponential random graph (p*) models for social networks. Social Networks 29 (2007), 192-215.


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

Michael Schweinberger and Tom A.B. Snijders (2007). Markov models for digraph panel data: Monte Carlo-based derivative estimation. Computational Statistics and Data Analysis 51, 4465-4483.

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 .

Tom A.B. Snijders, Christian E.G. Steglich, and Michael Schweinberger. Modeling the co-evolution of networks and behavior.
Pp. 41-71 in Longitudinal models in the behavioral and related sciences, edited by Kees van Montfort, Han Oud and Albert Satorra; Lawrence Erlbaum, 2007.

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 .

Pearson, Michael, Steglich, Christian, and Snijders, Tom. Homophily and assimilation among sport-active adolescent substance users.
Connections 27(1), 47-63. 2006.

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 .

Tom A.B. Snijders (2006). Statistical Methods for Network Dynamics. In: S.R. Luchini et al. (eds.), Proceedings of the XLIII Scientific Meeting, Italian Statistical Society, pp. 281-296. Padova: CLEUP.

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.

M.J. Lubbers, M.P.C. Van Der Werf, T.A.B. Snijders, B.P.M. Creemers, and H. Kuyper. The impact of peer relations on academic progress in junior high.
Journal of School Psychology 44 (2006), 491-512.

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.

Snijders, Tom A.B., Multi-level event history analysis for a sibling design: The choice of predictor variables.
In F.J. Yammarino and F. Dansereau (eds.), Research in Multi-level issues, vol. 5. Multi-level issues in social systems, p. 243-251 (2006).

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.

Christian E.G. Steglich, Tom A.B. Snijders, and Patrick West (2006). Applying SIENA: An Illustrative Analysis of the Coevolution of Adolescents' Friendship Networks, Taste in Music, and Alcohol Consumption. Methodology, 2 (2006), 48-56.

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 .

Tom A.B. Snijders, Philippa E. Pattison, Garry L. Robins, and Mark S. Handcock. New specifications for exponential random graph models. Sociological Methodology, 36 (2006), 99-153.


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.

Zijlstra, B.J.H., van Duijn, M.A.J., & Snijders, T.A.B. The multilevel p2 model - A random effects model for the analysis of multiple social networks. Methodology, 2 (2006), 42-47.

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.

Snijders, Tom A.B. (2005). Entries in Wiley Encyclopedia of Statistics in Behavioral Science.

The following entries in B.S. Everitt and D.C. Howell (eds.), Encyclopedia of Statistics in Behavioral Science. Chicester (etc.): Wiley, 2005:

Tom A.B. Snijders (2005). Models for Longitudinal Network Data. Chapter 11 in P. Carrington, J. Scott, & S. Wasserman (Eds.), Models and methods in social network analysis. New York: Cambridge University Press.

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.

Van der Gaag, Martin P.J. and Snijders, Tom A.B. (2005). The Resource Generator: Social capital quantification with concrete items. Social Networks, 27, 1-27.


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.

Snijders, Tom A.B., Explained Variation in Dynamic Network Models. Mathématiques, Informatique et Sciences Humaines / Mathematics and Social Sciences, 168, 2004(4), p. 31-41.

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 .

van Duijn, M.A.J., Snijders, T.A.B., & Zijlstra, B.H. p2: a random effects model with covariates for directed graphs. Statistica Neerlandica, 58 (2004), 234-254.

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.

Snijders, Tom A.B. (2003). Entries in SAGE Encyclopedia of Social Science Research Methods.

The following entries in M. Lewis-Beck, A.E. Bryman, and T.F. Liao (eds.), The SAGE Encyclopedia of Social Science Research Methods. Thousand Oaks, CA: Sage, 2003:

Schweinberger, Michael, and Snijders, Tom A.B. (2003). Settings in Social Networks: A Measurement Model
Pp. 307-341 in Sociological Methodology - 2003, edited by R.M. Stolzenberg. Boston and London: Basil Blackwell.

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.

Huisman, Mark, and Snijders, Tom A.B. (2003). Statistical analysis of longitudinal network data with changing composition.
Sociological Methods & Research, 32 (2003), 253-287.

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.

Van der Gaag, Martin P.J. and Snijders, Tom A.B. An approach to the measurement of individual social capital.
Pp. 199-218 in H. Flap and B. Völker (eds.), Creation and Returns of Social Capital. London: Routledge, 2003.

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.

Wittek, Rafael, van Duijn, Marijtje A.J., and Snijders, Tom A.B., Frame decay, informal power, and the escalation of social control in a management team. A Relational Signaling Perspective.
Research in the Sociology of Organizations, 20 (2003), 355-380.

Abstract. In a study of conflict in organizations, Lindenberg's relational signaling theory is used to develop hypotheses on the impact of relationship strength, network embeddedness, and organizational change on social escalation. Social escalation is defined as the involvement of one or more third parties in a conflict. An empirical test is conducted with data on 67 conflicts involving 22 managers, gathered during three years of ethnographic fieldwork and a longitudinal network study in a management team of a German Paper Factory. Multilevel analysis indicates that strong ties between conflicting parties decrease the level of social escalation, whereas informal power advantage of one party increases the chances for social escalation. Both effects disappear over time. It is argued that the dissolving impact of relationships and networks is due to the disappearance of so-called solidarity frame-stabilizing activities in the firm. The results highlight the context-dependence of network effects and escalation processes.

Maas, Cora J.M., and Snijders, Tom A.B., The multilevel approach to repeated measures for complete and incomplete data
Quality and Quantity, 37 (2003), 71-89.

Repeated measurements often are analyzed by multivariate analysis of variance (MANOVA). An alternative approach is provided by multilevel analysis, also called the hierarchical linear model (HLM), which makes use of random coefficient models. This paper is a tutorial which indicates that the HLM can be specified in many different ways, corresponding to different sets of assumptions about the covariance matrix of the repeated measurements. The possible assumptions range from the very restrictive compound symmetry model to the unrestricted multivariate model, and include polynomial and other types of trend models between these two extremes. Thus, the HLM can be used to steer a useful middle road between the two traditional methods for analyzing repeated measurements. Another important advantage of the multilevel approach to analyzing repeated measures is the fact that it can be easily used also if the data are incomplete. Thus it provides, e.g., a way to achieve a fully multivariate analysis of repeated measures with incomplete data. It is discussed also how the multilevel approach can be used for trend tests.

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.

Snijders, Tom A.B, Accounting for Degree Distributions in Empirical Analysis of Network Dynamics.
Pp. 146-161 in: R. Breiger, K. Carley, and P. Pattison (eds.), Dynamic Social Network Modeling and Analysis: Workshop Summary and Papers.
National Research Council of the National Academies, 2003. Washington, DC: The National Academies Press.

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.

Snijders, Tom A.B, and Baerveldt, Chris, A Multilevel Network Study of the Effects of Delinquent Behavior on Friendship Evolution.
Journal of Mathematical Sociology, 27 (2003), 123-151.

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).

Snijders, Tom A.B., and van Duijn, Marijtje A.J. (2002). Conditional Maximum Likelihood Estimation under Various Specifications of Exponential Random Graph Models.
Pp. 117-134 in Jan Hagberg (ed.), Contributions to Social Network Analysis, Information Theory, and Other Topics in Statistics; A Festschrift in honour of Ove Frank. University of Stockholm, Department of Statistics.

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.

David, Beata, and Tom A.B. Snijders. Estimating the size of the homeless population in Budapest, Hungary.
Quality and Quantity, 36 (2002), 291-303.

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.

Kampen, J. K., and T.A.B. Snijders. Estimation of the Wing-Kristofferson model for discrete motor responses.
British Journal of Mathematical & Statistical Psychology, 55 (2002), 159-168.

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.

Snijders, Tom A.B, Markov Chain Monte Carlo Estimation of Exponential Random Graph Models.
Journal of Social Structure, Vol. 3 (2002), No. 2.

Here is a direct link to this internet publication.

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.

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.

Berkhof, Johannes, and Snijders, Tom A.B., Variance Component Testing in Multilevel Models.
Journal of Educational and Behavioral Statistics, 26 (2001), 133-152.

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.

Snijders, Tom A.B, The Statistical Evaluation of Social Network Dynamics.
Sociological Methodology, 31 (2001), 361-395.


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.

Nowicki, Krzysztof, and Snijders, Tom A.B, Estimation and prediction for stochastic blockstructures.
Journal of the American Statistical Association, 96 (2001), 1077-1087.

The estimation procedure in this publication is available in the program BLOCKS.

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.

Snijders, T.A.B., Hypothesis Testing.
International Encyclopedia of the Social and Behavioral Sciences, vol. 10, 7121-7127. Amsterdam, etc.: Elsevier, 2001.

A test is a statistical procedure to obtain a statement on the truth or falsity of a proposition, on the basis of empirical evidence. After a brief historical account, attention is given to the contrasting approaches of R.A. Fisher's significance tests and J. Neyman and E. Pearson's tests of a null against an alternative hypothesis. The t-test is treated as a paradigmatic example and used to illustrate the role of assumptions. The p-value and the confidence interval are statistical procedures which are more informative than a test leading merely to the "reject"/"do not reject" dichotomy. A discussion is presented about problems in the interpretation and use of hypothesis tests, and it is argued that these result to a great extent from our human limitations in reasoning with uncertain evidence. Research results do usually not stand on their own (as assumed in the model of a hypothesis test), but are to be combined with other results, e.g., by meta-analysis.

Snijders, T.A.B., Asymptotic null distribution of person fit statistics with estimated person parameters.
Psychometrika, 66 (2001), 331-342.

Person fit statistics are considered for dichotomous item response models. The asymptotic null distribution is derived for statistics which are linear in the item responses, and in which the ability parameter is replaced by an estimate. This allows the asymptotically correct standardization of linear person fit statistics with estimated ability parameter. The fact that the ability parameter is estimated decreases the variance of this distribution.

Key words: item response theory, person fit, asymptotic approximations.

Snijders, T.A.B., and Hagenaars, J. Guest editors' introduction to the Special Issue on Causality at work.
Sociological Methods & Research, 30 (2001), 3-10.

This is the introduction to an issue of SMR on various issues of causal interpretations from statistical analyses.
The issues contains papers by Willem Saris on the causal relationship between living conditions and satisfaction; by Johannes van der Zouwen and Theo van Tilburg on reactivity in a panel study on personal networks; by Peter Abell on causality and low-frequency complex events; and by Patrick Doreian on causality in social network analysis.

Snijders, T.A.B., Sampling.
Chapter 11 (p. 159-174) in
A. Leyland and H. Goldstein (eds.) (2001) Multilevel Modelling of Health Statistics, Chichester etc.: Wiley.

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.

van Baarsen, B., Snijders, T.A.B., Smit, J.H., and van Duijn, M.A.J., Lonely but not alone: Emotional isolation and social isolation as two distinct dimensions of loneliness in older people.
Educational and Psychological Measurement, 61, 2001, 119-135.

This study addresses the validity of the De Jong-Gierveld loneliness scale. The internal properties of the scale scores were studied using item response theory (IRT), supplemented by an external validity study. Tests of the Rasch model using PML did not support the cumulativeness and unidimensionality of the scale. Correlational analyses investigating the relationships between concepts of loneliness and theoretically relevant external measures supported the bidimensionality of the loneliness scale. In line with attachment theory and the theory of relational loneliness, the results stress the significance of distinguishing between emotional loneliness and social loneliness.

Key words: loneliness, item response theory, Rasch model, dimensionality, aging.

A. Boomsma, M.A.J. van Duijn, and T.A.B. Snijders (eds.), Essays on Item Response Theory.
Lecture Notes in Statistics, 157. New York: Springer, 2001.

A volume of essays on IRT, dedicated to Ivo W. Molenaar on the occasion of his 65th birthday.

Snijders, T.A.B., Two-level non-parametric scaling for dichotomous data.
Pp. 319-338 in A. Boomsma, M.A.J. van Duijn, and T.A.B. Snijders (eds.), Essays on Item Response Theory. Lecture Notes in Statistics, 157. New York: Springer, 2001.

This paper and the accompanying program can be downloaded from my multilevel page.

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.

Van Yperen, Nico W., and Snijders, Tom A.B., Multilevel analysis of the Demands-Control Model,
Journal of Occupational Health Psychology, 5 (2000), 182-190.

This study explored the extent to which negative health-related outcomes are associated with differences between work groups and with differences between individuals within work groups using R.A. Karasek's (1979) demands-control model. The sample consisted of 260 employees in 31 working groups of a national bank in The Netherlands. Results suggest that job demands and job control should be conceptualized as having both group- and individual-level foundations. Support for Karasek's demands-control model was found only when these variables were split into the two parts, reflecting shared perceptions and employees' subjective assessment, respectively. One of the most appealing practical implications is that absence rates among homogeneous work groups could be reduced by enhancing actual control on the job.

Snijders, Tom A.B., and Bosker, Roel J.
Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling
London etc.: Sage Publishers, 1999

ISBN 0-7619-5889-4 (hardcover), ISBN 0-7619-5890-8 (pbk). ix + 266 p.

An extensive textbook on multilevel analysis.
Material about this book is available at a separate web page.


  1. Introduction
  2. Multilevel theories, multi-stage sampling, and multilevel models
  3. Statistical treatment of clustered data
  4. The random intercept model
  5. The hierarchical linear model
  6. Testing and model specification
  7. How much does the model explain?
  8. Heteroscedasticity
  9. Assumptions of the hierarchical linear model
  10. Designing multilevel studies
  11. Crossed random coefficients
  12. Longitudinal data
  13. Multivariate multilevel models
  14. Discrete dependent variables
  15. Software

Snijders, T.A.B. and Borgatti, S.P., Non-parametric standard errors and tests for network statistics.
Connections, 22(2) (1999), 61-70.

Two procedures are proposed for calculating standard errors for network statistics. Both are based on resampling of vertices: the first follows the bootstrap approach, the second the jackknife approach. In addition, we demonstrate how to use these estimated standard errors to compare statistics using an approximate t-test and how statistics can also be compared by another bootstrap approach that is not based on approximate normality.

Van Duijn, M.A.J., Van Busschbach, J.T., and Snijders, T.A.B., Multilevel analysis of personal networks as dependent variables.
Social Networks, 21 (1999), 187-209.

It is shown that multilevel methods are particularly well-suited for the analysis of relations in personal networks and the changes in these relations. Justice is done to the hierarchical nested structure of the data and the resulting dependence of these observations 'within egos'.
Multilevel techniques can also give more specific insight on why personal networks change: they allow to distinguish between the influence of individual and tie characteristics on the stability of personal networks as a whole and of specific ties within a personal network. This is illustrated by an application to changes in networks of four Dutch samples experiencing different life events.

Snijders, T.A.B., Prologue to the measurement of social capital.
The Tocqueville Review, 20.1 (1999), 27 - 44.

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.

Snijders, T.A.B., and Kenny, D., The Social Relations Models for family data: A multilevel approach.
Personal Relationships, 6 (1999), 471-486.

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.

Snijders, T.A.B., The transition probabilities of the reciprocity model
Journal of Mathematical Sociology, 23 (1999), 241-253.

The reciprocity model is a continuous-time Markov chain model used for modeling longitudinal network data. A new explicit expression is derived for its transition probability matrix. This expression can be checked relatively easily. Some properties of the transition probabilities are given, as well as a chi-squared goodness of fit test.

Key words: network dynamics, longitudinal social network data, continuous-time Markov chain.

Van De Bunt, G.G., Van Duijn, M.A.J., and Snijders, T.A.B., Friendship networks through time: An actor-oriented dynamic statistical network model.
Computational and Mathematical Organization Theory, 5 (1999), 167-192.

We propose a class of actor-oriented statistical models for closed social networks in general, and friendship networks in particular. The models are random utility models developed within a rational choice framework. Based on social psychological and sociological theories about friendship, mathematical functions capturing expected utility of individual actors with respect to friendship are constructed. Expected utility also contains a random (i.e., unexplained) component. We assume that, given their restrictions and contact opportunities, individuals evaluate their utility function and behave such that they maximize the expected amount of utility. The behavior under consideration is the expression of like and dislike (choice of friends). Theoretical mechanisms modelled are, e.g., the principle of diminishing returns, the tendency towards reciprocated choices, and the preference for friendship relations with similar others. Constraints imposed on individuals are, e.g., the structure of the existing network, and the distribution of personal characteristics over the respondents. The models are illustrated by means of a data set collected among university freshmen at 7 points in time during 1994 and 1995.

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.

Boahene, K., Snijders, T.A.B., and Folmer, H., An integrated socioeconomic analysis of innovation adoption: The case of hybrid cocoa in Ghana.
Journal of Policy Modeling, 21 (1999), 167-184.

This study employs a multidisciplinary model to explain the adoption of agricultural innovations in developing economies with reference to hybrid cocoa in Ghana. The empirical evidence shows that, in the adoption of hybrid cocoa, the support that small-scale farmers obtain via their social networks is more relevant than the advantage of farm size enjoyed by large-scale farmers. However, for large-scale farmers, access to a bank loan strongly increases their chance of adoption compared with small-scale farmers. Contacts with extension agents, education, and availability of hired labor also have positive effects on adoption. The social status of the farmers has only an indirect effect on adoption: farmers with higher social status are more likely to obtain a bank loan, and a bank loan has a positive impact on adoption.

Bonacich, Ph., Oliver, A., and Snijders, T.A.B., Controlling for size in centrality scores.
Social Networks, 20 (1998), 135-141.

All measures of centrality in graphs seem to be correlated with degree, the sheer number of connections of a position. There are occasions in which one wants a measure that is not necessarily related to degree but whose relationship to degree is an empirical finding. Existing corrections, which force a lack of correlation, or which have no statistical justification, are inadequate for this purpose. Based on an algorithm developed by Snijders (1991) for generating random graphs with fixed marginals, we suggest a measure of centrality that is logically but not necessarily empirically independent of degree.

Snijders, T.A.B., Methodological issues in studying effects of networks in organizations.
Computational and Mathematical Organization Theory, 4 (1998), 205-215.

Three methodological issues are discussed that are important for the analysis of data on networks in organizations. The first is the two-level nature of the data: individuals are nested in organizations. This can be dealt with by using multilevel statistical methods. The second is the complicated nature of statistical methods for network analysis. The third issue is the potential of mathematical modeling for the study of network effects and network evolution in organizations. Two examples are given of mathematical models for gossip in organizations. The first example is a model for cross-sectional data, the second is a model for longitudinal data tha reflect the joint development of network structure and individual behavior tendencies.

Key words: Multilevel analysis, network analysis, longitudinal models, mathematical modeling, gossip.

Snijders, T.A.B. & Van Duijn, M.A.J., Simulation for statistical inference in dynamic network models.
In: Conte, R., Hegselmann, R. Terna, P. (eds.), Simulating social phenomena , 493-512. Berlin: Springer (1997).

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.

Snijders, T.A.B. & Nowicki, K., Estimation and prediction for stochastic block models for graphs with latent block structure.
Journal of Classification, 14 (1997), 75 - 100.

A statistical approach to a posteriori blockmodeling for graphs is proposed.The model assumes that the vertices of the graph are partitioned into two unknown blocks and that the probability of an edge between two vertices depends only on the blocks to which they belong. Statistical procedures are derived for estimating the probabilities of edges and for predicting the block structure from observations of the edge pattern only. ML estimators can be computed using the EM algorithm, but this strategy is practical only for small graphs. A Bayesian estimator, based on Gibbs sampling, is proposed. This estimator is practical also for large graphs. When ML estimators are used, the block structure can be predicted based on predictive likelihood. When Gibbs sampling is used,the block structure can be predicted from posterior predictive probabilities. A side result is that when the number of vertices tends to infinity while the probabilities remain constant, the block structure can be recovered correctly with probability tending to 1.

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.

Ferrand, A. & Snijders, T.A.B., Social Networks and Normative Tensions.
In: Van Campenhoudt, L., Cohen, M., Guizzardi, G., Hausser, D.,
Sexual Interactions and HIV Risk: New Conceptual Perspectives in European Research.
Exeter: Taylor & Francis, 1997, pp. 6-21.

Approaches to studying sexuality have frequently been based on examination of individuals. This book argues that explanations of sexual behaviour should move away from individualistic approaches. This chapter proposes a view of sexual life as relational: the behaviour of partners is not only restricted by social context, but simultaneously influences and shapes the social context. There are five main postulates: that sexuality is dyadic (there is a focus on interactions between pairs of people); that a relation is viewed as a sequence of interactions, with an individual acting on the basis of the expected and perceived answers of another individual; that there is bargaining and change in the relationship over time; that relations are embedded in social networks (the relation is an element in each individual's system of interpersonal relationships as well as linking the two individuals); that norms and values are flexible (social norms are not rigid, and change in response to changes in situations). These postulates are explained in detail.

Gerlsma, C., Snijders, T.A.B., Van Duijn, M.A.J., & Emmelkamp, P.M.G., Parenting and Psychopathology: Differences in Family Members Perceptions of Parental Rearing Styles.
Personality and Individual Differences, 23 (1997), 271-282.

Psychiatric patients generally report more adverse recollections of their parents' rearing behaviour than individuals from the general community. It is, however, as yet unclear whether we can infer from this finding that the families of psychiatric patients differ from the families of healthy controls, that is, whether patients' adverse views are shared by their family members. This issue bears on the construct validity of reports about parental rearing styles: should these reports be interpreted to reflect characteristics of the family, of the parent-child relationship, or of the individual providing the reports? In this study, patterns of agreement and variability within families with regard to recalled parental behaviour were analysed in order to examine this aspect of the validity of parental representations. We examined whether families of psychiatric patients report less favourable parenting styles than families of healthy controls. Furthermore, we examined the level of agreement between all family members participating in the study, between the two members reporting on the same parent-child relationship, between parents, and between siblings. Finally, we examined what factors might be accountable for differences of opinion between family members. Results suggested that perceptions of parental rearing styles are primilary tales by individuals, and to a much smaller extend tales about families, parents of relationships. The implications of these findings for research with regard to the relationship between parental rearing behaviour and adult psychopathology are discussed.

T.A.B. Snijders, E.P.H. Zeggelink, and F.N. Stokman, Parameters in collective decision making models: estimation and sensitivity,
Mathématiques, Informatique et Sciences Humaines, 137 (1997), 81-99.

Simulation models for collective decision making are based on theoretical and empirical insight in the decision making process, but still contain a number of parameters of which the values are determined ad hoc. For the dynamic access model, some of such parameters are discussed, and it is proposed to extend the utility functions with a random term of which the variance also is an unknown parameter. These parameters can be estimated by fitting model predictions to data, where the predictions can refer to decision outcomes but also to network structure generated as a part of the decision making process. Given the stochastic nature of the model, this parameter estimation can be carried out with the Robbins Monro process. Such fitting is not completely straightforward: statistics must be chosen on which to base the parameter estimation, it is not certain a priori that there will be a solution to the estimating equation and that the Robbins Monro process will converge. The method is illustrated with data from the financial restructuring of a large company.

Key words. Dynamic access model, policy networks, computer simulation, method of moments, Robbins Monro process.

Snijders, T.A.B., and Spreen, M., Segmentation in personal networks.
Mathematiques, Informatique et Sciences Humaines, 35 (1997), 25-36.

A concept and several measures for segmentation of personal networks are proposed. It is argued that the implications of segmentation of personal networks are, in a sense, the opposite of those of segmentation of entire networks. The measures are illustrated by the example of the trust network in a civil service department. For the case where relations in the personal network are observed by a sample rather than completely, estimators for the segmentation measures are given.

Snijders, T.A.B., Stochastic actor-oriented dynamic network analysis.
Journal of Mathematical Sociology, 21 (1996), 149-172.

Also see Snijders (2001), Snijders and Van Duijn (1997), and the SIENA program .

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.

Snijders, T.A.B., What to do with the upward bias in R²: A comment on Huberty.
Journal of Educational and Behavioral Statistics, 21 (1996), 283-287.

A recent article by Huberty (1994) discusses significance testing of R² in linear regression and the definition of a corresponding effect size index. It recommends an adjustment to the standard null hypothesis, rho² = 0, in order to adjust for an upward bias in the statistic R². This note suggests that the adjustment proposed by Huberty has some conceptual shortcomings. Existing improvements on R² are described in some detail.

Snijders, T.A.B. (1995), Methods for Longitudinal Social Network Data: Review and Markov Process Models. In E.Tiit, T. Kollo, and H. Niemi (eds.), New Trends in Probability and Statistics. Vol. 3: Multivariate Statistics and Matrices in Statistics. Proceedings of the 5th Tartu Conference, p. 211-227. Vilnius, Lithuania: VSP/TEV.

Social network data pertain to social relations between individuals, or between other actors (countries, firms, etc.). The facts that each relation links two individuals, and that each individual can be related with multiple others, result in complicated stochastic dependence structures. Longitudinal studies about the change and development of social networks in time are important for progress in sociology and related disciplines, but the literature does not contain many satisfactory statistical models for longitudinal social network data. This is related to the difficulty of reconciling mathematical tractability with the possibility to reflect substantively interesting processes.
A review is given of models for longitudinal network data. Interesting developments are taking place in applying continuous time Markov processes to model network change. Such models can incorporate sociologically important effects. They are in many aspects similar to computer simulation models such as are now common in sociological theory, but with the extra ingredient of a component of random change. Difficulties of mathematical tractability can be resolved by using stochastic approximation methods to obtain estimators and tests. This is done, however, at the expense of introducing new problems: computer intense computations and questions about statistical efficiency. An example of this approach is given using a time series of personal preference relations in a closed group.

Snijders, T., Analysis of longitudinal data using the hierarchical linear model,
Quality & Quantity, 30 (1996), 405-426.

The hierarchical linear model is a linear model with nested random coefficients, fruitfully used for multilevel research. A tutorial is presented on the use of this model for the analysis of longitudinal data, i.e., repeated data on the same subjects. An important advantage of this approach is that differences across subjects in the numbers and spacings of measurement occasions do not present a problem, and that changing covariates can easily be handled. The tutorial approaches the longitudinal data as measurements on populations of (subject-specific) functions.

Key words: multilevel analysis, hierarchical linear model, random coefficients.

Snijders, T.A.B., Spreen, M. & Zwaagstra, R., The use of multilevel modelling for analysing personal networks (Networks of cocaine users in an urban area).
Journal of Quantitative Anthropology, 5 (1995), 85-105.

This paper explains how multilevel methods can be employed to analyze personal network data, when the dependent variable under consideration is a function of the relations contained in the personal networks. These methods take into account the mutual dependence of relations of the same respondent, and allow us to study the variability between respondents as well as the variability between different relations within respondents. As an illustration, multilevel models are applied to an analysis of personal networks of cocaine users, focusing on the significance of cocaine in their personal relations with other cocaine users.

Key words: Personal network, snowball sample, multilevel analysis,hierarchical linear model, random effects, cocaine.

Also see van Duijn, van Busschbach and Snijders (1999).

Snijders, T.A.B. & Bosker, R.J., Modeled variance in two-level models.
Sociological Methods and Research, 22 (1994), 342-363.

The concept of explained or modeled proportion of variance is reviewed in the situation of the random effects hierarchical two-level model. It is argued that the proportional reduction in (estimated) variance components is not an attractive parameter to represent the joint importance of explanatory variables for modeling the dependent variable. It is preferable instead to work with the proportional reduction in mean squared prediction error for predicting individual values (for the modeled variance at level 1) and for predicting group averages (for the modeled variance at level 2). It is shown that when predictors are added, the proportion of modeled variance defined in this way cannot go down in the population if the model is correctly specified, but can go down in a sample; the latter situation then points to the possibility of misspecification. This provides a diagnostic means for identifying misspecification.

Key words: R-squared, explained variance, coefficient of determination, multilevel analysis, misspecification.

Snijders, T.A.B., Dam, M. van & Weesie, J., Who contributes to public goods? With an application to local economic policies in the Netherlands.
Journal of Mathematical Sociology, 19 (1994), 149-164.

We present three models for the extent to which actors reduce their contributions to the production of a public good because of expected contributions by other actors. The first model is a simple game theoretic model, the second a spatial autocorrelation model, and the third is a hybrid of the first two models. Estimation of the three models from incomplete data is discussed. The three models are applied to data on economic policies of municipalities in the Netherlands. In particular, it is probed whether municipalities take a free ride on the measures of their neighbors.

Frank, O. & Snijders, T.A.B., Estimating hidden populations using snowball sampling.
Journal of Official Statistics, 10 (1994), 53-67.

Snowball sampling is a term used for sampling procedures that allow the sampled units to provide information not only about themselves but also about other units. This might be advantageous when rare properties are of interest. This article illustrates snowball sample situations and discusses various modelling and estimation problems in this context. The problem of estimating the size of a population is discussed for both design-based and model-based approaches. An application to a study of heroin use is included. Simulation results are provided for comparing and evaluating various estimators.

Key words: Network sampling; random graphs; link-tracing designs.

Baerveldt, C. & Snijders, T.A.B., Influences on and from the segmentation of networks: hypotheses and tests.
Social Networks, 16 (1994), 213-232.

This article discusses (a) the influence of network structure on the diffusion of (new) cultural behavior within the network and (b) the influence of external events, especially of social programs, on the diffusion of (new) cultural behavior, and on the network structure. Hypotheses are formulated and tested on data from a study on the diffusion of petty crime in pupils' networks in high schools. To test these hypotheses we propose and use a new measure of network structure: the segmentation index.

Post, W.J. & Snijders, T.A.B., Nonparametric unfolding models for dichotomous data.
Methodika 7 (1993), 130-156.

What are essential requirements, formulated in terms of item response theory, for unidimensional unfolding models for dichotomous data, if one does not wish to make specific assumptions concerning the form of the tracelines and of the population distribution of latent trait values? Tracelines should be unimodal, of course, but this requirement is not sufficient to derive empirically testable consequences. Two basic postulates are formulated concerning the inference about subjects' latent trait values on the basis of observed responses to items. These postulates are proven to be equivalent to total positivity of orders 2 and 3 for the traceline family. Given these postulates, unimodality of the tracelines leads to some empirically testable results. These are formulated as properties of the conditional adjacency matrix and of the correlation matrix.

Key words: Unfolding, item response theory, unimodal response models, total positivity, unidimensional scaling, measurement theory.

Snijders, T.A.B. & Bosker, R.J., Standard errors and sample sizes for two-level research
Journal of Educational Statistics, 18 (1993), 237-259.

The hierarchical linear model approach to a two-level design is considered, some variables at the lower level having fixed and others having random regression coefficients. An approximation is derived to the covariance matrix of the estimators of the fixed regression coefficients (for variables at the lower and the higher level) under the assumption that the sample sizes at either level are large enough. This covariance matrix is expressed as a function of parameters occurring in the model. If a research planner can make a reasonable guess as to these parameters, this approximation can be used as a guide to the choice of sample sizes at either level.
A PC program to carry out the calculations developed in this paper is available from my multilevel page.

Key words: hierarchical linear model, multilevel research, sample design.

Snijders, T.A.B., Estimation on the basis of snowball samples: how to weight?
Bulletin de Methodologie Sociologique, 36 (1992), 59-70.

What are the possibilities of snowball sampling, if one desires valid statistical inference without making probabilistic assumptions on the network structure? In a critical review of the possibilities of snowball sampling for a population of vertices connected by a network of arcs, it is argued that the snowball method is much more suitable for the estimation of parameters of the network structure (or parameters of the population of arcs) than to estimate parameters of the population of vertices. Further work needs to be done to relax the assumption of randomness of the initial sample of the snowball.

Key words: Snowball Sampling, Weighting, Parameter Estimations, Social Networks.

Muilwijk, J., Snijders, T.A.B. & Moors, J.J.A., Kanssteekproeven. Leiden: Stenfert Kroese, 1992.
(Probability Samples, in Dutch).

This is a textbook on sampling theory.

Jansen, M.G.M. & Snijders, T.A.B., Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction.
Statistica Neerlandica, 45 (1991), 51-65.

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared. It appears that gamma priors yield more tractable results than lognormal priors.

Key words: lognormal prior, Dirichlet prior, gamma prior, posterior mode, Rasch's multiplicative Poisson model, empirical Bayes estimation.

Snijders, T.A.B., Enumeration and simulation methods for 0-1 matrices with given marginals.
Psychometrika, 56 (1991), 397-417.

The algorithms in this publication are available in the program collection ZO.

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.

Snijders, T.A.B., Dormaar, M., Schuur, W.H. van, Dijkman, Ch. & Driessen, G., Distribution of some association coefficients for binary data in the case of two sets of operational taxonomic units and associated attributes.
Journal of Classification, 7 (1990), 5-31.

For three coefficients of similarity between pairs (dyads) of operational taxonomic units for multivariate binary data (presence/absence of attributes), parameters of their distribution under statistical independence are derived. These are applied to test independence for dyadic data. Association among attributes within operational taxonomic units is allowed. It is also allowed that the two units in the dyad are drawn from different populations having different presence probabilities of attributes. The variance of the distribution of the similarity coefficients under statistical independence is shown to be relatively large in many empirical situations. This result implies that the practical interpretation of these coefficients requires much care. An application using the Jaccard index is given for the assessment of consensus between psychotherapists and their clients.

Key words: Consensus, Dice coefficient, Jaccard coefficient, Simple Matching coefficient, Multivariate binary data, Observer agreement, Similarity coefficients, Beta distribution.

Snijders, T.A.B., Reliable counts: corrections of losses and gains for unreliability.
ISOR-Methodenreeks, Utrecht (1990).

Reliability of counts is investigated for situations where there may be underreporting because some of the elements to be counted are forgotten. An example of such a count is the size of a personal network. The reliability parameter is defined as the probability of duly reporting an element of the set to be counted. This parameter can be estimated from test-retest data. Estimation methods are given for the situation where one single reliability parameter value is estimated, for the situation where reliabilities vary randomly across individuals, and for the situation where the reliability parameter is modeled by linear regression on some independent variable. For data collected over time, relative losses and relative gains with respect to these counts can be corrected for unreliability.

Key words: Counts; personal networks; reliability; reliability of change; binomial distribution; random effects; empirical Bayes; regression.

Snijders, T.A.B., Testing for change in a digraph at two time points.
Social Networks, 12 (1990), 359-373.

The algorithms in this publication are available in the program collection ZO.

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).

Schweigman, C., Snijders, T.A.B. & Bakker, E.J., Operations Research as a tool for analysis of food security problems.
European Journal of Operations Research, 49 (1990), 211-221.

In the first part of the paper the role of operations research in analyzing daily life problems of farmers in developing countries is discussed. Experiences on village studies in Tanzania are reported which formed part of the training in operations research of students of the University of Dar es Salaam. In the second part, two examples of food security problems are worked out: risk of food shortage in subsistence farming in Tanzania and the use of rainfall-yield models to predict shortages of sorghum production at an early stage of the growing season in Burkina Faso. At the end of the paper, some discussion points are formulated.

Key words: Subsistence agriculture, risk, early warning.

Wilmink, F.W. & Snijders, T.A.B., Polytomous logistic regression analysis of the General Health Questionnaire and the Present State Examination.
Psychological Medicine , 19 (1989), 755-764.

First, two examples of dichotomous logistic regression analysis are presented. The probability of being a psychiatric case according to the Present State Examination is predicted from the total score on the General Health Questionnaire and from the general practitioner's judgement on the presence of a mental health problem. Subjects were 292 primary care attenders. Results are compared with those from prior studies. Next, the extension to the polytomous case is demonstrated. The probability of being at any given level of the Index of Definition (computed from PSE data) is estimated from the General Health Questionnaire total score by an ordered polytomous logistic regression model. Several applications of the polytomous logistic regression model are discussed. These range from estimating the proportion of psychiatric cases among individuals who refuse to be interviewed to the formulation of sampling schemes which can be expected to reduce costs while at the same time yielding optimal information for testing specific hypotheses.

Snijders, T.A.B. & Stokman, F.N., Extensions of triad counts to networks with different subsets of points and testing the underlying random graph distributions.
Social Networks, 9 (1987), 249-275.

Triad counts are defined for bipartite directed graphs, i.e., directed graphs where the set of points is partitioned into two subsets. Triads are considered with two points in the first, and one point in the second subset. The means, variances, and covariances of triad counts are given for various random digraph distributions.

Key words: bipartite graphs, conditionally uniform distribution.

Bhoj, D.S. & Snijders, T.A.B., Testing equality of correlated proportions with incomplete data on both responses.
Psychometrika, 51 (1986), 579-588.

Two test statistics are proposed for testing the equality of two correlated proportions when some observations are missing on both responses. The performance of these tests in terms of size and power is compared with other tests by means of Monte Carlo simulations. The proposed tests are easily computed and compare favorably with other tests.

Key words: combination of tests, equality of correlated proportions, incomplete data, asymptotically most powerful test, Monte Carlo study, antithetic variates, power comparison.

Snijders, T.A.B., Inter-station correlations and non-stationarity of Burkina Faso rainfall.
Journal of Climate and Applied Meteorology, 25 (1986), 524-531.

A study is presented of the rainfall regime for central and northern Burkina Faso over 1923-83. Interstation correlations for seasonal rainfall totals are rather low, with median 0.31. Sums of truncated daily rainfall values, with the number of rainy days as an extreme case, exhibit quite larger interstation correlations. An explanation is that factors determining the occurrence of rainfall in West Africa operate on a larger scale than those determining exact rainfall amounts. A new method is proposed for constructing regional rainfall indices from data for several locations in the presence of missing data. This method is applied in a study of (non-)stationarity of Burkina Faso rainfall. A highly significant departure of stationarity is found, which is especially expressed in earlier dates for the end of the rains, and smaller average rainfall amounts per day between the start and the end of the rains.

Snijders, T.A.B. & Schweigman, C., The stochastic nature of yields (Ch. 4).
Snijders, T.A.B. & Joosten, G. & Schweigman, C., Simulation (Ch. 5).
Snijders, T.A.B., Statistics (Appendix).

These are chapters in Operations research problems in agriculture in developing countries, ed. C. Schweigman, Khartoum University Press, Khartoum and Tanzania Publishing House, Dar-Es-Salaam (1985).

Snijders, T.A.B., Antithetic variates for Monte Carlo estimation of probabilities.
Statistica Neerlandica, 38 (1984), 55-74.

This paper explores some possibilities for variance reduction by the use of antithetic variates when estimating probabilities.

Key words: antithetic variates, Monte Carlo, variance reduction, change-point test, Wilcoxon test.

Snijders, T.A.B., The degree variance: an index of graph heterogeneity.
Social Networks, 3 (1981), 163-174.

In the analysis of empirically found graphs, the variance of the degrees can be used as a measure for the heterogeneity of (the points in) the graph. For several types of graphs, the maximum value of the degree variance is given, and the mean and variance of the degree variance under a simple stochastic null model are computed. These are used to produce normalized versions of the degree variance, which can be used as heterogeneity indices of graphs.

Key words: graph heterogeneity, graph centrality, random graphs, degree variance.

Snijders, T.A.B., Rank tests for bivariate symmetry.
Annals of Statistics, 9 (1981), 1087-1095.

The problem is considered of testing symmetry of a bivariate distribution L(X, Y) against "asymmetry towards high X-values," subject to the restriction of invariance under the transformations of (x,y) to (g(x),g(y)) for increasing bijections g. This invariance restriction prohibits the common reduction to the differences x - y. The intuitive concept of "asymmetry towards high X-values" is approached in several ways, and a mathematical formulation for this concept is proposed. Most powerful and locally most powerful invariant similar tests against certain subalternatives are characterized by means of a Hoeffding formula. Asymptotic normality and consistency results are obtained for appropriate linear rank tests.

Key words: Nonparametric tests, bivariate symmetry and asymmetry, locally most powerful tests, asymptotic normality.

ten Berge, J. M. F., Snijders, T.A.B. & Zegers, F.E., Computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis.
Psychometrika, 46 (1981), 201-213.

In the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum-trace communality solution in factor analysis have been developed. In this paper convergence properties of these methods are examined. Instead of using Lagrange multipliers, a new theorem is applied that gives a sufficient condition for a symmetric matrix to be Gramian. Whereas computational pitfalls for two methods suggested by Woodhouse and Jackson can be constructed it is shown that a slightly modified version of one method suggested by Bentler and Woodward can safely be applied to any set of data. A uniqueness proof for the solution desired is offered.

Key words: communality, internal consistency, Heywood case, positive definite.

Snijders, Tom. Maximum value and null moments of the degree variance.
Report TW-229, Department of Mathematics, University of Groningen, 1981.

This is a companion paper to "The degree variance: an index of graph heterogeneity", published in Social Networks, 1981, containing the proofs and derivations published there.
In the analysis of empirically found graphs, the variance of the degrees can be used as a measure for the heterogeneity of (the points in) the graph. For several types of graphs, the maximum value of the degree variance is derived, and the mean and variance of the degree variance under a simple stochastic null model are computed.

Key words: graph heterogeneity, graph centrality, random graphs, degree variance.

Snijders, T.A.B., Asymptotic optimality theory for testing problems with restricted alternatives.
Mathematical Centre Tracts 113, Mathematical Centre, Amsterdam 1979.

This monograph develops a theory of asymptotic optimality for testing problems where the alternative hypothesis is multidimensional and restricted by a finite number of linear inequalities. The optimality criterion is an asymptotic version of the "most stringent" property.

Snijders, T.A.B., Complete class theorems for the simplest empirical Bayes decision problems.
Annals of Statistics, 5 (1977), 164-171.

For the problem of empirical Bayes classification into two known probability distributions on a finite outcome space, an essentially complete class of procedures is determined. This class is proven to be minimal essentially complete if there are only two possible outcomes.

Key words: Empirical Bayes classification, complete class, monotone procedures.

Snijders, T.A.B., A test for randomness in behaviour.
Statistica Neerlandica, 29 (1975), 39-48.

In biological analysis of behaviour, transition matrices occur of which the diagonal entries are essentially zero. For such transition matrices, a model of randomness is constructed, with a test for the hypothesis that this model holds.

Key words: Markov chain, ethology, transition analysis.

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