Social Networks 32 (2010) 44–60
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Social NetworksWritten assignment
journal homepage: www.elsevier.com/locate/socnet
Introduction to stochastic actor-based models for network dynamics
Tom A.B. Snijders a,∗, Gerhard G. van de Bunt b, Christian E.G. Steglich c
a University of Oxford and University of Groningen, Grote Rozenstraat 31, 9712 TG Groningen, Netherlands
b Free University, Amsterdam, Netherlands
c University of Groningen, Netherlands
a r t i c l e i n f o
Keywords:
Statistical modeling
Longitudinal
Markov chain
Agent-based model
Peer selection
Peer influence
a b s t r a c t
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.
Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
1. Introduction
Social networks are dynamic by nature. Ties are established,
they may flourish and perhaps evolve into close relationships, and
they can also dissolve quietly, or suddenly turn sour and go with a
bang. These relational changes may be considered the result of the
structural positions of the actors within the network – e.g., when
friends of friends become friends –, characteristics of the actors
(‘actor covariates’), characteristics of pairs of actors (‘dyadic covariates’), and residual random influences representing unexplained
influences. Social network research has in recent years paid increasing attention to network dynamics, as is shown, e.g., by the three
special issues devoted to this topic in Journal of Mathematical Sociology edited by Patrick Doreian and Frans Stokman (1996, 2001, and
2003; also see Doreian and Stokman, 1997). The three issues shed
light on the underlying theoretical micro-mechanisms that induce
the evolution of social network structures on the macro-level. Network dynamics is important for domains ranging from friendship
networks (e.g., Pearson and Michell, 2000; Burk et al., 2007) to, for
We are grateful to Andrea Knecht who collected the data used in the example,
under the guidance of Chris Baerveldt. We also are grateful to Matthew Checkley
and two reviewers for their very helpful remarks on earlier drafts.
∗ Corresponding author.
E-mail address: [email protected] (T.A.B. Snijders).
example, organizational networks (see the review articles Borgatti
and Foster, 2003; Brass et al., 2004).
In this article we give a tutorial introduction to what we call
here stochastic actor-based models for network dynamics, which
are a type of models that have the purpose to represent network dynamics on the basis of observed longitudinal data, and
evaluate these according to the paradigm of statistical inference.
This means that the models should be able to represent network
dynamics as being driven by many different tendencies, such as the
micro-mechanisms alluded to above, which could have been theoretically derived and/or empirically established in earlier research,
and which may well operate simultaneously. Some examples of
such tendencies are reciprocity, transitivity, homophily, and assortative matching, as will be elaborated below. In this way, the models
should be able to give a good representation of the stochastic dependence between the creation, and possibly termination, of different
network ties. These stochastic actor-based models allow to test
hypotheses about these tendencies, and to estimate parameters
expressing their strengths, while controlling for other tendencies
(which in statistical terminology might be called ‘confounders’).
The literature on network dynamics has generated a large variety
of mathematical models. To describe the place in the literature of
stochastic actor-based models (Snijders, 1996, 2001), these models
may be contrasted with other dynamic network models.
Most network dynamics models in the literature pay attention
to a very specific set of micro-mechanisms – allowing detailed
0378-8733/$ – see front matter. Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.socnet.2009.02.004
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 45
analyses of the properties of thesemodels –, but lack an explicit estimation theory. Examples are models proposed by Bala and Goyal
(2000), Hummon (2000), Skyrms and Pemantle (2000), and Marsili
et al. (2004), all being actor-based simulation models that focus on
the expression of a single social theory as reflected, e.g., by a simple
utility function; those proposed by Jin et al. (2001) which represent a larger but still quite restricted number of tendencies; and
models such as those proposed by Price (1976), Barabási and Albert
(1999), and Jackson and Rogers (2007), which are actor-based, represent one or a restricted set of tendencies, and assume that nodes
are added sequentially while existing ties cannot be deleted, which
is a severe limitation to the type of longitudinal data that may be
faithfully represented. Since such models do not allow to control
for other influences on the network dynamics, and how to estimate
and test parameters is not clear for them, they cannot be used for
purposes of theory testing in a statistical model.
The earlier literature does contain some statistical dynamic network models, mainly those developed by Wasserman (1979) and
Wasserman and Iacobucci (1988), but these do not allow complicated dependencies between ties such as are generated by transitive
closure. Further there are papers that present an empirical analysis
of network dynamics which are based on intricate and illuminating
descriptions such as Holme et al. (2004) and Kossinets and Watts
(2006), but which are not based on an explicit stochastic model for
the network dynamics and therefore do not allow to control one
tendency for other (‘confounding’) tendencies.
Distinguishing characteristics of stochastic actor-based models
are flexibility, allowing to incorporate a wide variety of actor-driven
micro-mechanisms influencing tie formation; and the availability of procedures for estimating and testing parameters that also
allow to assess the effect of a given mechanism while controlling for the possible simultaneous operation of other mechanisms
or tendencies. We assume here that the empirical data consist of
two, but preferably more, repeated observations of a social network on a given set of actors; one could call this network panel
data. Ties are supposed to be the dyadic constituents of relations
such as friendship, trust, or cooperation, directed from one actor
to another. In our examples social actors are individuals, but they
could also be firms, countries, etc. The ties are supposed to be, in
principle, under control of the sending actor (although this will be
subject to constraints), which will exclude most types of relations
where negotiations are required for a tie to come into existence.
Actor covariates may be constant like sex or ethnicity, or subject to
change like opinions, attitudes, or lifestyle behaviors. Actor covariates often are among the determinants of actor similarity (e.g.,
same sex or ethnicity) or spatial proximity between actors (e.g.,
same neighborhood) which influence the existence of ties. Dyadic
covariates likewise may be constant, such as determined by kinship or formal status in an organization, or changing over time,
like friendship between parents of children or task dependencies
within organizations. This paper is organized as follows. In the next
section, we present the assumptions of the actor-based model. The
heart of the model is the so-called objective function, which determines probabilistically the tie changes made by the actors. One
could say that it captures all theoretically relevant information the
actors need to ‘evaluate’ their collection of ties. Some of the potential components of this function are structure-based (endogenous
effects), such as the tendency to form reciprocal relations, others
are attribute-based (exogenous effects), such as the preference for
similar others. In Section 3, we discuss several statistical issues,
such as data requirements and how to test and select the appropriate model. Following this we present an example about friendship
dynamics, focusing on the interpretation of the parameters. Section
4 proposes some more elaborate models. In Section 5, models for
the coevolution of networks and behavior are introduced and illustrated by an example. Section 6 discusses the difference between
equilibrium and out-of-equilibrium situations, and how these longitudinal models relate to cross-sectional statistical modeling of
social networks. Finally, in Section 7, a brief discussion is given, the
Siena software is mentioned which implements these methods, and
some further developments are presented.
2. Model assumptions
A dynamic network consists of ties between actors that change
over time. A foundational assumption of the models discussed in
this paper is that the network ties are not brief events, but can be
regarded as states with a tendency to endure over time. Many relations commonly studied in network analysis naturally satisfy this
requirement of gradual change, such as friendship, trust, and cooperation. Other networks more strongly resemble ‘event data’, e.g.,
the set of all telephone calls among a group of actors at any given
time point, or the set of all e-mails being sent at any given time
point. While it is meaningful to interpret these networks as indicators of communication, it is not plausible to treat their ties as
enduring states, although it often is possible to aggregate event
intensity over a certain period and then view these aggregates as
indicators of states.
Given that the network ties under study denote states, it is further assumed, as an approximation, that the changing network can
be interpreted as the outcome of a Markov process, i.e., that for any
point in time, the current state of the network determines probabilistically its further evolution, and there are no additional effects
of the earlier past. All relevant information is therefore assumed to
be included in the current state. This assumption often can be made
more plausible by choosing meaningful independent variables that
incorporate relevant information from the past.
This paper gives a non-technical introduction into actor-based
models for network dynamics. More precise explanations can be
found in Snijders (2001, 2005) and Snijders et al. (2007). However,
a modicum of mathematical notation cannot be avoided. The tie
variables are binary variables, denoted by xij. A tie from actor i to
actor j, denoted i → j, is either present or absent (xij then having values 1 and 0, respectively). Although this is in line with traditional
network analysis, an extension to valued networks would often be
theoretically sound, and could make the Markov assumption more
plausible. This is one of the topics of current research. The tie variables constitute the network, represented by its n × n adjacency
matrix x = (xij) (self-ties are excluded), where n is the total number of actors. The changes in these tie variables are the dependent
variables in the analysis.
2.1. Basic assumptions
The model is about directed relations, where each tie i → j has
a sender i, who will be referred to as ego, and a receiver j, referred
to as alter. The following assumptions are made.
1. The underlying time parameter t is continuous, i.e., the process
unfolds in time steps of varying length, which could be arbitrarily
small. The parameter estimation procedure, however, assumes
that the network is observed only at two or more discrete points
in time. The observations can also be referred to as ‘network
panel waves’, analogous to panel surveys in non-network studies.
This assumption was proposed already by Holland and
Leinhardt (1977), and elaborated by Wasserman (1979 and other
publications) and Leenders (1995 and other publications)—but
their models represented only reciprocity, and no other structural dependencies between network ties. The continuous-time
assumption allows to represent dependencies between network
ties as the result of processes where one tie isformed as a reaction
46 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
to the existence of other ties. If, for example, three actors who
at the first observation are mutually disconnected form at the
second observation a closed triangle, where each is connected
to both of the others, then a discrete-time model that has the
observations as its time steps would have to account for the fact
that this closed triangle structure is formed ‘out of nothing’, in
one time step. In our model such a closed triangle can be formed
tie by tie, as a consequence of reciprocation and transitive closure. Thus, the appearance of closed triangles may be explained
based on reciprocity and transitive processes, without requiring
a special process specifically for closed triangles.
Since many small changes can add up to large differences
between consecutively observed networks, this does not preclude that what is observed shows large changes from one
observation to the next.
2. The changing network is the outcome of a Markov process. This
was explained above. Thus, the total network structure is the
social context that influences the probabilities of its own change.
The assumption of a Markov process has been made in practically all models for social network dynamics, starting by Katz and
Proctor’s (1959) discrete Markov chain model. This is an assumption that will usually not be realistic, but it is difficult to come up
with manageable models that do not make it. We could say that
this assumption is a lens through which we look at the data—it
should help but it also may distort. If there are only two panel
waves, then the data have virtually no information to test this
assumption. For more panel waves, there is in principle the possibility to test this assumption and propose models making less
restrictive assumption about the time dependence, but this will
require quite complicated models.
3. The actors control their outgoing ties. This means not that actors
can change their outgoing ties at will, but that changes in ties
are made by the actors who send the tie, on the basis of their
and others’ attributes, their position in the network, and their
perceptions about the rest of the network. This assumption is
the reason for using the term ‘actor-based model’. This approach
to modeling is in line with the methodological approach of
structural individualism (Udehn, 2002; Hedström, 2005), where
actors are assumed to be purposeful and to behave subject to
structural constraints. The assumption of purposeful actors is
not required, however, but a question of model interpretation
(see below). It is assumed formally that actors have full information about the network and the other actors. In practice, as
can be concluded from the specifications given below, the actors
only need more limited information, because the probabilities
of network changes by an actor depend on the personal network
(including actors’ attributes) that would result from making any
given change, or possibly the personal network including those
to whom the actor has ties through one intermediary (i.e., at
geodesic distance two).
4. At a given moment one probabilistically selected actor – ‘ego’
– may get the opportunity to change one outgoing tie. No
more than one tie can change at any moment—a principle first
proposed by Holland and Leinhardt (1977). This principle decomposes the change process into its smallest possible components,
thereby allowing for relatively simple modelling. This implies
that tie changes are not coordinated, and depend on each other
only sequentially, via the changing configuration of the whole
network. For example, two actors cannot decide jointly to form
a reciprocal tie; if two actors are not tied at one observation
and mutually tied at the next, then one of the must have taken
the initiative and extended a one-sided tie, after which, at some
later moment, the other actor reciprocated and formed a reciprocal tie. This assumption excludes relational dynamics where
some kind of coordination or negotiation is essential for the creation of a tie, or networks created by groups participating in
some activity, such as joint authorship networks. For directed
networks, this usually is a reasonable simplifying assumption.
In most cases, panel data of directed networks have many tie
differences between successive observations and do not provide
information about the order in which ties were created or terminated, so that this is an assumption about which the available
data contain hardly any empirical evidence.
Summarizing the status of these four basic assumptions: the
first (continuous-time model) makes sense intuitively; the second
(Markov process) is an as-if approximation and it would be interesting in future to construct models going beyond this assumption;
the third (actor-based) is often a helpful theoretical heuristic; and
the fourth (ties change one by one) is an assumption which limits
the applicability to a wide class of panel data of directed networks
for which this assumption seems relatively harmless.
The actor-based network change process is decomposed into
two sub-processes, both of which are stochastic.
5. The change opportunity process, modeling the frequency of tie
changes by actors. The change rates may depend on the network
positions of the actors (e.g., centrality) and on actor covariates
(e.g., age and sex).
6. The change determination process, modeling the precise tie
changes made when an actor has the opportunity to make a
change. The probabilities of tie changes may depend on the network positions, as well as covariates, of ego and the other actors
(‘alters’) in the network. This is explained below.
The actor-based model can be regarded as an agent-based simulation model (Macy and Willer, 2002). It does not deviate in
principle from other agent-based models, only in ways deriving
from the fact that the model is to be used for statistical inference, which leads to requirements of flexibility (enough parameters
that can be estimated from the data to achieve a good fit between
model and data) and parsimony (not more fine detail in the model
than what can be estimated from the data). The word ‘actor’ rather
than ‘agent’ is used, in line with other sociological literature (e.g.,
Hedström, 2005), to underline that actors are not regarded as subservient to others’ interests in any way.
The actor-based model, when elaborated for a practical application, contains parameters that have to be estimated from observed
data by a statistical procedure. Since the proposed stochastic models are too complex for the straightforward application of classical
estimation methods such as maximum likelihood, Snijders (1996,
2001) proposed a procedure using the method of moments implemented by computer simulation of the network change process.
This procedure uses the basic principle that the first observed network is itself not modeled but used only as the starting point of the
simulations. In statistical terminology: the estimation procedure
conditions on the first observation. This implies that it is the change
between two observed periods time points that is being modeled,
and the analysis does not have the aim to make inferences about
the determinants of the network structure at the first time point.
2.2. Change determination model
The first step in the model is the choice of the focal actor (ego)
who gets the opportunity to make a change. This choice can be
made with equal probabilities or with probabilities depending on
attributes or network position, as elaborated in Section 4.1. This
selected focal actor then may change one outgoing tie (i.e., either
initiate or withdraw a tie), or do nothing (i.e., keep the present status quo). This means that the set of admissible actions contains n
elements: n – 1 changes and one non-change. The probabilities for
a choice depend on the so-called objective function. This is a func
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 47
tion of the network, as perceived by the focal actor. Informally, the
objective function expresses how likely it is for the actor to change
her/his network in a particular way. On average, each actor ‘tries to’
move into a direction of higher values of her/his objective function,
subject to the constraints of the current network structure and the
changes made by the other actors in the network; and subject to
random influences. The objective function will depend in practice
on the personal network of the actor, as defined by the network
between the focal actor plus those to whom there is a direct tie (or,
depending on the specification, the focal actor plus those to whom
there is a direct or indirect – i.e., distance-two – tie), including the
covariates for all actors in this personal network. Thus, the probabilities of changes are assumed to depend on the personal networks
that would result from the changes that possibly could bemade, and
their composition in terms of covariates, via the objective function
values of those networks.
The precise interpretation is given by Eq. (4) in Appendix A. This
is the core of model and it must represent the research questions
and relevant theoretical and field-related knowledge. The objective
function is explained in more detail in the next section.
The name ‘objective function’ was chosen because one possible
interpretation is that it represents the short-term objectives (net
result of preferences and structural as well as cognitive constraints)
of the actor. Which action to choose out of the set of admissible
actions, given that ego has the opportunity to act (i.e., change a
network tie), follows the logic of discrete choice models (McFadden,
1973; Maddala, 1983) which have been developed for modeling situations where the dependent variable is a choice made from a finite
set of actions.
2.3. Specification of the objective function
The objective function determines the probabilities of change
in the network, given that an actor has the opportunity to make a
change. One could say it represents the ‘rules for network behavior’
of the actor. This function is defined on the set of possible states
of the network, as perceived from the point of view of the focal
actor, where ‘state of the network’ refers not only to the ties but
also to the covariates. When the actor has the possibility of moving
to one out of a set of network states, the probability of any given
move is higher accordingly as the objective function for that state is
higher.
Like in generalized linear statistical models, the objective function is assumed to be a linear combination of a set of components
called effects,
fi(ˇ, x) =
k
ˇkski(x). (1)
In this section and elsewhere, the symbol i and the term ‘ego’ are
ways of referring to the focal actor. Here fi(ˇ, x) is the value of the
objective function for actor i depending on the state x of the network; the functions ski(x) are the effects, functions of the network
that are chosen based on theory and subject-matter knowledge, and
correspond to the ‘tendencies’ mentioned in the introductory section; and the weights ˇk are the statistical parameters. The effects
represent aspects of the network as ‘viewed’ from the point of view
of actor i. As examples, one can think of the number of reciprocated
ties of actor i, representing tendencies toward reciprocity, or the
number of ties from i toward actors of the same gender, representing tendencies toward gender homophily. Many more examples are
presented below. The effects ski(x) depend on the network x but
may also depend on actor attributes (actor covariates), on variables
depending on pairs of actors (dyadic covariates), etc. If ˇk equals
0, the corresponding effect plays no role in the network dynamics;
if ˇk is positive, then there will be a higher probability of moving
into directions where the corresponding effect is higher, and the
converse if ˇk is negative.
For the model selection, an essential part is the theory-guided
choice of effects included in the objective function in order to test
the formulated hypotheses. A good approach may be to progressively build up the model according to the method of decreasing
abstraction (Lindenberg, 1992). An additional consideration here
is, however, that the complexity of network processes, and the limitations of our current knowledge concerning network dynamics,
imply that model construction may require data-driven elements
to select the most appropriate precise specification of the endogenous network effects. For example, in the investigation of friendship
networks one might be interested in effects of lifestyle variables and
background characteristics on friendship, while recognizing the
necessity to control for tendencies toward reciprocation and transitive closure. As discussed below in the section on triadic effects,
multiple mathematical specifications are available (as ‘effects’ ski(x)
to be included in Eq. (1)) expressing the concept of transitive closure. Usually there are no prior theoretical or empirical reasons for
choosing among these specifications. It may then be best to use theoretical considerations for deciding to include lifestyle-related and
background variables as well as tendencies toward reciprocation
and transitive closure in the model, and to choose the best specification for transitive closure, by one or several specific effects, in a
data-driven way.
In the following we give a number of effects that may be considered for inclusion in the objective function. They are described
here only conceptually, with some brief pointers to empirical results
or theories that might support them; the formulae are given in
Appendix A. Effects depending only on the network are called
structural or endogenous effects, while effects depending only on
externally given attributes are called covariate or exogenous effects.
The complexity of networks is such that an exhaustive list cannot
meaningfully be given. To simplify formulations, the presentation
shall assume that the relation under study is friendship, so the existence of a tie i → j will be described as i calling j a friend. Higher
values of the objective function, leading to higher tendencies to
form ties, will sometimes be interpreted in shorthand as preferences.
2.3.1. Basic effects
The most basic effect is defined by the outdegree of actor i, and
this will be included in all models. It represents the basic tendency
to have ties at all, and in a decision-theoretic approach its parameter
could be regarded as the balance of benefits and costs of an arbitrary
tie. Most networks are sparse (i.e., they have a density well below
0.5) which can be represented by saying that for a tie to an arbitrary
other actor – arbitrary meaning here that the other actor has no
characteristics or tie pattern making him/her especially attractive
to i–, the costs will usually outweigh the benefits. Indeed, in most
cases a negative parameter is obtained for the outdegree effect.
Another quite basic effect is the tendency toward reciprocity,
represented by the number of reciprocated ties of actor i. This is
a basic feature of most social networks (cf. Wasserman and Faust,
1994, Chapter 13) and usually we obtain quite high values for its
parameter, e.g., between 1 and 2.
2.3.2. Transitivity and other triadic effects
Next to reciprocity, an essential feature in most social networks
is the tendency toward transitivity, or transitive closure (sometimes
called clustering): friends of friends become friends, or in graphtheoretic terminology: two-paths tend to be, or to become, closed
(e.g., Davis, 1970; Holland and Leinhardt, 1971). In Fig. 1 a, the twopath i → j → h is closed by the tie i → h.
The transitive triplets effect measures transitivity for an actor i
by counting the number of pairs j, h such that there is the transitive
48 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
Fig. 1. (a) Transitive triplet (i, j, h) and (b) three-cycle.
triplet structure of Fig. 1 a. However, this is just one way of measuring transitivity. Another one is the transitive ties effect, which
measures transitivity for actor i by counting the number of other
actors h for which there is at least one intermediary j forming a
transitive triplet of this kind. The transitive triplets effect postulates
that more intermediaries will add proportionately to the tendency
to transitive closure, whereas the transitive ties effect expects that
given that one intermediary exists, extra intermediaries will not
further contribute to the tendency to forming the tie i → h.
An effect closely related to transitivity is balance (cf. Newcomb,
1962), which in our implementation is the same as s tructural
equivalence with respect to out-ties (cf. Burt, 1982), which is the
tendency to have and create ties to other actors who make the
same choices as ego. The extent to which two actors make the same
choices can be expressed simply as the number of outgoing choices
and non-choices that they have in common.
Transitivity can be represented by still more effects: e.g., negatively, by the number of others to whom i is indirectly tied but not
directly (geodesic distance equal to 2). The choice between these
representations of transitivity may depend both on the degree to
which the representation is theoretically convincing, and on what
gives the best fit.
A different triadic effect is the number of three-cycles that actor
i is involved in (Fig. 1b). Davis (1970) found that in many social
network data sets, there is a tendency to have relatively few threecycles, which can be represented here by a negative parameter ˇk
for the three-cycle effect. The transitive triplets and the three-cycle
effects both represent closed structures, but whereas the former is
in line with a hierarchical ordering, the latter goes against such
an ordering. If the network has a strong hierarchical tendency,
one expects a positive parameter for transitivity and a negative
for three-cycles. Note that a positive three-cycle effect can also be
interpreted, depending on the context of application, as a tendency
toward generalized exchange (Bearman, 1997).
2.3.3. Degree-related effects
In- and outdegrees are primary characteristics of nodal position
and can be important driving factors in the network dynamics.
One pair of effects is degree-related popularity based on indegree
or outdegree. If these effects are positive, nodes with higher
indegree, or higher outdegree, are more attractive for others to
send a tie to. This can be measured by the sum of indegrees of
the targets of i’s outgoing ties, and the sum of their outdegrees,
respectively. A positive indegree-related popularity effect implies
that high indegrees reinforce themselves, which will lead to a
relatively high dispersion of the indegrees (a Matthew effect in
popularity as measured by indegrees, cf. Merton, 1968; Price,
1976). A positive outdegree-related popularity effect will increase
the association between indegrees and outdegrees, or keep this
association relatively high if it is high already.
Another pair of effects is degree-related activity for i ndegree or
outdegree: when these effects are positive, nodes with higher indegree, or higher outdegree respectively, will have an extra propensity
to form ties to others. These effects can be measured by the indegree of i times i’s outdegree; and, respectively, the outdegree of
i times i’s outdegree, that is, the square of the outdegree.1 The
outdegree-related activity effect again is a self-reinforcing effect:
when it has a positive parameter, the dispersion of outdegrees will
tend to increase over time, or to be sustained if it already is high.
The indegree-related activity effect has the same consequence as
the outdegree-related popularity effect: positive parameters lead
to a relatively high association between indegrees and outdegrees.
Therefore these two effects will be difficult, or impossible, to distinguish empirically, and the choice between them will have to be
made on theoretical grounds. These four degree-related effects can
be regarded as the analogues in the case of directed relations of
what was called cumulative advantage by Price (1976) and preferential attachment by Barabási and Albert (1999) in their models for
dynamics of non-directed networks: a self-reinforcing process of
degree differentiation.
These degree-related effects can represent hierarchy between
nodes in the network, but in a different way than the triadic effects
of transitivity and 3-cycles. The degree-related effects represent
global hierarchy while the triadic effects represent local hierarchy.
In a perfect hierarchy, ties go from the bottom to the top, so that
the bottom nodes have high outdegrees and low indegrees and the
top nodes have low outdegrees and high indegrees. This will be
reflected by positive indegree popularity and negative outdegree
popularity, and by positive outdegree activity and negative indegree activity. Therefore, to differentiate between local and global
hierarchical processes, it can be interesting to estimate models with
triadic and degree-related effects, and assess which of these have
the better fit by testing the triadic parameters while controlling for
the degree-related parameters, and vice versa.
Other degree-related effects are assortativity-related: actors
might have preferences for other actors based on their own and the
other’s degrees (Morris and Kretzschmar, 1995; Newman, 2002). In
settings where degrees reflect status of the actors, such preferences
may be argued theoretically based on status-specific preferences,
constraints, or matching processes. This gives four possibilities,
depending on in- and outdegree of the focal actor and the potential
friend.
Together, this list offers eight degree-related effects. The
outdegree-related popularity and indegree-related activity effects
are nearly collinear, and it was already mentioned that theory, not
empirical fit, will have to decide which one is a more meaningful
representation. Some of the other effects also may be confounded,
but this depends on the data set. The four effects described as
degree-related popularity and activity are more basic than the
assortativity effects (cf. the relation between main effects and interactions in linear regression). Because of this, when testing any
assortativity effects, one usually should control for three of the
degree-related popularity and activity effects.
2.3.4. Covariates: exogenous effects
For an actor variable V, there are three basic effects: the ego
effect, measuring whether actors with higher V values tend to nominate more friends and hence have a higher outdegree (which also
can be called covariate-related activity effect or sender effect);
the alter effect, measuring whether actors with higher V values
will tend to be nominated by more others and hence have higher
indegrees (covariate-related popularity effect, receiver effect); and
the similarity effect, measuring whether ties tend to occur more
often between actors with similar values on V (homophily effect).
Tendencies to homophily constitute a fundamental characteristic
1 Experience has shown that for the degree-related effects, often the ‘driving force’
is measured better by the square roots of the degrees than by raw degrees. In some
cases this may be supported by arguments about diminishing returns of increasingly
high degrees. See the formulae in Appendix A.
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 49
of many social relations, see McPherson et al. (2001). When the
ego and alter effects are included, instead of the similarity effect
one could use the ego–alter interaction effect, which expresses that
actors with higher V values have a greater preference for other
actors who likewise have higher V values.
For categorical actor variables, the same V effect measures the
tendency to have ties between actors with exactly the same value
of V.
For a dyadic covariate, i.e., a variable defined for pairs of actors,
there is one basic effect, expressing the extent to which a tie
between two actors is more likely when the dyadic covariate is
larger.
2.3.5. Interactions
Like in other statistical models, interactions can be important to
express theoretically interesting hypotheses. The diversity of functions that could be used as effects makes it difficult to give general
expressions for interactions. The ego–alter interaction effect for an
actor covariate, mentioned above, is one example.
Another example is given by de Federico (2004) as an interaction
of a covariate with reciprocity. In her analysis of a friendship network between exchange students, she found a negative interaction
between reciprocity and having the same nationality. Having the
same nationality has a positivemain effect, reflecting that it is easier
to become friends with those coming from the same country. The
negative interaction effect was unexpected, but can be explained
by regarding reciprocation as a response to an initially unreciprocated tie, the latter being a unilateral invitation to friendship. Since
contacts between those with the same nationality are easier than
between individuals from different nationalities, extending a unilateral invitation to friendship is more remarkable (and perhaps
more costly) between individuals of different nationalities than
between those of the same nationality. Therefore it will be noticed
and appreciated, and hence reciprocated, with a higher probability. Thus, the rarity of cross-national friendships leads to a stronger
tendency to reciprocation in cross-national than same-nationality
friendships.
As a further class of examples, note that in the actor-based
framework it may be natural to hypothesize that the strength of
certain effects depends on attributes of the focal actor. For example, girls might have a greater tendency toward transitive closure
than boys. This can be modeled by the interaction of the ego effect
of the attribute and the transitive triplets, or transitive ties effect.
Other interactions (and still other effects) are discussed in
Snijders et al. (2008). As the selection presented here already illustrates, the portfolio of possible effects in this modeling approach
is very extensive, naturally reflecting the multitude of possibilities
by which networks can evolve over time. Therefore, the selection
of meaningful effects for the analysis of any given data set is vital.
This will be discussed now.
3. Issues arising in statistical modeling
When employing these models, important practical issues are
the question how to specify the model – boiling down mainly to the
choice of the terms in the objective function – and how to interpret
the results. This is treated in the current section.
3.1. Data requirements
To apply this model, the assumptions should be plausible in an
approximate sense, and the data should contain enough information. Although rules of thumb always must be taken with many
grains of salt, we first give some numbers to indicate the sizes of
data sets which might be well treated by this model. These rules of
thumb are based on practical experience.
The amount of information depends on the number of actors,
the number of observation moments (‘panel waves’), and the total
number of changes between consecutive observations. The number of observation moments should be at least 2, and is usually
much less than 10. There are no objections in principle against analyzing a larger number of time points, but then one should check
the assumption that the parameters in the objective function are
constant over time, or that the trends in these parameters are well
represented by their interactions with time variables (see point 10
below).
If one has more than two observation points, then in practice
one may wish to start by analyzing the transitions between each
consecutive pair of observations (provided these provide enough
information for good estimation—see below). For each parameter
one then can present the trend in estimated parameter values, and
depending on this one can make an analysis of a larger stretch of
observations if the parameters appear approximately constant, or
do the same while including for some of the parameters an interaction with a time variable.
The number of actors will usually be larger than 20—but if the
data contain many waves, a smaller number of actors could be
acceptable. The number of actors will usually not be more than
a few hundred, because the implicit assumption that each actor is a
potential network partner for any other actor might be implausible
for networks with so many actors that not all actors are aware of
each others’ existence.
The total number of changes between consecutive observations should be large enough, because these changes provide the
information for estimating the parameters. A total of 40 changes
(cumulated over all successive panel waves) is on the low side. More
changes will give more information and, thereby, allow more complicated models to be fitted. Between any pair of consecutive waves,
the number of changes should not be too high, because this would
call into question the assumption that the waves are consecutive
observations of a gradually changing network; or, if they were, the
consecutive observations would be too far apart.
This implies that, when designing the study, the researcher has
to have a reasonable estimate of how much change to expect. For
instance, if one is interested in the development of the friendship
network of a group of initially mutual strangers (e.g., university
freshmen), it may be good to plan the observation moments to
be separated by only a few weeks, and to enlarge the period
between observations after a couple of months. On the other hand,
if one studies inter-firm network dynamics, given the time delays
involved for firms in the planning and executing of their ties to other
firms, it may be enough to collect data once every year, or even less
frequently.
To express quantitatively whether the data collection points are
not too far apart, one may use the Jaccard (1900) index (also see
Batagelj and Bren, 1995), applied to tie variables. This measures the
amount of change between two waves by

N11 N11 + N01 + N10

,

(2)

where N11 is the number of ties present at both waves, N01 is
the number of ties newly created, and N10 is the number of ties
terminated. Experience has shown that Jaccard values between
consecutive waves should preferably be higher than 0.3, and –
unless the first wave has a much lower density than the second
– values less than 0.2 would lead to doubts about the assumption
that the change process is gradual, compared to the observation
frequency. If the network is in a period of growth and the second
network has many more ties than the first, one may look instead
at the proportion, among the ties present at a given observation,
of ties that have remained in existence at the next observation
(N11/(N10 + N11) in the preceding notation). Proportions higher
50 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
than 0.6 are preferable, between 0.3 and 0.6 would be low but may
still be acceptable. If the data collection was such that values of
ties (ranging from weak to strong) were collected, then these numbers may be used as rough rules of thumb and give some guidance
for the decision where to dichotomize the tie values—although, of
course, substantive concerns related to the interpretation of the
results have primacy for such decisions.
The methods require in principle that network data are complete. However, it is allowed that some of the actors enter the
network after the start, or leave before the end of the panel waves
(Huisman and Snijders, 2003), and a limited amount of missing
data can be accommodated (Huisman and Steglich, 2008). Another
option to represent that some actors are not yet, or no more, present
in the network, is to specify that certain ties cannot exist (‘structural zeros’) or that some ties are prescribed (‘structural ones’),
see Snijders et al. (2008). The use of structural zeros allows, e.g.,
to combine several small networks into one structure (with structural zeros forbidding ties between different networks), allowing to
analyze multiple independent networks that on themselves would
not yield enough information for estimating parameters, under the
extra assumption that all network follow dynamics with the same
parameter values in the objective function.
3.2. Testing and model selection
It turns out (supported by computer simulations) that the distributions of the estimates of the parameters ˇk in the objective
function (1), representing the importance of the various terms
mentioned in Section 2.3, are approximately normally distributed.
Therefore these parameters can be tested by referring the t-ratio,
defined as parameter estimate divided by standard error, to a standard normal distribution.
For actor-based models for network dynamics, informationtheoretic model selection criteria have not yet generally been
developed, although Koskinen (2004) presents some first steps
for such an approach. Currently the best possibility is to use ad
hoc stepwise procedures, combining forward steps (where effects
are added to the model) with backward steps (where effects are
deleted). The steps can be based on significance test for the various
effects that may be included in the model. Guidelines for such procedures are the following. We prefer not to give a recipe, but rather
a list of considerations that a researcher might have in mind when
constructing a strategy for model selection.
1. Like in all statistical models, exclusion of one effect may mask
the existence of another effect, so that pure forward selection
may lead to overlooking some effects, and it is advisable to start
with a model including all effects that are expected to be strong.
2. Fitting complicated models may be time-consuming and lead to
instability of the algorithm, and a resulting failure to obtain good
estimates. Therefore, forward selection is technically easier than
backward selection, which is unfortunately at variance with the
preceding remark.
3. The estimation algorithm (Snijders, 2001) is iterative, and the
initial value can determine whether or not the algorithm converges. For relatively simple models, a simple standard initial
value usually works fine. For complicated models, however, the
algorithm may converge more easily if started from an initial
value obtained as the estimate for a somewhat simpler model.
Estimates obtained from a more complicated model by simply
omitting the deleted effects sometimes do not provide good
starting values. Therefore, forward selection steps often work
better from the algorithmic point of view than backward steps.
This implies that, to improve the performance of the algorithm,
it is advisable to retain copies of the parameter values obtained
from good model fits, for use as possible initial values later on.
4. Network statistics can be highly correlated just because of
their definition. This also implies that parameter estimates can
be rather strongly correlated, and high parameter correlations
do not necessarily imply that some of the effects should be
dropped. For example, the parameter for the outdegree effect
often is highly correlated with various other structural parameters. This correlation tells us that there is a trade-off between
these parameters and will lead to increased standard errors of
the parameter for the outdegree effect, but it is not a reason for
dropping this effect from the model.
5. Parameters can be tested by a so-called score-type test without estimating them, as explained in Schweinberger (submitted
for publication). Since estimating many parameters can make
the algorithm instable, and in forward selection steps it may be
necessary to have tests available for several effects to choose
the most important one to include, the score-type tests can be
very helpful in model selection. In this procedure, a model (null
hypothesis) including significant and/or theoretically relevant
parameters is tested against a model (alternative hypothesis)
extended by one or several parameters one is also interested
in. Under the null hypothesis, those parameters are zero. The
procedure yields a test statistic with a chi-squared null distribution, along with standard normal test statistics for each
separate parameter. The parameters for which a significant test
result was found, then may be added to the model for a next
estimation round.
6. It is important to let the model selection be guided by theory,
subject-matter knowledge, and common sense. Often, however,
theory and prior knowledge are stronger with respect to effects
of covariates – e.g., homophily effects – than with respect to
structure. Since a satisfactory fit is important for obtaining generalizable results, the structural side of model selection will of
necessity often be more of an inductive nature than the selection of covariate effects. The newness of this method implies
that we still need to accumulate more experience as to what
is a ‘satisfactory’ fit, and how complicated models should be in
practice.
7. Among the structural effects, the outdegree and reciprocity
effect should be included by default. In almost all longitudinal
social network data sets, there also is an important tendency
toward transitivity (Davis, 1970). This should be modeled by
one, or several, of the transitivity-related effects described
above.
8. Often, there are some covariate effects which are predicted by
theory; these may be control effects or effects occurring in
hypotheses. It is good practice to include control effects from
the start. Non-significant control effects might be dropped provisionally and then tested again as a check in the presumed final
model; but one might also retain control effects independent of
their statistical significance. We do not think there are unequivocal rules whether or not to include from the start the effects
representing the main hypotheses in a given study.
9. The degree-based effects (popularity, activity, assortativity) can
be important structural alternatives for actor covariate effects,
and can be important network-level alternatives for the triadlevel effects. It is advisable, at some moment during the model
selection process, to check these effects; note that the squareroot specification usually works best.
10. If the data have three or more waves and the model does not
include time-changing variables, then the assumption is made
that the time dynamics is homogeneous, which will lead to
smooth trajectories of the main statistics from wave to wave. It
is good as a first general check to consider how average degree
develops over the waves, and if this development does not follow a rather smooth curve (allowing for random disturbances),
to include time-varying variables that can represent this devel
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 51
opment. Another possibility is to analyze consecutive pairs of
waves first, which will show the extent of inhomogeneity in the
process (cf. the example in Snijders, 2005).
11. The model assumes that the ‘rules for network change’ are the
same for all actors, except for differences implied by covariates or network position. This leaves only moderate room for
outlying actors, such as are indicated by relatively very large
outdegrees or indegrees. Very high or very low outdegrees or
indegrees should be ‘explainable’ from the model specification;
if they are only explainable from earlier observations (‘path
dependence’), they will have a tendency to regress toward the
mean. The model specification may be able to explain outliers
by covariates identifying exceptional actors, but also by degreerelated endogenous effects such as the self-reinforcing Matthew
effect mentioned above. It is good as a first general check to
inspect the indegrees and outdegrees for outliers. If there are
strong outliers then it is advisable to seek for actor covariates
which can help to explain the outlying values, or to investigate
the possibility that degree-related effects, explainable or at least
interpretable from a theoretical point of view, may be able to
represent these outlying degrees. If these cannot be found, then
one solution is to use dummy variables for the actors concerned,
to represent their outlying behavior. Such an ad hocmodel adaptation, which may improve the fit dramatically, is better than
the alternative of working with a model with a large unmodeled heterogeneity. In case there is a theoretical argument to
expect certain outliers, these can be pointed out by including a
dummy variable, but of a different kind. In contrast to the former type of outliers, the latter one is expected, so should be
captured in advance by a covariate. If one is not capable to make
a difference between the two types, one has to rely on the ad
hoc model adaptation.
3.3. Example: friendship dynamics
By way of example, we analyze the evolution of a friendship
network in a Dutch school class. The data were collected between
September 2003 and June 2004 as part of a study reported in
Knecht (2008). The 26 students were followed over their first year
at secondary school during which friendship networks as well as
other data were assessed at four time points at intervals of three
months. There were 17 girls and 9 boys in the class, aged 11–13 at
the beginning of the school year. Network data were assessed by
asking students to indicate up to 12 classmates which they considered good friends. The average number of nominated classmates
ranged between 3.6 and 5.7 over the four waves, showing a moderate increase over time. Jaccard coefficients for similarity of ties
between waves are between 0.4 and 0.5, which is somewhat low
(reflecting fairly high turnover) but not too low.
Some data were missing due to absence of pupils at the moment
of data collection. This was treated by ad-hoc model-based imputation using the procedure explained in Huisman and Steglich (2008).
One pupil left the classroom. Such changes in network composition
can also be treated by the methods of Huisman and Snijders (2003),
but this simple case was treated here by using structural zeros: starting with the first observation moment where this pupil was not a
member of the classroom any more, all incoming and outgoing tie
variables of this pupil were fixed to zero and not allowed to change
in the simulations.
Considering point 1 above, effects known to play a role in
friendship dynamics, such as basic structural effects and effects of
basic covariates, are included in the baseline model. From earlier
research, it is known that friendship formation tends to be reciprocal, shows tendencies towards network closure, and in this age
group is strongly segregated according to the sexes. The model
includes, for each of these tendencies, effects corresponding to
these expectations. Structural effects included are reciprocity; transitive triplets and transitive ties, measuring transitive closure that
is compatible with an informal local hierarchy in the friendship
network; and the three-cycles effect measuring anti-hierarchical
closure. Homophily based on the sexes is included as the same sex
effect. All variables are centered. For example, the dummy variable for sex (boys = 1, girls = 0) has mean 0.346 (9 boys and 17
girls), which leads to the centered values vi = –0.346 for girls and
vi = 0.654 for boys.
As exogenous control variables, we include sender and receiver
effects of sex, and a dyadic covariate indicating friendship in primary school reflecting relationship history. In addition, several
degree-related endogenous effects are included as control effects:
in- and outdegree-related popularity, and outdegree-related activity, explained above. Estimates for this model are given in Table 1 as
Model 0. All calculations were done using Siena version 3.2 (Snijders
et al., 2008).
The parameters reported for the rate function in periods 1–3
are defined in the simulation model as the expected frequencies,
between successive waves, with which actors get the opportunity
to change a network tie. For these parameters no p-values are given
in the tables, as testing that they are zero is meaningless (if these
would be zero there would be no change at all). These estimated rate
parameters will be higher than the observed numbers of changes
per actor, however, because in the model an actor may get the
opportunity to change a tie but choose not to make any change, and
because actors may add a tie during the simulations, and withdraw
the same tie before the next observation moment.
This analysis confirms, for this data set, several of the known
properties of friendship networks: there is a high degree of reciprocity, as seen in the significant reciprocity parameter; there is
segregation according to the sexes, as seen in the significant same
sex parameter; there is an almost equally strong effect of having been friends at primary school already, and there is evidence
for transitive closure, as seen in the significant effects of transitive triplets and transitive ties. A direct comparison of the size of
parameter estimates is possible, given that they occur in the same
linear combination in the objective function, but it should be kept in
mind that these are unstandardized coefficients. Other significant
effects are the negative 3-cycles parameter, which indicates that the
tendencies toward closure are not completely egalitarian (as one
might have thought based on the reciprocity parameter), but do
show some evidence for local hierarchization in the network. This
also is suggested by the marginally significant negative effect of the
outdegree-related popularity which indicates that active pupils, i.e.,
those who nominate particularly many friends, are less likely to be
chosen as friends – this could be a status effect negatively associated with nomination activity. Also significant is the sender effect
of sex (sex (M) ego), which in our coding of the variable means that
the boys tend to be more active in the classroom friendship network
than the girls.
Rate parameters, finally, suggest that the amount of friendship change seems to peak in the second period (perhaps due to
a higher friendship turnover after the Christmas break) and slow
down towards the end of the school year. These differences are
small, however. The same descriptive conclusion can be drawn also
by inspecting the observed amounts of change, without needing to
refer to a statistical model.
In a subsequent model (Model 1 in Table 1), more parsimony is
obtained by eliminating the non-significant effects in a backward
selection procedure. The sex alter effect was retained in spite of
its non-significance, because the three sex-related effects belong
together as a representation of sex-related friendship preferences.
One by one, the least significant of the insignificant effects were
dropped from the model. While doing so, score-type tests were
made for the earlier omitted parameters (now constrained to zero)
52 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
Table 1
Parameter estimates of friendship evolution models, with standard errors and two-sided p-values.
Model 0 Model 1 Model 2 Model 3
Estim. S.E. p Estim. S.E. p Estim. S.E. p Estim. S.E. p
Objective function

Outdegree Reciprocity (evaluation) Reciprocity (endowment) Transitive triplets Transitive ties 3-cycles Indegree popularity (sqrt) Outdegree popularity (sqrt) Outdegree activity (sqrt) Sex (M) ego Sex (M) alter Same sex Primary school	–1.41 1.34	0.43 0.22	<0.001 <0.001	–1.67 1.42	0.38 0.20	<0.001 <0.001	–1.62 1.45	0.34 0.20	<0.001 <0.001	–1.59 0.71 1.42 0.20 0.67 –0.22	0.33 0.48 0.80 0.03 0.20 0.10	<0.001 0.14 0.076 <0.001 <0.001 0.028 0.29 <0.001 0.19 <0.001 0.25 <0.001 0.020
0.23 0.74 –0.32 –0.18 –0.40 0.01 0.35 0.10 0.49 0.37	0.03 0.21 0.10 0.12 0.24 0.07 0.13 0.13 0.13 0.13	<0.001 <0.001 <0.001 0.13 0.093 0.93 0.008 0.46 <0.001 0.006	0.21 0.74 –0.26	0.03 0.23 0.09	<0.001 0.001 0.005 0.23 0.013 0.18 0.002 0.24 <0.001 0.013	0.21 0.78 –0.25	0.03 0.22 0.09	<0.001 <0.001 0.007 0.23 0.002 0.12 0.002 0.20 <0.001 0.016
–0.56	0.23	–0.62	0.20	–0.61	0.20
0.39 0.15 0.54 0.35	0.13 0.13 0.12 0.14	0.44 0.18 0.56 0.34	0.14 0.14 0.14 0.14	0.41 0.16 0.56 0.35	0.13 0.14 0.13 0.15
Rate function
Rate period 1 Rate period 2 Rate period 3 Effect sex (M) on rate	10.01 10.67 9.51	2.05 1.85 1.53	9.76 10.44 8.91	1.91 1.83 1.39	9.69 10.28 8.81 –0.42	1.96 1.82 1.37 0.28	10.93 11.13 9.38	2.26 1.93 1.52	0.13

The p-values are based on approximate normal distributions of the t-ratios (estimate divided by standard error). When rendered for non-estimated parameters, they refer to
score-type tests.

to check whether the parameter does not become significant upon dropping other effects from the model. This is possible in models with correlated effects like ours, but it did not occur for our data	A, the joint contribution of these V-related effects to the objective
set. Estimates of Model 1 give the same qualitative results as those of Model 0. The parameters dropped due to insignificance were the outdegree-related activity effect (suggesting that the value for ego of an individual friendship does not depend on how many other	j xijI{vi = vj}
function is
ˇe xijvi + ˇa xijvj + ˇs
j	j
where I{vi = vj} = 1 if vi = vj, and 0 otherwise. This means that the

friends the friend currently has) and the indegree-related popularity effect (suggesting that receiving many friendship nominations

is not a self-reinforcing process).

ˇevi + ˇavj + ˇsI{vi = vj} = 0.35vi + 0.10vj + 0.49I{vi = vj} Substituting the values –0.346 for females and 0.654 for males

3.4. Parameter interpretation

For the general understanding of the numerical values of the parameters, it may be kept in mind that the parameters ˇk in the objective function are unstandardized coefficients of the statistics	Ego	Alter	M
F

of which the mathematical formulae are given in Appendix A.
The parameters in the objective function can be interpreted in
two ways. In the first place, by interpreting this function as the
“attractiveness” of the network for a given actor. For getting a feeling of what are small and large values, it may be noted (see the
interpretation in terms of myopic optimization in Snijders, 2001)
that the objective functions are used to compare how attractive
various different tie changes are, and for this purpose random disturbances are added to the values of the objective function with
standard deviations equal2 to 1.28.
The objective function is a weighted sum of effects sik(x); their
mathematical definitions are given in Appendix A. In most cases the
contribution of a single tie variable xij is just a simple component
of this formula.
For example, consider the actor variable sex, denoted as V, and
originally with values 1 for girls and 2 for boys. All variables are
centered. The global mean of this variable is 1.346 (9 boys and 17
girls), which leads to the centered values vi = –0.346 for girls and
vi = 0.654 for boys. For this variable the model includes the ‘ego’
effect, the ‘alter’ effect, and the ‘same’ effect. Let us denote the
parameters by ˇe, ˇa, and ˇs. Then, using the formulae in Appendix
2 More exactly, the value is2/6, the standard deviation of the Gumbel distribution.
contribution of the single tie xij to the objective function, considering only the sex-related effects, is given by
yields the following table.
F 0.33 –0.06
M 0.19 0.78
This table shows that girls as well as boys prefer friendships with
same-sex alters, but for boys the difference is more pronounced
than for girls.
A second interpretation is that when actor i has the opportunity
to make a change in her outgoing ties (where no change also is an
option), and xa and xb are two possible results of this change, then
fi(xb, ˇ) – fi(xa, ˇ) is the log odds ratio for choosing between these
two alternatives—so that the ratio of the probability of xb and xa as
next states is
exp(fi(xb, ˇ) – fi(xa, ˇ)) = exp(fi(xb, ˇ))
exp(fi(xa, ˇ)) .
Note that, when the current state is x, the possibilities for xa and
xb are x itself (no change), or x with one extra outgoing tie from i,
or x with one less outgoing tie from i. Explanations about log odds
ratios can be found in texts about logistic regression and loglinear
models, e.g., Agresti (2002). A further elaborated example of this is
given in Section 4.2.
4. More complicated models
This section treats two generalizations of the model sketched
above.
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 53
4.1. Differential rates of change: the rate function
Depending on actor attributes or on positional characteristics
such as indegree or outdegree, actors might change their ties at
differential frequencies. This can be the case, e.g., in networks
between organizations with clear differences in degrees, where
the outdegrees reflect the importance to the organizations of the
network under study, and the resources they devote to positioning themselves in it. The average frequency at which actors get
the opportunity to change their outgoing ties then is called the
rate function, depending on attributes and network position of the
actors.
Model 2 in Table 1 gives an example of such an analysis. It
extends Model 1 by adding an effect of sex on the rate function.
The estimated negative effect indicates that in the data set under
study, boys change their network ties less frequently than girls, but
the difference is not significant (p = 0.13).
To interpret the parameter values, one should know that a socalled exponential link function is used (Snijders, 2001; Snijders et
al., 2008), which means that the variables have an effect on the rate
function after an exponential transformation, with a multiplicative
effect. For example, the parameter estimate of –0.42 for the effect
of sex on the rate function implies that the estimated rate function
is the base rate multiplied by exp(–0.42vi). Recall that the values
of the variable ‘sex’ are, centered, vi = –0.346 for females and vi =
0.654 for males. Thus, for period 1, for girls the expected number of
opportunities for change is 9.69 × exp(–0.42 × (–0.346)) = 11.2,
and for boys it is 9.69 × exp(–0.42 × (0.654)) = 7.4. The difference
seems rather large but is not significant in view of the small sample
size.
4.2. Differences between creating and terminating ties: the
endowment function
In the treatment given above, terminating a tie is just the opposite of creating one. This is not always a good representation of
reality. It is conceivable, for example, that the loss when terminating a reciprocal tie is greater than the gain in creating one; or
that transitive closure works especially for the creation of new ties,
but hardly guards against termination of existing ties. This can be
modeled by having two components of the objective function: the
evaluation function, which considers only the network that will be
the case as a result of the change to be made; and the endowment
function, which is a component that operates only for the termination of ties and not for their creation. Everything discussed above
about the objective function concerned the evaluation function—in
other words, in those discussions and the example, the endowment
function was nil. The endowment function gives contributions to
the objective function that do not play a role when creating ties,
but that are lost when dissolving ties.
Model 3 in Table 1 gives the results of an analysis that includes,
in addition to the effects of Model 1, also an endowment effect
related to reciprocity. It was estimated as significant and positive,
while the corresponding evaluation function effect of reciprocity
dropped in size and significance. To interpret this result, jointly consider the reciprocity evaluation effect with parameter 0.71 and the
reciprocity endowment effect with parameter 1.42. The contribution of a tie being reciprocated then is 0.71 for the creation of the
tie and 0.71 + 1.42 = 2.13 against the termination of the tie. Thus,
reciprocity here is more important against terminating friendships
– that is, for maintaining friendships – than for creating friendships.
To elaborate this example, consider how the friendship choices
of a girl towards other girls depend on reciprocity (Fig. 2). Suppose
that actor i can change one of her ties, while there are two girls j1
and j2, both of them choosing i as a friend, and two others j3 and j4
not choosing i. In addition, suppose that currently i chooses j1 and
Fig. 2. Four options for actor i.
j3 as friends, but not the other two. Assume finally (artificially, for
the sake of explanation) that these four girls do not choose each
other and further also are isolated from i’s network so that other
structural effects besides reciprocity do not matter. Since the actor
variable ‘sex’ has centered value vi = –0.346 for girls, the parameter
estimates for Model 3 give as the total contribution of the three
sex-related effects for girl–girl ties 0.41vi + 0.16vj + 0.56I{vi = vj} =
(0.41 + 0.16) × (–0.346) + 0.56 = 0.36. With the outdegree effect
of –1.59, this yields –1.59 + 0.36 = –1.23 as the basic contribution
of a tie to the evaluation function.
When girl i can change a tie variable, using this value of –1.23 for
the combined effect of outdegree and the three sex-related effects
for a girl–girl tie, five of the options for i are the following:
(A): drop reciprocated friendship tie to j1: –(–1.23) – 2.13 =
–0.90;
(B): reciprocate friendship tie from j2 : –1.23 + 0.71 = –0.52;
(C): drop non-reciprocated friendship tie to j3 : –(–1.23) = 1.23;
(D): initiate friendship tie to j4 : –1.23;
(E): do nothing: 0.0.
Since these are contributions to logarithms of probabilities, the
proportionality factors between the probabilities of these events
must be calculated as the exponential transformations of these values, which are, respectively, e–0.90 = 0.41, 0.59, 3.42, 0.29, and 1.
These are the relative probabilities of changes toward any given
other girl. One should note, however, that there may be different
numbers of the four cases (A–D) for a given girl, and the probability
of severing any reciprocated tie, or of creating any non-reciprocated
tie, depends also on these numbers for the ‘ego’ girl under consideration. Since the friendship network is sparse, with average degrees
between 3.6 and 5.7, the cases of type (D) will be most numerous.
Consider, for instance, a girl with 3 mutual girlfriends (A), who has
2 non-reciprocated friendships to girls (C), 1 other girl who mentions her as a friend without reciprocation (B), and 10 girls without
a friendship either way (D). Suppose that in addition she has no
friendships with any of the 9 boys, and denote the option of establishing a friendship to a boy by (F). Retain the unrealistic simplifying
assumption that all her network members are mutually unrelated,
also after adding one hypothetical new friend, so that transitivity
does not influence probabilities of change. The baseline value of a
tie from a girl to a boy is –1.59 + 0.41 × (–0.346) + 0.16 × 0.654 =
–1.63, with exponential transform 0.20. Taking into account the
fact that the number of opportunities for options (A) to (F) are
3, 1, 2, 10, 1, and 9, the six proportionality factors have to be
divided by the denominator (3 × 0.41) + (1 × 0.59) + (2 × 3.42) +
(10 × 0.29) + (1 × 1) + (9 × 0.20) = 14.36. Thus, for this girl, the
probabilities are:
(A): of dropping any of the three reciprocated friendship ties:
(3 × 0.41)/14.36 = 0.09;
(B): of reciprocating the incoming friendship tie:
(1 × 0.59)/14.36 = 0.04;
54 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
(C): of dropping one of the non-reciprocated friendship ties:
(2 × 3.42)/14.36 = 0.48;
(D): of initiating some new friendship tie to a girl:
(10 × 0.29)/14.36 = 0.20;
(E): of doing nothing: 0.07;
(F): and of extending a new friendship tie to a boy:
(9 × 0.20)/14.36 = 0.13.
Thus, in line with theories about reciprocation such as balance
theory, the probability is slightly larger than 0.5 that the proportion of reciprocity in friendships will be increased. There are many
random influences, however, that would decrease reciprocity—but
most of these are proposals of new ties which could be seen by the
other party as an invitation toward future reciprocation.
5. Dynamics of networks and behavior
Social networks are so important also because they are relevant for behavior and other actor-level outcomes: related actors
may influence one another (e.g., Friedkin, 1998), and ties will be
selected in part based on the similarity between ego and potential
relational partners (homophily, see McPherson et al., 2001). This
means that not only is the network changing as a function of itself
and of the actor variables, but likewise the actor variables are changing as a function of themselves and of the network. We use the
term behavior as shorthand for endogenously changing actor variables, although these could also refer to attitudes, performance,
etc.; there could be one or more of such variables. It is assumed
here that the behavior variables are ordinal discrete variables, with
values 1, 2, etc., up to some maximum value, for instance, several
levels of delinquency, several levels of smoking, etc. The dependence of the network dynamics on the total network-behavior
configuration will be also called the social selection process, while
the dependence of the behavior dynamics on the total networkbehavior configuration will be called the social influence process.
Both social influence and social selection can lead to similarity between tied actors, which is often observed. A fundamental
question then is whether this similarity is caused mainly by influence or mainly by selection, as discussed by Ennett and Bauman
(1994) for smoking behavior and Haynie (2001) for delinquent
behavior.
This combination of selection and influence can be modeled by
an extension of the actor-based model to a structure where the
dependent variables consist not only of the tie variables but also of
the actors’ behavior variables, as specified in Snijders et al. (2007)
and Steglich et al. (submitted for publication). Of course there
usually will be, in addition, also exogenous actor and/or dyadic
variables in the role of independent variables.
The assumptions for the actor-based model for the dynamics
of networks and behavior are extensions of the assumptions for
network dynamics. The extended formulations are as follows, given
without the background explanations which were given above and
which apply also for this case.
1. As above, the underlying time parameter is continuous.
2. The changing system consisting of network and behavior is the
outcome of a Markov process. Thus, the probabilities of change of
the network as well as those of the behavioral variables depend,
at each moment, on the current combination of network structure and behavior variables for all actors.
3. At a given moment either one probabilistically selected actor
may change a tie, or one actor may change his/her behavior by
going one unit up or down (recall that the behavior variables
are assumed to be integer-valued). This excludes coordination
between changes in the network and in the behavior.
The fact that changes in behavior are assumed to be by one unit
in a single time point imply that a ‘natural’ application of the
model requires that the total number of ordinal scale values is
not too large; in practice applications mostly have had 2–5, and
sometimes up to 10, scale values.
4. The actors control their outgoing ties as well as their own behavior. This is meant not in the sense of conscious control, but in the
sense that the explanation of the actor’s outgoing ties and behavior is based in the actor and the structural and other limitations
provided by the actor and his/her social context.
5. The moments were actors get the opportunity for a tie change
or a behavior change are modeled as distinct processes, so these
are governed by a priori unrelated parameters.
6. There are distinct processes also for tie changes and behavior
changes, conditional on the possibility to make the respective
type of change, so these are governed by a priori unrelated
parameters.
The changes in behavior depend on an objective function similar
to the objective function for network changes. However, this function will be different because it needs to represent primarily the
actor’s behavior rather than his/her network position, and because
choices of behavior changes may be framed differently from choices
of tie changes, depending on different goals and restrictions.
The model assumptions imply that the dependent behavior
variable will change endogenously during the simulations, representing the endogenous social influence process. Since the network
and the behavior variables both influence the dynamics of the network ties and of the actors’ behavior, the sequence of changes in
the network and in the behavior, reacting on each other, generates a mutual dependence between the network dynamics and the
behavior dynamics.
5.1. The objective function for behavior
We only consider models where increasing the behavior variable
has just the opposite effect of decreasing it, and the objective function for behavior is the same as the evaluation function (a separate
endowment function is not considered). The objective, or evaluation, function can be represented, analogously to (1), as

fZ

ki Z (x, z),

(3)

i(ˇ, x, z) =
k
ˇkZswhere sZ
ki(x, z) are functions depending on the behavior of the focal
actor i, but also on the behavior of his network partners, his network
position, etc. The strength of the effects of these functions on behavior choices are represented by the parameters ˇkZ. The superscript
Z is used to distinguish the effects and parameters for behavior
change from those for network change (which could be given the
superscript X). The main possible terms of the evaluation function
are as follows.
5.1.1. Basic shape effects
We first discuss basic tendencies determining behavior change
that are independent of actor attributes and network position.
A baseline definition for the evaluation function will be a curve,
depending on the actor’s own behavior zi, that can be loosely interpreted as the relative preference for the specific value zi of the
behavior. The term ‘prefer’ should be taken with much reservation,
as a shorthand ‘as-if’ term—we could just as well see this, e.g., as a
matter of constraints. When the behavior variable is dichotomous,
then a linear function suffices, as each function of two values can
be represented by a linear function; but for three or more possible
values a unimodal ‘preference’ function will often be reasonable,
so that a specification will be required that allows the function
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 55
Fig. 3. Basic shape of the evaluation function for behavior; in this case, with the
maximum at z = 2.
to be curvilinear. Thus, in Fig. 3, a simple evaluation function is
drawn for a behavior variable with range 1–4, which is maximal at
the value z = 2, indicating that when actors have a possibility for
change they will be drawn toward the value z = 2: if their current
value for Z is higher than 2 then the probability is higher that they
will decrease their value, if the current value is lower than 2 then
the probability is higher that they will increase their value of Z. To
represent this mathematically, a quadratic function can be used.
The linear and quadratic coefficients in this function are called the
linear shape effect and the quadratic shape effect. Note that the latter
is superfluous for a dichotomous behavior variable.
The quadratic shape effect can also be called the effect of Z
on itself, and is a kind of feedback effect. When this parameter
is negative there is negative feedback, or a self-correcting mechanism: when the value of the actor’s behavior increases, the further
push toward higher values of the behavior will become smaller
and when it decreases, the further push toward lower values of
the behavior will become smaller. Conversely, when the coefficient
of the quadratic term is positive, the feedback will be positive, so
that changes in the behavior are self-reinforcing. This can be an
indication of addictive behavior. Negative values, or values not significantly different from 0, are more often seen than positive values.
When the coefficient for the shape effect is denoted ˇ1Z and
the coefficient for the quadratic shape effect is ˇ2Z, then the total
contribution of these two effects is ˇ1Z zi + ˇ2Z zi2. With a negative
coefficient ˇ2Z, this is a unimodal preference function. The maximum then is attained for zi = –2ˇ1Z /ˇ2Z, or more precisely, the
integer value within the prescribed range that is closest to this number. (Of course additional effects will lead to a different picture; but
if the additional effects are linear as a function of zi in the permitted
range – which can be checked from the formulae in Appendix A, and
which is not the case for all similarity effects as defined below! –,
this will change the location of the maximum but not the unimodal
shape of the function.)
5.1.2. Influence and position-dependent effects
The tendencies expressed in the shape parameters affect every
actor in the same way, irrespective of his/her characteristics or
network position. To capture social network effects, additional
terms in the behavior evaluation function are needed, differentiating between actors on the basis of their network position and the
behavior of the others to whom they are tied.
The actor-based model can represent social influence, i.e., influence from alters’ behavior on ego’s behavior, in various ways,
because there are several different ways to measure and aggregate
the influences from different alters. Three different representations
are as follows.
1. The average similarity effect, expressing the preference of actors
to be similar in behavior to their alters, in such a way that the
total influence of the alters is the same regardless of the number
of alters (i.e., ego’s outdegree).
2. The total similarity effect, expressing the preference of actors to
be similar in behavior to their alters, in such a way that the total
influence of the alters is proportional to the number of alters.
3. The average alter effect, expressing that actors whose alters have
a higher average value of the behavior, also have themselves a
stronger tendency toward high values on the behavior.
The choice between these three will be made on theoretical
grounds and/or on the basis of statistical significance.
In addition to this type of social influence, network position itself
could also have an effect on the dynamics of the behavior. Indeed,
the actor’s outdegree or indegree may be terms in the objective
function; in the case of positive parameters, this expresses that
those who are more active (higher outdegree) or more popular
(higher indegree) have a stronger tendency to display higher values
of the behavior.
5.1.3. Effects of other actor variables
For each actor-dependent covariate as well as for each of the
other dependent behavior variables (if any), a main effect on the
behavior can be included, representing the influence of this actor
variable on changes in the behavior. In addition, it is possible that
such a variable will moderate the influence effect, leading to an
interaction between the variable and the influence effect.
5.2. Specification of models for dynamics of networks and
behavior
There is a natural advantage of the network part of the model
over the behavior part in terms of the amount of information
available on the two dimensions. For n actors, there are n measurements of the behavior variable, while there are as many as
n(n – 1) measurements of network tie variables. In statistical terms,
there will be less power to detect determinants of behavioral evolution than there is to detect determinants of network evolution.
Therefore, backward model selection is not a good route to follow for specifying a model of network-behavior co-evolution: weak
and unsystematic effects on behavioral change, when estimated,
subtract from the ability to identify the stronger and more systematically occurring ones. Instead of starting with an extensive model,
it is better to start with a small one, and proceed by way of forward
model selection to arrive at a good model.
Since it is more difficult to estimate a model of network-behavior
co-evolution than to estimate a model of only network evolution, it
makes sense first to estimate a good model for the dynamics of only
the network according to the approach of the preceding section, and
to use this as a baseline for the network part of the co-evolution
model.
It has become clear above that there are several specifications
of the social selection part, such as Z-similarity and Z-ego × alter;
like for any actor variable, also for behavior variable Z the ego and
alter effects may be relevant. The similarity effect in the network
dynamics part is directly interpretable, whereas the Z-ego × alter
interaction effect needs also the ego and alter effects to be well
interpretable. For the social influence part likewise there are several
possible specifications, such as average similarity, total similarity,
and average alter.
A difficulty is that we often have no clear theoretical clue as to
which of these three specifications is better in a particular case;
the alternative specifications are defined by effects that may be
highly mutually correlated and therefore are not readily estimated
jointly in the same model; and the power of detecting selection and
influence will depend on the specification chosen. If we do have
prior information as to the best specification, then it is preferable
to work with this specification. If we do not have such information, and we wish to avoid the chance capitalization inherent in
56 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60
using the ‘most significant’ effect without taking into account that
it was chosen exactly because it was the most significant, then we
could proceed along something like the following lines. An example of this approach is given in the next subsection. This procedure
uses score-type tests (Schweinberger, submitted for publication)
for several parameters simultaneously, which are chi-squared tests
with number of degrees of freedom equal to the number of tested
effects, and which have the advantage that parameters can be tested
without estimating them.
1. Specify and estimate a baseline model ‘BM’ for network-behavior
co-evolution in which the network and behavior dynamics are
independent; that is, the model for network dynamics contains
no effects dependent on the behavior variable, and vice versa.
2. Choose a number of candidate social selection effects ‘SEL’ and a
number of candidate social influence effects ‘INF’ on theoretical
grounds, without considering the data.
3. Test the effects in the sets SEL and INF by score-type tests in the
baseline model BM. This gives the statistical evidence about the
existence of influence and selection, not controlling each effect
for the other.
4. Select the effects that are individually most significant in either
set, and denote these effects by SEL1 and INF1.
(In this formulation, the effects are selected based on their significance when hypothesized to be added to the baseline model.
Another possibility is to select the effects based on the following
two models.)
5. To test influence effects while controlling for selection, estimate
the model BM + SEL1 (i.e., the baseline model extended with the
most significant selection effect) and within this model test all
of the effects INF jointly by a score-type test.
6. To test selection effects while controlling for influence, estimate
the model BM + INF1 and test all of the effects SEL jointly by a
score-type test.
7. Finally, to estimate a model with influence and selection, estimate the model BM + SEL1 + INF1.
8. It often will be sensible to conduct some further checks to guard
against the danger of overlooking important effects. This can be
done again with score-type tests. Candidate effects to be checked
include the indegree and outdegree effects on the behavior variable.
5.3. Example: dynamics of friendship and delinquency
Substantively, what we address is again the dynamics of
the friendship network in the school class of 11–13 year old
pupils investigated above, now co-evolving with their delinquency
(Knecht, 2008). This variable is defined as a rounded average over
four types of minor delinquency (stealing, vandalism, graffiti, and
fighting), measured in each of the four waves of data collection.
The five-point scale ranged from ‘never’ to ‘more than 10 times’,
and the distribution is highly skewed, most students reporting no
delinquency. In a range of 1–5, the mode was 1 at all four waves, the
average rose over time from 1.4 to 2.0, and the value 5 was never
observed.
The question addressed is, whether the data provide evidence
for network influence processes playing a role in the spread of delinquency through the group defined by the classroom. Analyses were
carried out by Siena version 3.17 (Snijders et al., 2008). Reported
results all were taken from runs in which all ‘t-ratios for convergence’ (see the Siena manual) were less than 0.1 in absolute value,
indicating good convergence of the algorithm.
For the model selection we follow the steps laid out in the preceding section. The baseline model is the model which for the
friendship dynamics is model 3 in Table 1, and for the delinquency
dynamics includes the linear and quadratic shape effects and the
effect of sex (as a control variable). The effects potentially modelling social selection based on delinquency (the set SEL in the
preceding section) are delinquency ego, delinquency alter, delinquency similarity, and delinquency ego × alter. Delinquency ego
and delinquency alter are included here as control variables for
delinquency ego × delinquencyalter. The effects potentially modelling social influence with respect to delinquency (the set INF) are
average similarity, total similarity, and average delinquency alter.
Score tests for the selection and influence effects tested with the
baseline model as the null hypothesis yielded the following pvalues. For both parts of the model, first the results of the overall
score test (of SEL and INFL, respectively) are given, and then the
results for the separate degrees of freedom of which this test is
composed.
Effect p
Friendship dynamics

Overall test for social selection (4 d.f.) Delinquency ego Delinquency alter Delinquency similarity Delinquency ego × delinquency alter Delinquency dynamics Overall test for social influence (3 d.f.) Average similarity Total similarity Average delinquency alter	< 0.001 < 0.001 0.32 < 0.001 0.02
0.04 0.03 0.12 0.48

The overall tests show evidence for social selection (p < 0.001)
as well as for social influence (p < 0.05), when these are not being
controlled for each other. The separate tests suggest that strongest
effects are delinquency similarity for the network dynamics and
average similarity for the behavior dynamics. These are the effects
denoted above as SEL1 and INF1, respectively.
To test selection while controlling for influence, the baseline
model was extended with influence operationalized as average similarity, and the four selection effects (delinquency ego, delinquency
alter, delinquency similarity, and delinquency ego × delinquency
alter) were jointly tested by a score-type test. The test result was
highly significant, p < 0.001.
To test influence while controlling for selection, the baseline model was extended with selection operationalized as
delinquency similarity, and the three influence effects (average similarity, total similarity, and average delinquency alter)
were jointly tested by a score-type test. This led to p = 0.04.
Thus, for this classroom there is clear evidence (p < 0.001) for
delinquency-based friendship selection, and evidence (p = 0.04)
for influence from pupils on the delinquent behavior of their
friends.
To estimate a model incorporating selection and influence,
the baseline model was extended with delinquency similarity for
the network dynamics and average similarity for the behavior
dynamics. This model is estimated and presented in Table 2. The
delinquency ego and delinquency alter effects were also tested for
inclusion in the network dynamics model, and the indegree and
outdegree effects were tested for inclusion in the behavior dynamics model, but none of these were significant.
It can be concluded for this data set that there is evidence for
delinquency-based friendship selection, expressed most clearly by
the delinquency similarity measure; and for influence from pupils
on the delinquent behavior of their friends, expressed best by the
average similarity measure. The delinquent behavior does not seem
to be influenced by sex. The model for network dynamics yields
estimates that are quite similar to those for the model without
the simultaneous delinquency dynamics, except that the reciprocity effect has shifted more strongly towards the endowment
effect.
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 57
Table 2
Estimates of model for co-evolution of friendship and delinquency, with standard
errors and two-sided p-values.
Effect Estimate S.E. p
Network objective function

Outdegree Reciprocity (evaluation) Reciprocity (endowment) Transitive triplets Transitive ties 3-cycles Outdegree based popularity (sqrt) Sex (M) ego Sex (M) alter Same sex Primary school Delinquency similarity	–1.91 0.25 2.10 0.22 0.67 –0.27 –0.52 0.48 0.19 0.61 0.46 3.22	0.41 0.44 0.81 0.03 0.23 0.11 0.25 0.16 0.16 0.15 0.18 1.66	<0.001 0.57 0.010 <0.001 0.004 0.014 0.034 0.002 0.23 <0.001 0.010 0.053
Network rate function
Network rate period 1 Network rate period 2 Network rate period 3 Delinquency linear Delinquency quadratic Sex (M) Average similarity	9.94 10.86 9.39 0.00 0.12 –0.19 6.08	2.12 2.00 1.49 0.27 0.16 0.42 3.06	1.00 0.48 0.66 0.047
Behavior rate function
Delinquency rate period 1 Delinquency rate period 2 Delinquency rate period 3	1.50 3.50 2.64	0.70 2.48 1.39

The p-values are based on approximate normal distributions of the t-ratios (estimate
divided by standard error).
6. Cross-sectional and longitudinal modeling
For a further understanding of this actor-based model, it may
be helpful to reflect about equilibrium and out-of-equilibrium
social systems. Equilibrium is understood here not as a fixed
state but as dynamic equilibrium, where changes continue but
may be regarded as stochastic fluctuations without a systematic
trend. This can be combined with discussing the relation between
cross-sectional and longitudinal statistical modeling of social
networks.
For cross-sectional modeling the exponential random graph
model (‘ERGM’), or p∗ model, is similar to the model of this paper
in its statistical approach to network structure (cf. Wasserman and
Pattison, 1996; Robins et al., 2007, and the references cited there).
A telling characteristic of the ERGM is that the only feasible way
to obtain random draws from such a probability distribution is to
simulate a process longitudinally until it may be assumed to have
reached dynamic equilibrium, and then take samples from the process (Snijders, 2002). Thus, the ERGM can be best understood as a
model of a process in equilibrium. If we would have longitudinal
data from a process in dynamic equilibrium, then modeling them
by the approach of this paper would give roughly the same results
as modeling its cross-sections by an ERGM. It would not be exactly
the same because the ERGM is not actor-based; a tie-based version
of our longitudinal model (Snijders, 2006) is possible, which does
correspond exactly to the ERGM. On the contrary, if we would have
cross-sectional data which may be assumed to have been observed
far from an equilibrium situation, then it is difficult to see what
would be the precise meaning of results of an ERGM analysis, based
as this is on an implicit equilibrium assumption.
If one has observed a longitudinal network data set of which the
consecutive cross-sections have similar descriptive properties – no
discernible trends or important fluctuations in average degree, in
proportion of reciprocated ties, in proportion of transitive closure
among all two-paths, etc. –, then it would be a mistake to infer
that the development is not subject to structural network tendencies just because the descriptive network indices are stationary. For
example, if the network shows a persisting high extent of transitive closure, in a process which is dynamic in the sense that quite
some ties are dissolved while other new ties appear, then it must
be concluded that the dynamics of the network contains an aspect
which sustains the observed extent of transitive closure against
the random influences which, without this aspect, would make the
transitive closure tend to attenuate and eventually to disappear.
The longitudinal actor-based model is more general than the
ERGM in that it does not require that the observed process be in
equilibrium. Given a sequence of consecutively observed networks,
if one were to make an analysis of the first one by an ERGM and of
the further development by an actor-based model, then in theory
it is possible to obtain opposite results for these two analyses, and
this would point toward a non-equilibrium situation. For example,
it would be possible that the first observed network shows no transitive closure at all, but the dynamics does show a transitivity effect;
then over time the extent of transitive closure would increase, perhaps to reach some dynamic equilibrium later on. Conversely, it is
possible that there is a strong transitivity effect at the first observation but no transitivity in the longitudinal model, which means that
the observed extent of transitive closure in repeated cross-sectional
analyses would eventually peter out to nil.
The advantage of longitudinal over cross-sectional modeling is
that the parameter estimates provide a model for the rules governing the dynamic change in the network, which often are better
reflections of social rules and regularities than what can be derived
from a single cross-sectional observation. This also is an argument
for not modeling the first observation but using it only as an initial condition for the network dynamics (as mentioned at the end
of Section 2.1). In many situations the first observation cannot be
regarded as coming from a process in equilibrium, and then it is
unclear what the first observation by itself can tell us about social
rules and regularities.
7. Discussion
This paper has given a tutorial introduction in the use of actorbased models for analyzing the dynamics of directed networks –
expressed by the usual format of a directed graph – and of the
joint interdependent dynamics of networks and behavior – where
‘behavior’ is an actor variable which may refer to behavior, attitudes, performance, etc., measured as an ordinal discrete variable.
The purpose of these models is to be used to test hypotheses concerning network dynamics and represent the strength of various
tendencies driving the dynamics by estimated parameters. To be
useful in this way for statistical inference, the models must be able
to give a good representation of the dependencies between network
ties, and between network positions and behavior of the actors.
The models also have to contain parameters that can express theoretical considerations about tendencies driving network change.
Further, the models must be flexible enough to represent several different explanations of change – which may be competing
but also potentially complementary – and test these against each
other, or controlling for each other. These goals are accomplished
by the stochastic actor-driven model in which the central object of
modeling is the objective function (1), (3), analogous to the linear
predictor in generalized linear modeling (e.g. Long, 1997).
These models can be estimated by software called Siena
(‘Simulation Investigation for Empirical Network Analysis’), of
which the manual is Snijders et al. (2008). Program and documentation can be downloaded free from the Siena web-page,
http://www.stats.ox.ac.uk/siena/. Some examples of applications
of this model are van de Bunt et al. (1999) and Burk et al. (2007).
Further applications of these models are presented in some of the
papers in this special issue.
58 T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60

These models are relatively new, and more complicated than many other statistical models to which social scientists are used. Applications are starting to be published, but more experience is needed to get a better understanding of their applicability and the interpretation of the results. Especially important will be the fur ther development of ways to assess the goodness of fit of these models and to diagnose what in the data-model combination may be mainly responsible for a possible lack of fit. Another issue is the assessment of the robustness of results with respect to misspecified	These are the same formulae as used in multinomial logistic regres
models, and the development of other models for network dynam ics which may serve as potential alternatives for cases where the models presented here do not fit to a satisfactory degree. A case in point would be the development of models permitting a more elaborate temporal dependence than Markovian dependence. All this should lead to better knowledge about how to fit longitudinal models to network data, and hence to more reliable results. In this tutorial we have tried to represent the current knowledge about the specification of longitudinal network models in a concise but more or less complete way, but we hope that this knowledge will expand rapidly. Various extensions of the model are the topic of recent and cur	(6)
sion.
A.2. Effects
Some formulae for effects sik(x) are as follows. Replacing an index by a + sign denotes summation over this index. Exogenous actor covariates are denoted by vi and dyadic covariates by wij.
Reciprocity j xijxji,
transitive triplets j,h
xihxijxjh,	(7)
transitive ties h
xihmaxj(xijxjh),	(8)
three-cycles j,h
xijxjhxhi,	(9)

rent research. Models for the dynamics of non-directed networks,
e.g., alliance networks, have been developed and were applied in
Checkley and Steglich (2007) and van de Bunt and Groenewegen
(2007). Extensions to valued ties are in preparation. Other estimation procedures have been proposed: Bayesian inference by
Koskinen and Snijders (2007) and Schweinberger (2007),Maximum
Likelihood estimation by Snijders et al. (submitted for publication).
All these developments will be tracked at the Siena website mentioned above.
Appendix A
This appendix contains some formulae to support the understanding of the verbal descriptions in the paper.
A.1. Objective function
When actor i has the opportunity to make a change, he/she can
choose between some set C of possible new states of the network.
Normally this will be set consisting of the current network and all
other networks where one outgoing tie variable of i is changed.
The probability of going to some new state x in this set is given
by
exp(fi(ˇ, x))
x∈C
exp(fi(ˇ, x))
. (4)
In words: the probability that an actor makes a specific change
is proportional to the exponential transformation of the objective
function of the new network, that would be obtained as the consequence of making this change. Similarly, when actor i can make a
change in the behavior variable z and the current value is z0, then
the possible new states are z0 – 1, z0, and z0 + 1 (unless the first or
last of these three falls outside the range of the behavior variable).
Denoting this allowed set also by C, the probability of going to some
new state z in this set is given by
exp(fiZ(ˇZ, x, z))
z∈C
exp(fiZ(ˇ, x, z))
. (5)
balance 1
n – 2
n j=1
xij
n h=1
h =/ i,j
(b0 – |xih – xjh|), (10)
where b0 is the mean of |xih – xjh|;
indegree popularity (sqrt)

j
j,	(11)
outdegree popularity (sqrt) j xijx
j+,	(12)
indegree activity (sqrt) x+ixi+, outdegree activity (sqrt) xi1+.5,	(13)
(14)

xijx+out – outdegree assortativity (sqrt)
j
xijxi+xj+ (15)
(other assortativity effects similar)
V-ego

j
xijvi,	(16)
V-alter xijvj,	(17)

j

V-similarity j xij(simij – sim), where sim ij = (1 – |vi – vj|/), with  = maxij|vi – vj|; same V xijI{vi = vj},	(18)
(19)

j
where I{vi = vj} = 1 if vi = vj, and 0 otherwise;
V-ego × alter
j

xijvivj, dyadic covariate W j xijwij, dyadic cov. W × reciprocity j	(20)
(21)

xijxjiwij, (22)
T.A.B. Snijders et al. / Social Networks 32 (2010) 44–60 59

actor cov. V × transitive triplets vi j,h xijxjhxih, Some formulae for behavior effects sZ ik(x, z) are the following: linear shape zI, quadratic shape zi2, outdegree zixi+, indegree zix+i, average similarity xi-+1 j xij(simz ij – simz), where simz ij = (1 – |zi – zj|/Z) with Z = maxij|zi – zj|;	(23)
(24)
(25) (26) (27) (28)

total similarity
j
xij(simz ij – simz), (29)
average alter zi ⎛ ⎝j xijzj⎞ ⎠⎛ ⎝j xij⎞ ⎠, (30)
main effect covariate V zivi. (31)
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