The model is completely described by probabilities of changes of single tie, depending on the state of. The probability that the next opportunity for change is for actor i is given by P{Next opportunity for change is for actor i}= λi(x;ρ). 2) This formula corresponds to a 'first past the post' model, where all actors have stochastic waiting times as in (5.), the first gets the opportunity to make a change, and then everything starts over but in a new state .
Specification of the Actor-based Model
Therefore, the role of the graduation effect in the model is the contribution of 2β1 in favor of tie creation vs. Theoretical arguments for this effect were formulated by Simmel (1950), who discussed the consequences of triadic embeddedness on the bargaining power of the social actors. about the possibilities of conflicts. They reflect access to other actors and are often quite directly linked to opportunities as well as costs of the network position of the actors.
Levels can be indicators of influence potential, success (de Solla Price, 1976), prestige (Hafner-Burton and Montgomery, 2006), search potential (Scholz et al., 2008), etc., depending on the context. Accordingly, the probabilities of tie-making and tie-breaking may depend on the degree of actors involved. In addition to these effects based on the network structure itself, research questions will naturally lead to effects that depend on the attributes of the actors – indicators of goals, constraints and resources, etc., defined outside the network.
Here the word "ego" is used for the central actor or sender of the tie; while 'alter' is used for a potential candidate to receive a draw.
Parameter Estimation
Many other effects that can be used for model specification are mentioned in the RSiena manual (Ripley et al., 2016). Sometimes this is at given intervals (eg annually), sometimes at irregular times depending on what is convenient for data collection. The definition of the model implies that irregular observation times pose no problem at all; these will be absorbed in panel wave specific parametersρm (see (6)) without affecting the other parameters.
For panel data, this is a generalized linear model with a lot of missing data (ie, unobserved time of linkage changes). The method simulates the network dynamics many times with test parameter values, updating them, until the averages of a suitable set of network descriptors, reflecting the estimated parameters, are close enough to the observed values. This method may be called an MCMC method, but it is frequentist in nature, not Bayesian, and thus does not require the specification of a prior distribution.
Recently, other and potentially more efficient estimators were developed: a Bayesian estimator by Koskinen and Snijders (2007), a Maximum Likelihood estimator by Snijders et al. 2010a), and a generalized method of moments estimator by Amati et al.
Changing actor sets
There the two actors on either side of the tie a priori have a say in its existence, and assumptions must be made about the negotiation or coordination between the two actors involved in creating and ending the tie. In game-theoretic network models, it is usually assumed that for a tie to exist, the consent of both actors is involved. This is the basis of the definition of pairwise stability proposed by Jackson and Wolinsky (1996): a network is pairwise stable if no pair of actors can benefit from the creation of a new link between them and if no single actor may benefit from termination. of one of the relationships in which this actor is involved.
In our statistical approach, such a stability concept has no place, but the basic idea that both actors should benefit from the tie is translated to our probabilistic framework. Several models are presented here, all based on a two-step process of opportunity and choice, and which make different assumptions about the combination of choices between the two actors involved in a tie.
Two-sided Choices
Mutual: Both actors must agree that a relationship exists between them, according to Jackson and Wolinsky (1996). Compensator: The two actors decide on the basis of their combined objective function, which may represent reaching a mutual agreement. The combination with the unilateral initiative here seems somewhat artificial and we elaborate this option only for the bilateral initiative.
Model M.1, unilateral initiative with mutual confirmation, is in most cases the most attractive simple representation of the coordination required to create and maintain undirected ties. Models D.1 and D.2 have potential for modeling military conflicts, and model C.2 may be useful in a joint negotiation situation.
Mathematical elaboration
The probability that the next chance for change is for the pair (i, j) is given by. 8) The selection process has the three options D (ictatorial), M (utual) and. As in the directed case, actor i chooses to change the single binding variableXij given the objective function fi(x(0), x;β) using (3), and actor j simply has to accept. For the one-sided initiative, the probability that the link variable is changed is Xij, so that the network x changes to x(±ij), is given by.
In the case of a unilateral initiative, actor i selects the tie variable to be changed with probabilities (9) according to i's objective function. The combination with one-sided initiative is somewhat artificial here and we only develop this option for the two-sided initiative. The binary decision on the existence of the tie i↔j is based on the sum of the objective functions of actors i and j.
The probability that networkx changes to x(±ij) is now given by . 13) Combining this into the transition stages defined above gives the following results.
Model specification for undirected networks
It should be noted that switching the variable Xij is the same as switching Xji and that the rules described above give different roles for the first and second actor in the pair (i, j). If all ski effects included in model (7) are such that the tie contributions are the same for both participating actors (which is the case, for example, for the degree effect s1i and the similarity effect s12i), then the compensatory dyadic model C.2 is identical to the dictatorial dyadic to the D.2 model, except that it is For general models, this identity will not hold, but in a first-order approximation, the parameters βk in model C.2 can still be expected to be about twice as small as those in D.2, and ρm.
Estimation and Examples
For general models this identity will not hold, but in a first-order approximation it can still be expected that the βk parameters in model C.2 are about twice as small as those in D.2, and ρm. the network of interaction partners. Studying the intertwining of networks and outcomes at the actor level is difficult due to the endogeneity of both: the network affects the outcomes, while the outcomes affect the network. One way to get a handle on this is to model these dynamic dependencies both ways in studies of the co-evolution of network and nodal attributes.
A method for modeling this, using grid panel data and attributes, was proposed by Steglich et al. 2010), using an elaboration of the Stochastic Actor Oriented Model. This methodology does not claim to provide causal conclusions based on statistical analysis; What the method provides, if applied with a well-defined model, is an insight into the temporal sequence: to what extent there is evidence that changes in attributes depend on the state of the network ('first the network, then changes in attributes'); and to what extent there is evidence that network changes depend on the state of attributes ('attributes first, then network changes').
A more extensive discussion of the serious problems in attempting to establish causality in social network research is presented in Robins (2015, Chapter 10).
Dynamics of Networks and Behavior
The behavior objective function fiZh(z(0), z;x, βZh) for the behavior variable Zh determines the probability of subsequent behavior change by actor i,. Here the option set for the change decision is different, however, in the following way. For notational simplicity, we give the formulas only for the case of H= 1 dependent variable, omitting the index h.
In a process driven by rate functions λZi (x, z;ρZ), player i is given the opportunity to change the value of his behavior Zi at stochastic moments. If the current value is at the minimum or maximum range, one of these options is disabled. The choice probabilities again have a multinomial logit form, where the choice probability is z (with allowed values z(0)−1, z(0), z(0)+ 1).
Specification of Behavior Dynamics
The behavioral dynamics can also depend directly on the network position, for example on the degrees of the actor. The three main approaches currently available for statistically analyzing network dynamics – which in addition to the SAOM are temporal Exponential Random Graph Models ('ERGMs') and latent Euclidean space models (for an overview see Desmarais and Cranmer, 2016) – use very different resources to express network and time dependencies, that is, the correlation structure between the relationship variables observed at the same and at different times. However, the simulation setup is limited in the sense that observed changes are considered as the result of a series of changes to individual connections, each implying a – usually small – change in the network context for all actors.
It will be interesting to compare the interpretive value of the spatial representation in latent space models with the value of the dependence. It should be noted that covariates 'remove' some of the dependency, leaving the 'rest'. This means that the network consists of the pattern of ties between all actors in a well-defined group, and that ties of these actors to others outside the group may be ignored: the network boundary problem (Marsden, 2005) is assumed to have been solved in a previous phase of the research.
This means, in terms of Huckfeldt's (2009, p. 928) statement that "(p)olitical communication networks are created as a complex product of this intersection between human choices and the possibilities imposed by the environment", that the methods discussed here focus on the component 'human choice', while the determination of a set of nodes is considered 'environmentally imposed'. Defining SAOM in terms of the choices of individual actors means that changing dyadic and monadic variables can be analyzed coherently. The interpretation of the multinomial logistic model in terms of myopic choices does not exclude strategic considerations, but means that these must be represented by short-term goals through which actors try to achieve their long-term goals.
