3.1 Conceptualizing Roles in Longitudinal Networks

Within cross-sectional networks, roles are treated as static. In the aforementioned application presented by Box-Steffensmeier et al. (Reference Box-Steffensmeier, Campbell, Christenson and Navabi2018), the temporal dynamics informing role assignment in the environmental lobbying coalition are not considered. This does not imply such dynamics do not exist, however. Their theory argues that to overcome the collective action problem associated with political lobbying, interest groups adopt roles to establish behavioral expectations. These roles may become consolidated with time, or vary significantly from year to year as salient issues evolve or group resources change.

One might be interested in two essential temporal dynamics associated with roles: stability and change.Footnote ⁴ While these two dynamics are different sides of the same coin, they must be considered as analytically distinct from a modeling perspective. Role stability refers to the process wherein the roles adopted within a network are static and do not vary over time. This might be a particular phenomenon associated with the roles of a network, or a modeling assumption. Consider the network of military alliances during the Cold War. America’s role as Chief Balancer is fairly stable and unlikely to have changed from the late 1940s to the late 1980s. Role stability may also be a modeling assumption akin to the implicit assumption of a time-invariant data generating process invoked by many statistical models that pool observations over time. The ego-TERGM presented in the following section is a pooled model that invokes this assumption, positing that roles are assumed to be stable and time-invariant within a clearly defined temporal domain. Role stability may also be invoked to produce parsimonious role assignments in a network with little autoregressive tendencies.

Role change refers to a process wherein the roles actors adopt within a network vary with time. Similar to role stability, role change may be a phenomenon associated with the roles of a network, or an explicit modeling assumption. The interstate conflict network is fairly dynamic, experiencing significant variation from year to year. A highly central node that may be a Pariah and involved in many conflicts in one year may not be a Pariah in the following year. In 1914 and 1915, Germany might have been considered a Pariah, only to enter the “Golden Era” and join many any countries less than a decade later as an Isolate in the conflict network. Role change may also be a modeling assumption, where a model explicitly attempts to account for the variation of role assignments from period to period.Footnote ⁵

3.2 Extant Strategies

While techniques to explicitly infer roles from longitudinally observed networks are yet to be developed, there exist general techniques for cross-sectional networks that have been repeated for each period in a time series of networks. Those typically interested in inferring roles from longitudinal networks have calculated structural or regular equivalency scores for nodes at every time step. Structural equivalence refers to whether or not two actors occupy the same structural position within a network, or in other words, whether two nodes are connected to the same set of alters (Lorrain and White Reference Lorrain and White1971). If actors are structurally equivalent, then they are thought to occupy the same role. A simple example is shown in Figure 1(a), where four structurally equivalent sets exist: $\{1\},\{2\},\{3,4\}$ , and $\{5,6,7,8,9\}$ . In one featured application, Maoz et al. (Reference Maoz, Kuperman, Terris and Talmud2006) argue that actors occupying the same role or structural position should share an affinity toward one another, and as such, structurally equivalent states in international networks may be less likely to engage in conflict.

Figure 1.

Structural and Regular Equivalence. Equivalent sets are share the same color.

An alternative to structural equivalence is regular equivalence, a method of graph partitioning based upon general patterns of connectivity (Borgatti and Everett Reference Borgatti and Everett1992; Everett and Borgatti Reference Everett and Borgatti1994). Formally, actors that have similar relations to members of other regular equivalent sets are said to be in the same set. This is a fairly intuitive concept—assume that there exist three colors of nodes: black nodes are connected to gray nodes, gray nodes are connected to black and white nodes, and white nodes are only connected to gray nodes. In this particular case, all black nodes are regularly equivalent, as are gray and white nodes. This is illustrated through the simple example in Figure 1(b), where the regularly equivalent sets are $\{1\},\{2,3,4\}$ , and $\{5,6,7,8,9\}$ . Similar to Maoz et al. (Reference Maoz, Kuperman, Terris and Talmud2006), Braithwaite et al. (Reference Braithwaite, Dasandi and Hudson2016) compute regular equivalence scores for countries within the trade network to detect the roles states adopt in the core–periphery structure of the international economic network. Using these positions, Braithwaite et al. (Reference Braithwaite, Dasandi and Hudson2016) find that the roles adopted within this core–periphery structure predict whether a state is more or less likely to experience civil conflict.

The aforementioned strategies are not without their limitations. First, and foremost, allowing roles to vary every year may make interpretation difficult. This is particularly true in highly dynamic networks where an actor’s position in the network changes frequently. Second, in the previous models, only network position informs role assignment. This is problematic as both endogenous topological features and exogenous node or dyad characteristics in a node’s ego-network may inform role assignments. The ego-TERGM provides a flexible modeling approach to detect time-invariant roles conditional upon a variety of network statistics, including both endogenous dependencies and exogenous covariates.

4.1 Deriving the Ego-TERGM

The ego-TERGM is based upon a similar model derived for cross-sectional networks by Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) and introduced by Box-Steffensmeier et al. (Reference Box-Steffensmeier, Campbell, Christenson and Navabi2018), the ego-ERGM. The ego-ERGM, as a finite mixture model, assumes that each cross-sectional ego-network has probabilities associated with belonging to a user-specified and finite number of clusters, or roles, $G$ . For each possible role $g\in G$ there is some probability that the node belongs to that particular role. For all nodes, the naive probability of belonging to $g$ , $\unicode[STIX]{x1D70F}_{g}$ , sums to one across all $g\in G$ , creating a vector $\text{}\underline{\unicode[STIX]{x1D70F}}$ .Footnote ⁷ These probabilities are conditional upon a vector of role specific parameters, $\text{}\underline{\unicode[STIX]{x1D703}}_{g}$ , modeled and estimated through an ERGM. Note that $\boldsymbol{\unicode[STIX]{x1D703}}$ refers to the $G\times H$ matrix of model parameters where $H$ is equal to the number of terms specified and $\text{}\underline{\unicode[STIX]{x1D703}}_{g}$ refers to the $g^{th}$ row of matrix $\boldsymbol{\unicode[STIX]{x1D703}}$ . Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) model the prior mixture of data generating process as the probability of an ego-network $\mathbf{Y}_{i}$ :

(1) $$\begin{eqnarray}{\mathcal{P}}(\mathbf{Y}_{i}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}})=\mathop{\sum }_{g=1}^{G}\unicode[STIX]{x1D70F}_{g}\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{i})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}.\end{eqnarray}$$

In Equation (1), $h(\mathbf{Y}_{i})$ refers to a series of endogenous and/or exogenous statistics computed on the network and $\text{}\underline{\unicode[STIX]{x1D703}}_{g}$ refers to the vector of model parameters for group $g$ . Those familiar with ERG-family models will acknowledge that the largest headache for using these models is the normalizing constant, $\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})$ , which refers to the sum of all possible permutations of the network $\mathbf{Y}_{i}$ where the number of nodes remains constant and the number of ties varies from a perfectly empty network to a perfectly full network. This normalizing constant, necessary to calculate the probability of the observed network conditional on model estimates, becomes computationally intractable as the number of nodes in a network increases. As such, it is typically approximated through maximum pseudolikelihood estimation (MPLE) or Markov chain maximum likelihood estimation (MCMLE).Footnote ⁸

The mixture model for a cross-sectional network can be extended to a single time slice in a longitudinally observed network by conditioning on $q^{th}$ order autoregression. This produces the probability of observing a particular ego-network $\mathbf{Y}_{i}$ at time $t$ as a function of the mixing proportions for all clusters $\text{}\underline{\unicode[STIX]{x1D70F}}$ , model-based parameters of $\boldsymbol{\unicode[STIX]{x1D703}}$ and the prior $q$ observations of that particular ego-network:

(2) $$\begin{eqnarray}{\mathcal{P}}(\mathbf{Y}_{i,\,t}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{it-q})=\mathop{\sum }_{g=1}^{G}\unicode[STIX]{x1D70F}_{g}\,\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\mathbf{Y}_{it-1},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}.\end{eqnarray}$$

This mixture model in Equation (2) sums over the ERGM for all groups $g\in G$ , where the probability of the network for group $g$ is a function of group-level parameters $\text{}\underline{\unicode[STIX]{x1D703}}_{g}$ , a series of statistics computed on the ego-network, and previous iterations of the ego-network normalized by $\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})$ . From this conditional probability, the likelihood of observing a time slice of the broader network, $\mathbf{Y}_{t}$ , is derived by making the assumption that each ego-network is independently distributed and then taking the product of each ego-network’s mixture model from Equation (2):Footnote ⁹

(3) $$\begin{eqnarray}{\mathcal{P}}(\mathbf{Y}_{t}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{t-q})=\mathop{\prod }_{i=1}^{N}\left[\mathop{\sum }_{g=1}^{G}\unicode[STIX]{x1D70F}_{g}\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\mathbf{Y}_{it-1},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}\right].\end{eqnarray}$$

This likelihood is then pooled over all time periods $t\in T$ by taking the product of each time-slice likelihood from Equation (3) to produce the likelihood of observing the full time series of networks $\mathbf{Y}_{T}$ . To produce independence across time periods, one need only condition upon a $q^{th}$  order autoregressive process (Cranmer and Desmarais Reference Cranmer and Desmarais2011; Desmarais and Cranmer Reference Desmarais and Cranmer2012):

(4) $$\begin{eqnarray}\displaystyle {\mathcal{P}}(\mathbf{Y}_{T}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{t-q})=\mathop{\prod }_{i=1}^{N}\left[\mathop{\sum }_{g=1}^{G}\mathop{\prod }_{t=1}^{T}\unicode[STIX]{x1D70F}_{g}\,\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\mathbf{Y}_{it-1},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}\right]. & & \displaystyle\end{eqnarray}$$

Note that in Equations (3) and (4) the $i$ subscript on the left-hand side is removed to aggregate this up to the full network observed at a single time slice in Equation (3) and the full time series of networks in Equation (4). While this model is pooled, it also accounts for $q^{th}$ order autoregression for each network observed while iteratively incrementing $t$ for each time step. To reiterate, when moving from Equation (3) to Equation (4), the subscript $t$ becomes $T$ as the probability of all $\mathbf{Y}_{t}\in \mathbf{Y}_{T}$ is modeled in Equation (4).

As previously noted, there are two forms of independence that must be considered: the independence of ego-networks and the independence of time slices. First, it seems unclear whether each ego-network is independent as they are, almost by definition, overlapping to some degree. The inclusion of a node $j$ in the undirected ego-network of node $i$ necessarily implies that node $i$ is included in the undirected ego-network of node $j$ . Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) note this problem in their derivation of the cross-sectional ego-ERGM, noting that the model is more of a pseudolikelihood model than a true likelihood model. The consequence of violating this independence assumption, as noted by Brandes and Lerner (Reference Brandes and Lerner2007), Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015), and Box-Steffensmeier et al. (Reference Box-Steffensmeier, Campbell, Christenson and Navabi2018), is that these nodes $i$ and $j$ are more likely to be assigned to the same cluster. In more general terms, these overlapping and adjacent ego-networks will lead ERG-based mixture models like the ego-ERGM and ego-TERGM to cluster proximal nodes at higher rates than distal nodes (Salter-Townshend and Murphy Reference Salter-Townshend and Murphy2015, 525). This phenomenon is a well documented feature, not a bug, of many other node clustering methods which do not sort nodes in dense subgraphs (Brandes and Lerner Reference Brandes and Lerner2007, 12). In many cases, this particular consequence may not be problematic if analysts expect structurally similar nodes to adopt the same role (Box-Steffensmeier et al. Reference Box-Steffensmeier, Campbell, Christenson and Navabi2018, 220).

Consider a simple four node network illustrated in Figure 2 that contains nodes $h$ , $i$ , $j$ , and $k$ where $k$ is of some arbitrary type (partisan identification, etc.) that is distinct from other nodes in the network. A simple homophily model would assign nodes $h$ , $i$ , and $j$ all to the same role as their presence in each other’s ego-network necessarily implies that they each have the same covariate portfolio: three homophilous edges. In certain applications this implication may be more problematic than in others, any users of this technique must consider a violation of this assumption accordingly.

Figure 2.

Simple example of overlapping ego-centric networks. Nodes are colored by some arbitrary type.

Second, by taking the product across all time slices, as done in Equation (4), one is assuming that conditioned upon $q^{th}$ order autoregression, each temporally observed ego-network is independent (Cranmer and Desmarais Reference Cranmer and Desmarais2011; Desmarais and Cranmer Reference Desmarais and Cranmer2012).Footnote ¹⁰ One is certainly not required to treat temporal dependence as a nuisance. The roles actors adopt may be informed by temporal dynamics associated with their ego-network such as a tendency toward tie stability, creation, or loss. Given the outstanding concern of potential dependence across ego-networks, one should view the likelihood previously presented in the vein of a pseudolikelihood similar to Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015).

An expectation–maximization (EM) algorithm is then used to find maximum likelihood estimates for $\boldsymbol{\unicode[STIX]{x1D703}}$ as role assignments are unknown and missing parameters. These role assignments are captured through an unobserved parameter vector $\text{}\underline{Z}_{i}=(Z_{i1},Z_{i2},\ldots ,Z_{iG})$ where $Z_{ig}$ is the probability that node $i$ belongs to role $g$ . $\mathbf{Z}$ is an $N\times G$ matrix of latent and unobserved cluster assignment probabilities comprised row-wise by all vectors $\text{}\underline{Z}_{i}$ . It is noted here that these indicators are not subscripted by time, and thus, are assumed to reflect a process where a node maintains the same role $g$ over all time periods. As the number of possible roles $G$ is time-invariant, this is a pooled version of the model.Footnote ¹¹ When this indicator is treated as missing data, the complete data log-likelihood for a particular time slice of the longitudinal network $\mathbf{Y}_{t}$ is derived as follows:

(5) $$\begin{eqnarray}{\mathcal{P}}(\mathbf{Y}_{t},\mathbf{Z}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{t-q})=\mathop{\prod }_{i=1}^{N}\mathop{\prod }_{g=1}^{G}[\unicode[STIX]{x1D70F}_{g}\,\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\mathbf{Y}_{it-1},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}]^{Z_{ig}}.\end{eqnarray}$$

For the complete pooled model, one must simply take the product across all iterations of the network across time to get the ego-TERGM likelihood. To reiterate, in this case, $\mathbf{Y}_{T}$ refers to the time series of networks, producing the complete data likelihood for the pooled model:

(6) $$\begin{eqnarray}{\mathcal{P}}(\mathbf{Y}_{T},\mathbf{Z}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{t-q})=\mathop{\prod }_{i=1}^{N}\mathop{\prod }_{g=1}^{G}\mathop{\prod }_{t=1}^{T}[\unicode[STIX]{x1D70F}_{g}\,\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\mathbf{Y}_{it-1},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}]^{Z_{ig}}.\end{eqnarray}$$

From this, the log-likelihood can be derived in Equation (7):

(7) $$\begin{eqnarray}\displaystyle \log [{\mathcal{P}}(\mathbf{Y}_{T},\mathbf{Z}\mid \text{}\underline{\unicode[STIX]{x1D70F}},\boldsymbol{\unicode[STIX]{x1D703}},\mathbf{Y}_{t-q})] & = & \displaystyle \mathop{\sum }_{i=1}^{N}\mathop{\sum }_{g=1}^{G}\mathop{\sum }_{t=1}^{T}Z_{ig}\,\log [\unicode[STIX]{x1D70F}_{g}\,\exp \{\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\ldots ,\mathbf{Y}_{it-q})-\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}]\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=1}^{N}\mathop{\sum }_{g=1}^{G}\mathop{\sum }_{t=1}^{T}Z_{ig}\{\log \,\unicode[STIX]{x1D70F}_{g}+\text{}\underline{\unicode[STIX]{x1D703}}_{g}^{\prime }h(\mathbf{Y}_{it},\ldots ,\mathbf{Y}_{it-q})-\log \,\unicode[STIX]{x1D6FE}(\text{}\underline{\unicode[STIX]{x1D703}}_{g})\}.\end{eqnarray}$$

4.2 Model Estimation

As $\mathbf{Z}$ is a matrix of latent and unobserved cluster assignment probabilities, and the estimation of group-level parameters is contingent upon its composition, ego-TERGM estimation reflects a recursive process that requires an iterative estimation technique. To estimate the ego-TERGM, an EM algorithm is used with a two-step initialization procedure similar to that used by Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) to uncover the following:

• ∙ $N\times G$ matrix of estimated probabilities for actor-level $n_{i}$ role assignments ( $\hat{\mathbf{Z}}$ ). Note that this matrix is not indexed by time, indicating that cluster assignments are time-invariant.

• ∙ $G$ length vector of estimated baseline probabilities of assignment ( $\hat{\text{}\underline{\unicode[STIX]{x1D70F}}}$ ).

• ∙ $G\times H$ matrix of estimated group-level parameters ( $\hat{\boldsymbol{\unicode[STIX]{x1D703}}}$ ). As previously noted, $H$ refers to the number of statistics computed on each longitudinally observed ego-network.Footnote ¹²

The estimation routine begins using a two-step initialization process. First, for each node that ever appears in $\mathbf{Y}_{T}$ , a TERGM is estimated using bootstrapped MPLE (Cranmer and Desmarais Reference Cranmer and Desmarais2011; Desmarais and Cranmer Reference Desmarais and Cranmer2012). To adjust for the influence of network size on parameter estimates, an offset term is included to adjust for network size (Krivitsky, Handcock and Morris Reference Krivitsky, Handcock and Morris2011). Second, once the parameters for each TERGM are estimated, they are clustered using a $k$ -means algorithm (Hartigan and Wong Reference Hartigan and Wong1979). This provides initial estimates for $\mathbf{Z}$ and $\text{}\underline{\unicode[STIX]{x1D70F}}$ which can then be passed on to the EM algorithm and used to maximize Equation (7) (Dempster et al. Reference Dempster, Laird and Rubin1977). Role parameters $\boldsymbol{\unicode[STIX]{x1D703}}$ are then set equal to the cluster means for the group-based parameters.

From this initialization procedure, an EM algorithm is used to find the estimates for $\text{}\underline{\unicode[STIX]{x1D70F}}$ , $\mathbf{Z}$ , and $\boldsymbol{\unicode[STIX]{x1D703}}$ which were most likely to produce $\mathbf{Y}_{T}$ . EM algorithms, introduced by Dempster et al. (Reference Dempster, Laird and Rubin1977), are conventionally used to extrapolate unknown model information including missing data or parameters (Gill Reference Gill2014). As noted by Gill (Reference Gill2014), EM finds unbiased parameter estimates assuming that the likelihood function is unimodal. This assumption may vary according to the complexity of model specification, and as such, must be considered by the user.

Broadly, EM algorithms proceed in two steps. The first general step of an EM algorithm is to generate temporary data that represent a reasonable guess for unknown parameter values. In the case of the algorithm utilized here, mirroring that developed by Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015), role assignments $\mathbf{Z}$ for iteration $u$ are computed based upon the pooled TERGM parameters for all observations $i$ at iteration $u-1$ . In the second step, parameter estimation proceeds as if there is complete data. For the ego-TERGM algorithm, the complete data log-likelihood presented in Equation (7) is maximized to update pooled ego-TERGM estimates. In more formal terms, the EM algorithm is written as follows:Footnote ¹³

1. 1. Let $u=0$ reflect the estimates from the initial $k$ -means clustering initialization, where $\text{}\underline{\hat{\unicode[STIX]{x1D70F}}}^{(0)}$ and $\hat{\boldsymbol{\unicode[STIX]{x1D703}}}^{(0)}$ represent the initial estimates of the mixing proportions and centroids.

2. 2. E-Step: This step is repeated for all nodes and for all groups. Compute the expected role assignments for a particular node in a particular group, $\hat{Z}_{ig}$ , based upon current model parameters where $P(\mathbf{Y}_{iT}\mid \text{}\underline{\unicode[STIX]{x1D703}}_{g})$ can be understood as the conditional probability of observing the longitudinally observed ego-network $\mathbf{Y}_{iT}$ conditioned on group-level parameters and mixing parameters:

   (8) $$\begin{eqnarray}\hat{Z}_{ig}^{(u)}=\frac{\hat{\unicode[STIX]{x1D70F}_{g}}^{(u-1)}P(\mathbf{Y}_{iT}\mid \text{}\underline{\hat{\unicode[STIX]{x1D703}}}_{g}^{(u-1)})}{\mathop{\sum }_{j}\hat{\unicode[STIX]{x1D70F}}_{j}^{(u-1)}P(\mathbf{Y}_{iT}\mid \hat{\text{}\underline{\unicode[STIX]{x1D703}}}_{j}^{(u-1)})}.\end{eqnarray}$$

3. 3. M-Step: Maximize the expected complete data log-likelihood to yield new and updated parameter estimates:

   (9) $$\begin{eqnarray}(\text{}\underline{\unicode[STIX]{x1D70F}}^{(u)},\hat{\boldsymbol{\unicode[STIX]{x1D703}}}^{(u)})=\text{argmax}_{\text{}\underline{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D703}}\mathop{\sum }_{i=1}^{N}\mathop{\sum }_{g=1}^{G}\mathop{\sum }_{t=1}^{T}\hat{Z}_{ig}^{(u)}\log [\unicode[STIX]{x1D70F}_{g}P(\mathbf{Y}_{it}\mid \text{}\underline{\unicode[STIX]{x1D703}}_{g})]].\end{eqnarray}$$

4. 4. Check for convergence, if it has converged then stop, otherwise, move to the next iteration and increment $u$ , returning to Step 2. Convergence occurs if the log-likelihood changes by less than $10^{-6}$ .

4.3 Model Specification and Fit

The selection of terms to include in the sufficient network statistics for $h(\mathbf{Y}_{i})$ or the number of groups $G$ to fit is not clear cut. It is recommended that the analyst rely upon a combination of model selection tools (including the Bayesian Information Criterion or BIC), substantive intuition, and sensitivity analysis. The use of BIC may not always produce the best fitting model when used to select model terms as it is based upon pseudolikelihood and highly collinear ERGM terms (Hunter, Goodreau and Handcock Reference Hunter, Goodreau and Handcock2008). However, BIC has been found to be more effective in selecting appropriate values for $G$ (Salter-Townshend and Murphy Reference Salter-Townshend and Murphy2015). Regardless, caution is required and Kass and Raftery (Reference Kass and Raftery1995) recommend using a difference of ten or greater as suggestive evidence for the preferability of one model to another. BIC for the ego-TERGM is calculated using the estimated log-likelihood ( $\hat{{\mathcal{L}}}$ ), the number of roles specified ( $G$ ), the number of model terms ( $H$ ), the number of egos included in the network ( $N$ ), and the number of time periods ( $T$ ):

(10) $$\begin{eqnarray}BIC=2\hat{{\mathcal{L}}}-(GH+G-1)\log (NT).\end{eqnarray}$$

Unsurprisingly, model specification should be theoretically motivated. Presumably the analyst has some number of roles they expect to be present within the network, and this number of clusters should inform the value of $G$ chosen. Additionally, selecting a higher value for $G$ need not imply that $G$ clusters are actually fit. If the EM algorithm determines that assigning nodes to three roles produces the maximum likelihood, three roles will be fit, even if a $G>3$ is chosen. Alternatively, selecting a value $G=3$ will constrain the number of roles fit to at most three. The specification of terms for $h(\mathbf{Y}_{i})$ should also be theoretically motivated. These terms and the parameters associated with them are used to distinguish between roles and influence role assignments.

To assess the relative importance of $G$ values or particular terms to the prevalence of roles, a sensitivity based approach is recommended (Box-Steffensmeier et al. Reference Box-Steffensmeier, Campbell, Christenson and Navabi2018). The intuition for $G$ selection is quite simple, if $G$ is changed but the cluster assignments remain unchanged, then $G$ might reflect a well-fitting value. For term selection, the intuition is still simple. Within an $h$ -dimensional Euclidean space, the inclusion of a given term $h\in H$ should transform the space and influence the probability that a node is assigned to a particular role.

This sensitivity analysis may be conducted two ways. First, one may look at the role assignments, defined as a node’s most likely cluster assignment, to determine if the substantive implications of clustering change by adjusting model specification. This is what is typically referred to as the role assignment through this manuscript. Second, one may look closer at the changes in the probability of assignments to assess if the role assignment becomes more or less certain by excluding a covariate.Footnote ¹⁴

4.4 Assessing the Role-Generating Process

While the ego-TERGM is first and foremost a clustering algorithm, designed to sort nodes into finite groups based upon a set of model terms, interpretable coefficients can be estimated that describe the role-generating process. Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) find that the centroids used to sort nodes cannot be interpreted as parameters describing the role-generating process. However, one may use pooled TERGMs to assess the structure of ego-networks associated with a role easily. Networks assigned to the same role are thought to be of a shared data generating process. Using cluster assignments from the ego-TERGM, networks are pooled into sets by common membership. A bootstrapped TERGM (Cranmer and Desmarais Reference Cranmer and Desmarais2011; Desmarais and Cranmer Reference Desmarais and Cranmer2012), specified as the analyst desires, is then estimated on the pooled networks. The SI Appendix contains a deeper discussion and proof of concept for this routine.

Unbiased likelihood estimates of the role’s generative structure will be estimated when the networks are independent of one another. In this case, two forms of dependence may induce bias. First, time slices belonging to the same ego-network may possess autoregressive tendencies and produce temporal dependence (Cranmer and Desmarais Reference Cranmer and Desmarais2011). As such, an analyst must condition out any autoregression prior to pooling. This is easily accomplished through including a memory term, such as tie stability, loss, or innovation, when calculating change statistics for a time slice prior to pooling and bootstrap sampling (Leifeld, Cranmer and Desmarais Reference Leifeld, Cranmer and Desmarais2017). Second, there may be dependence between overlapping ego-networks that include the same actors. This form of dependence may be difficult to condition out and will imply that TERGM estimates are highly influenced by high-degree nodes that appear in many networks. However, this may not necessarily be a significant problem or a “bug” as these high-degree nodes could be an important feature of the role’s generative structure. In other words, even if these high-degree nodes are included in multiple networks, these nodes may not be oversampled. In fact, it is possible their higher rate of inclusion is substantively expected. Regardless, an analyst must carefully consider these forms of dependence before using this routine—while estimates will be returned, they may not reflect the true likelihood estimates and must be interpreted accordingly.

4.5 Simulation Study

To assess whether the ego-TERGM is capable of detecting roles according to similarity in TERGM model parameters, a Monte Carlo simulation study was conducted as a proof of concept and to examine the conditions that influence model performance.Footnote ¹⁵ This study approximates the equivalent cross-sectional study of Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015): 30 longitudinally observed ego-networks were simulated over five distinct time periods according to three distinct sets of model parameters or clusters.Footnote ¹⁶ This Monte Carlo has a $3^{2}$ factorial design, yielding nine treatment combinations. First, the size of each ego-network takes on one of three values per treatment combination: ${\mathcal{N}}_{ego}\in \{20,30,40\}$ . Ego-network size is important to examine as it produces a multiplicative increase in the prevalence of triadic closure in the network and should produce greater consistency in model parameter estimates. Second, the mixture of data generating processes is allowed to vary according to three sets of parameters. The first mixture uses the simulation parameters from Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015), presented in Table 1. This should represent the easiest experimental condition.Footnote ¹⁷ The second mixture uses the same terms as the prior mixture (Edges, GWESP ( $\unicode[STIX]{x1D6FC}=0.8$ ), GW Degree ( $\unicode[STIX]{x1D6FC}=0.8$ )) but allows the degree of triadic closure to vary by a minuscule margin of $0.5$ while all other model parameters are held constant. These simulation parameters are presented in Table 2. The third and final set of parameters uses the same terms as the prior DGPs but allows the relative density of each DGP to vary by a minor amount, as evidenced by a change in the value of the edges term by $1$ . The density condition simulation parameters are presented in Table 3. One set of 90 networks observed over five time periods and three DGPs is simulated per experimental condition.Footnote ¹⁸

Table 1.

ERGM Parameter Values for Conventional DGP. Rows refer to roles and columns refer to parameters.

Table 2.

ERGM Parameter Values for Triadic Closure Condition. Rows refer to roles and columns refer to parameters.

Table 3.

ERGM Parameter Values for Density Condition. Rows refer to roles and columns refer to parameters.

Overall, the results from the Monte Carlo, presented in Figure 3, are favorable. When considering the conventional Salter-Townshend and Murphy (Reference Salter-Townshend and Murphy2015) DGP with the simulation parameters presented in Table 1, ego-networks with a size between 20 and 30 nodes are accurately clustered 100% of the time. Problems appear to emerge when the size of the ego-network is 40, with a slight imbalance in the proportion of role assignments. This is likely a result of minuscule heterogeneity between mixtures that are amplified by the size of the network, which is to say, that Roles 1 and 2 appear to be relatively similar under the conventional DGP, and these similarities are amplified by a larger network and collinearity between model terms.

Figure 3.

Ego-TERGM Monte Carlo Simulation Study Results. Each plot refers to a different mixture of data generating processes while the shape of points refers to different ego-network sizes. Dashed line at 0.33 refers to perfect classification.

For the second data generating process, presented in Table 2 with results in the second panel of Figure 3, the ego-TERGM appears to do well. The model fares exceptionally well when the size of each ego-network ranges between 20 and 30 nodes, achieving an accurate classification rate of 100%. However, when the size of the ego-networks becomes larger (40 nodes), only two roles are retrieved, with nodes that would otherwise be classified as Role 1 being classified as Role 2 nodes. This appears to be a function of the relative similarity of the GWESP simulation parameters for Roles 1 and 2 and the multiplicative nature of the GWESP statistic in large networks. A great degree of triadic closure creates problems for any clustering method as ego-networks approaching full connectivity and density are difficult to distinguish.

Finally, the model appears to perform exceptionally well when the only factor distinguishing between roles is the relative density of the ego-networks. For the density condition, with a DGP presented in Table 3 and results presented in the third panel of Figure 3, network size does not appear to matter. Regardless of ego-network size, nodes are accurately clustered with 100% accuracy.

The prior Monte Carlo study has demonstrated that the ego-TERGM fares well even under very difficult and potentially uncommon conditions. The model performs best with ego-networks between 20 and 30 nodes and when the key factor thought to influence heterogeneity in DGPs is relative network density. Alternatively, the model does not appear to do well in large networks where nodes are distinguished between minuscule differences in the rate of triadic closure. This is to be expected given the multiplicative relationship between network size and the possibility for triadic closure. This news is not damning as ego-networks with a size greater than 20 or 30 are relatively rare. Nevertheless, when analysts encounter such networks they must be careful when they expect a great degree of triadic closure. Overall, this demonstrates the promise of using network topology to cluster nodes according to similarity in DGPs. In the following section, the substantive importance of the ego-TERGM are presented in the form of a pedagogical application to the Kapferer (Reference Kapferer1972) strike network.