Bodong Chen

Crisscross Landscapes

Notes: Snijders2010-ua-Introduction to stochastic actor-based models for network dynamics

2017-09-01


References

Citekey: @Snijders2010-ua

Snijders, T. A. B., van de Bunt, G. G., & Steglich, C. E. G. (2010). Introduction to stochastic actor-based models for network dynamics. Social Networks, 32(1), 44–60. https://doi.org/10.1016/j.socnet.2009.02.004

Notes

Summarize: A such nicely written intro to SIENA. Should be read for more than once.

Assess:

Reflect:

Highlights

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 net- work 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 theo- retically derived and/or empirically established in earlier research, and which may well operate simultaneously. (p. 1)

Social networks are dynamic by nature. (p. 1)

Some examples of such tendencies are reciprocity, transitivity, homophily, and assor- tativematching,aswillbeelaboratedbelow.Inthisway,themodels shouldbeabletogiveagoodrepresentationofthestochasticdepen- dence 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’). (p. 1)

butlackanexplicitesti- mation theory. (p. 2)

  1. Model assumptions (p. 2)

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. (p. 2)

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. (p. 2)

Given that the network ties under study denote states, it is fur- ther 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 proba- bilistically 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. (p. 2)

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. (p. 2)

Distinguishing characteristics of stochastic actor-based models are flexibility, allowing to incorporate a wide variety of actor-driven micro-mechanisms influencing tie formation (p. 2)

micro-mechanism is a key idea here. Does the current literature connect it to self-oraganization? (p. 2)

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 (p. 2)

We assume here that the empirical data consist of two, but preferably more, repeated observations of a social net- work on a given set of actors; one could call this network panel data. (p. 2)

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. (p. 2)

important assumptions discussed here (p. 2)

2.1. Basic assumptions (p. 2)

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. (p. 2)

The heart of the model is the so-called objective function, which deter- mines probabilistically the tie changes made by the actors. (p. 2)

  1. The underlying time parameter t is continuous, i.e., the process unfoldsintimestepsofvaryinglength,whichcouldbearbitrarily 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 panelwaves’,analogoustopanelsurveysinnon-networkstudies. (p. 2)

Some of the poten- tial 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. (p. 2)

The continuous-time assumption allows to represent dependencies between network tiesastheresultofprocesseswhereonetieisformedasareaction (p. 2)

to the existence of other ties. (p. 3)

  1. 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. (p. 3)

The actor-based network change process is decomposed into two sub-processes, both of which are stochastic. (p. 3)

The assumption of a Markov process has been made in practi- callyallmodelsforsocialnetworkdynamics,startingby Katzand Proctor’s (1959) discrete Markov chain model. (p. 3)

  1. 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). (p. 3)

We could say that this assumption is a lens through which we look at the data—it should help but it also may distort. (p. 3)

  1. 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 net- work positions, as well as covariates, of ego and the other actors (‘alters’) in the network. This is explained below. (p. 3)

The actor-based model can be regarded as an agent-based sim- ulation 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 infer- ence,whichleadstorequirementsofflexibility(enoughparameters 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 sub- servient to others’ interests in any way. (p. 3)

interesting.. distinctions between agent and actor. also good to know these two clusters share similar traits/interests. (p. 3)

  1. 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’. T (p. 3)

he methodological approach of structural individualism (Udehn, 2002; Hedström, 2005), where actors are assumed to be purposeful and to behave subject to structural constraints. (p. 3)

It is assumed formally that actors have full infor- mation about the network and the other actors. (p. 3)

This procedure uses the basic principle that the first observed net- work 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. (p. 3)

  1. 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— (p. 3)

principledecom- poses 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. (p. 3)

2.2. Change determination model (p. 3)

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 (p. 3)

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 sta- tus 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. (p. 3)

This assumption excludes relational dynamics where some kind of coordination or negotiation is essential for the cre- ation of a tie (p. 3)

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 progres- sively 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 lim- itations 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 endoge- nousnetworkeffects. (p. 4)

Thus, the proba- bilities of changes are assumed to depend on the personal networks thatwouldresultfromthechangesthatpossiblycouldbemade,and their composition in terms of covariates, via the objective function values of those networks. (p. 4)

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. (p. 4)

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 networktie),followsthelogicofdiscretechoicemodels (McFadden, 1973; Maddala, 1983) which have been developed for modeling sit- uations where the dependent variable is a choice made from a finite set of actions. (p. 4)

i am thinking a lot about a student’s ‘objectives’ when choosing to read, write, and respond to something in KF. The fundamental theoretical alignment is there but needs to be carefully unpacked. (p. 4)

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. (p. 4)

2.3. Specification of the objective function (p. 4)

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. (p. 4)

2.3.1. Basic effects (p. 4)

The most basic effect is defined by the outdegree of actor i, and this will be included in all models. (p. 4)

Like in generalized linear statistical models, the objective func- tion is assumed to be a linear combination of a set of components called effects, (p. 4)

Another quite basic effect is the tendency toward reciprocity, represented by the number of reciprocated ties of actor i. (p. 4)

the functions s ki(x) are the effects, functions of the network thatarechosenbasedontheoryandsubject-matterknowledge,and correspond to the ‘tendencies’ mentioned in the introductory sec- tion (p. 4)

the weights ˇ k are the statistical parameters. (p. 4)

2.3.2. Transitivity and other triadic effects (p. 4)

Next to reciprocity, an essential feature in most social networks isthetendencytowardtransitivity,ortransitiveclosure(sometimes called clustering): friends of friends become friends, or in graph- theoretic terminology: two-paths tend to be, or to become, closed (p. 4)

The transitive triplets effect measures transitivity for an actor i by counting the number of pairs j, h such that there is the transitive (p. 4)

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. (p. 5)

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 prefer- ential attachment by Barabási and Albert (1999) in their models for dynamics of non-directed networks: a self-reinforcing process of degree differentiation. (p. 5)

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. (p. 5)

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. (p. 5)

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 inde- gree activity. (p. 5)

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. (p. 5)

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 (p. 5)

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 three- cycles, which can be represented here by a negative parameter ˇ k 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. (p. 5)

he four effects described as degree-related popularity and activity are more basic than the assortativityeffects(cf.therelationbetweenmaineffectsandinter- actions 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. (p. 5)

2.3.3. Degree-related effects (p. 5)

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. (p. 5)

2.3.4. Covariates: exogenous effects (p. 5)

For an actor variable V, there are three basic effects: the ego effect, measuring whether actors with higher Vvalues tend to nom- inate more friends and hence have a higher outdegree (p. 5)

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. (p. 5)

the alter effect, measuring whether actors with higher V values will tend to be nominated by more others and hence have higher indegrees (p. 5)

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 (p. 5)

Another pair of effects is degree-related activity for i ndegree or outdegree: when these effects are positive, nodes with higher inde- gree,orhigheroutdegreerespectively,willhaveanextrapropensity to form ties to others. (p. 5)

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 num- ber of observation moments should be at least 2, and is usually much less than 10. (p. 6)

instead of the similarity effect one could use the ego–alter interaction effect (p. 6)

There are no objections in principle against ana- lyzing 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). (p. 6)

2.3.5. Interactions (p. 6)

Like in other statistical models, interactions can be important to express theoretically interesting hypotheses. The diversity of func- tions that could be used as effects makes it difficult to give general expressions for interactions. (p. 6)

AnotherexampleisgivenbydeFederico(2004) asaninteraction of a covariate with reciprocity. In her analysis of a friendship net- work between exchange students, she found a negative interaction between reciprocity and having the same nationality. (p. 6)

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. (p. 6)

The total number of changes between consecutive observa- tions should be large enough, because these changes provide the information for estimating the parameters. A total of 40 changes (cumulatedoverallsuccessivepanelwaves)isonthelowside.More changes will give more information and, thereby, allow more com- plicatedmodelstobefitted.Betweenanypairofconsecutivewaves, 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. (p. 6)

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 (p. 6)

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 exam- ple, girls might have a greater tendency toward transitive closure than boys. (p. 6)

interesting. gender could also be considered in CSCL data. (p. 6)

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 (p. 6)

  1. Issues arising in statistical modeling (p. 6)

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. (p. 6)

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. (p. 6)

important to note (p. 6)

3.1. Data requirements (p. 6)

the data should contain enough informa- tion. (p. 6)

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 (N 11/ (N 10 + N 11) in the preceding notation). Proportions higher (p. 6)

than 0.6 are preferable, between 0.3 and 0.6 would be low but may still be acceptable. (p. 7)

  1. 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. (p. 7)

The methods require in principle that network data are com- plete. However, it is allowed that some of the actors enter the network after the start, or leave before the end of the panel waves (p. 7)

Another optiontorepresentthatsomeactorsarenotyet,ornomore,present in the network, is to specify that certain ties cannot exist (‘struc- tural 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 struc- tural 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. (p. 7)

will this allow multidimensional network as well? (p. 7)

3.2. Testing and model selection (p. 7)

It turns out (supported by computer simulations) that the dis- tributions of the estimates of the parameters ˇ k 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 (p. 7)

estimation round. (p. 7)

  1. 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. (p. 7)

For actor-based models for network dynamics, information- theoretic 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). (p. 7)

  1. 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. (p. 7)

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. (p. 7)

  1. 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. (p. 7)

  2. Fitting complicated models may be time-consuming and lead to instabilityofthealgorithm,andaresultingfailuretoobtaingood estimates.Therefore,forwardselectionistechnicallyeasierthan backward selection, which is unfortunately at variance with the preceding remark. (p. 7)

  3. 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 triad- level effects. It is advisable, at some moment during the model selection process, to check these effects; note that the square- root specification usually works best. (p. 7)

  4. The estimation algorithm (Snijders, 2001) is iterative, and the initial value can determine whether or not the algorithm con- verges. (p. 7)

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. (p. 7)

  1. 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. (p. 7)

  2. The model assumes that the ‘rules for network change’ are the same for all actors, except for differences implied by covari- ates or network position. (p. 8)

As exogenous control variables, we include sender and receiver effects of sex, and a dyadic covariate indicating friendship in pri- mary school reflecting relationship history. In addition, several degree-related endogenous effects are included as control effects: in- and outdegree-related popularity, and outdegree-related activ- ity, explained above. Estimates for this model are given in Table 1 as Model0.AllcalculationsweredoneusingSienaversion3.2 (Snijders et al., 2008). (p. 8)

This analysis confirms, for this data set, several of the known properties of friendship networks: there is a high degree of reci- procity, 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 hav- ing been friends at primary school already, and there is evidence for transitive closure, as seen in the significant effects of transi- tive triplets and transitive ties. (p. 8)

3.3. Example: friendship dynamics (p. 8)

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 (p. 8)

Other significant effectsarethenegative 3-cyclesparameter,whichindicatesthatthe 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. (p. 8)

the marginally significant negative effect of the outdegree-relatedpopularitywhichindicatesthatactivepupils,i.e., those who nominate particularly many friends, are less likely to be chosen as friends – this could be a status effect negatively associ- ated with nomination activity. (p. 8)

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. (p. 8)

Also significant is the sender effect of sex (sex (M) ego), which in our coding of the variable means that theboystendtobemoreactiveintheclassroomfriendshipnetwork than the girls. (p. 8)

Some data were missing due to absence of pupils at the moment of data collection. This was treated by ad-hoc model-based imputa- tion using the procedure explained in Huisman and Steglich (2008). (p. 8)

In a subsequent model (Model 1 in Table 1), more parsimony is obtained by eliminating the non-significant effects in a backward selection procedure. (p. 8)

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 recip- rocal, shows tendencies towards network closure, and in this age group is strongly segregated according to the sexes. (p. 8)

3.4. Parameter interpretation (p. 9)

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 feel- ing 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 dis- turbances are added to the values of the objective function with standard deviations equal2 to 1.28. (p. 9)

Thistableshowsthatgirlsaswellasboyspreferfriendshipswith same-sex alters, but for boys the difference is more pronounced than for girls. (p. 9)

  1. More complicated models (p. 9)

4.1. Differential rates of change: the rate function (p. 10)

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. (p. 10)

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. (p. 10)

To interpret the parameter values, one should know that a so- called 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. (p. 10)

4.2. Differences between creating and terminating ties: the endowment function (p. 10)

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 termina- tion of ties and not for their creation. (p. 10)

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 droppedinsizeandsignificance.Tointerpretthisresult,jointlycon- sider the reciprocity evaluation effect with parameter 0.71 and the reciprocity endowment effect with parameter 1.42. The contribu- tion 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. (p. 10)

  1. Dynamics of networks and behavior (p. 11)

I plan to revisit this section later when getting more familiar with SIENA. (p. 11)

Social networks are so important also because they are rele- vant 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 andoftheactorvariables,butlikewisetheactorvariablesarechang- ing as a function of themselves and of the network. (p. 11)

The changes in behavior depend on an objective function similar to the objective function for network changes. However, this func- tion will be different because it needs to represent primarily the actor’s behavior rather than his/her network position, and because choicesofbehaviorchangesmaybeframeddifferentlyfromchoices of tie changes, depending on different goals and restrictions. (p. 11)

We use the term behavior as shorthand for endogenously changing actor vari- ables, 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. (p. 11)

The depen- dence 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 network- behavior configuration will be called the social influence process. (p. 11)

terms: social selection, social influence. (p. 11)

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 (p. 11)

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 (p. 11)

The longitudinal actor-based model is more general than the ERGM in that it does not require that the observed process be in equilibrium. (p. 14)

  1. Cross-sectional and longitudinal modeling (p. 14)

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. (p. 14)

  1. Discussion (p. 14)

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 pro- cess (Snijders, 2002). Thus, the ERGM can be best understood as a model of a process in equilibrium. (p. 14)

The purpose of these models is to be used to test hypotheses con- cerning 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 togiveagoodrepresentationofthedependenciesbetweennetwork ties, and between network positions and behavior of the actors. (p. 14)

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). (p. 14)

Further applications of these models are presented in some of the papers in this special issue. (p. 14)

check out the special issue (p. 14)

These models are relatively new, and more complicated than many other statistical models to which social scientists are used. (p. 15)