Day 9: Actor Effects and Relational Roles

What Changes Today

The Syrian Cooperation Case

Gade et al. (2019) study armed organizations operating in the Syrian rebellion from July 2012 through June 2015. The network includes organizations such as Al-Nusrah Front (ANF) and Ahrar al-Sham Islamic Movement (ASIM).

• Why did some pairs cooperate more than others?
• Were some organizations unusually active across many partnerships?
• Was cooperation spread broadly, divided into separate camps, or organized around a small tactical core?

What Counts as Cooperation?

The outcome is the number of claimed joint tactical operations between two organizations. We model its square root, following the published application.

Use netify to Build the Network Inputs

gade_net <- netify(
  input = gade,
  actor1 = "actor1",
  actor2 = "actor2",
  symmetric = TRUE,
  weight = "sqrt_coop",
  nodal_vars = c("averageId_actor1", "size_actor1", "spons_actor1"),
  dyad_vars = c("ideol_diff", "powerdiff", "loc", "spons"),
  dyad_vars_symmetric = rep(TRUE, 4),
  missing_to_zero = TRUE
)

gade_lame <- to_lame(gade_net, family = "normal")

netify turns the pair-level table into one symmetric outcome matrix, one dyadic predictor array, and one row of actor attributes per organization.

Start With the Expectations

The Models Answer Different Questions

Why Independent Dyads Are Not Credible Here

The pairs ANF-ASIM and ANF-ISIL both contain ANF.

If ANF has unmeasured organizational capacity, both observations inherit it.

Let $\epsilon_{ij}$ be what the independent regression still gets wrong for pair $ij$. Then

$$\epsilon_{\text{ANF,ASIM}} \not\perp \epsilon_{\text{ANF,ISIL}}$$

The crossed independence symbol, $\not\perp$, means the two remaining errors are not statistically independent.

Put the Residuals Back Into the Network

For each organization, average the residuals from all of its partnerships.

The Symmetric SRM Adds One Actor Effect

$$\begin{aligned} y_{ij}&=\mathbf{x}_{ij}^{\mathsf T}\boldsymbol\beta+a_i+a_j+\epsilon_{ij},\\ a_i&\stackrel{\mathrm{iid}}{\sim}\mathcal N(0,\sigma_a^2). \end{aligned}$$

$\mathbf x_{ij}$ contains the measured pair attributes, $\boldsymbol\beta$ contains their coefficients, and $\epsilon_{ij}$ is what remains unique to the pair.

The label $\mathrm{iid}$ means independent and identically distributed across organizations: the actor effects use the same distribution but are separate draws.

Directed SRM: Same Logic, More Pieces

$$\begin{aligned} y_{ij}&=\mathbf{x}_{ij}^{\mathsf T}\boldsymbol\beta+a_i+b_j+\epsilon_{ij},\\ \begin{pmatrix}a_i\\b_i\end{pmatrix} &\stackrel{\mathrm{iid}}{\sim}\mathcal N\!\left[ \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}\sigma_a^2&\sigma_{ab}\\\sigma_{ab}&\sigma_b^2\end{pmatrix} \right],\qquad \operatorname{Corr}(\epsilon_{ij},\epsilon_{ji})=\rho . \end{aligned}$$

• $a_i$: how often actor $i$ appears on the source side, beyond the measured predictors
• $b_i$: how often the same actor appears on the target side
• $\sigma_a^2$ and $\sigma_b^2$: how much the adjusted source-side and target-side scores differ across actors
• $\sigma_{ab}$: whether states that are often sources are also often targets
• $\rho$: whether an unexpectedly high $i\!\rightarrow\!j$ outcome accompanies an unexpectedly high $j\!\rightarrow\!i$ outcome

Omitted Actor Structure and the SRM

Suppose the relationship really follows:

$$y_{ij}=\beta x_{ij}+a_i+a_j+\epsilon_{ij}.$$

If we omit the actor effects:

$$\operatorname{plim}(\widehat\beta_{\text{naive}}) =\beta+\frac{\operatorname{Cov}(x_{ij},a_i+a_j)} {\operatorname{Var}(x_{ij})}.$$

The standard random-effects SRM assumes that the remaining actor effect has conditional mean zero given the included predictors:

$$\operatorname{E}(a_i\mid X)=0.$$

Positive Covariance Case

Let $x$ be ideological distance. Suppose organizations with high positive actor effects are disproportionately involved in ideologically distant pairs.

Then $\operatorname{Cov}(x_{ij},a_i+a_j)>0$, so the independent model credits ideological distance with some of the broad organizational activity.

Ideological distance can look less negatively associated with cooperation than it does after recurring actor activity enters, appear unrelated, or even appear positively associated.

Negative Covariance Case

Keep $x$ as ideological distance. Suppose organizations with high positive actor effects are concentrated in ideologically similar pairs.

Then $\operatorname{Cov}(x_{ij},a_i+a_j)<0$, so the independent-model ideology coefficient is pulled downward.

Ideological distance can look more negatively associated with cooperation than it does after broad recurring organization activity enters.

Zero Covariance Case

If $\operatorname{Cov}(x_{ij},a_i+a_j)=0$, the independent-model ideology coefficient can still be centered on $\beta$.

The errors are still dependent because relationships sharing organizations inherit the same actor effects.

What the Random-Effects SRM Assumes

Adding actor effects represents relationships that share actors, but the usual SRM still assumes the remaining actor effects are unrelated to the included predictors after conditioning on the model.

• If an actor effect and a covariate remain correlated, $\operatorname{E}(a_i\mid X)=0$ is violated.
• The coefficient can then blend stable actor differences with the measured predictor.
• With one cross-sectional network, we cannot compare the same actor with itself over time.
• A badly chosen control variable remains a problem after actor effects enter.

A Concise Mundlak Remedy

For a time-varying predictor, separate within-actor change from stable between-actor differences:

$$x_{it}=(x_{it}-\bar x_i)+\bar x_i.$$

Include the actor mean $\bar x_i$ in the model, with separate source and target means in a directed network. The remaining random effect is then assumed unrelated to the within-actor deviation rather than to the raw predictor.

Also measure the missing mechanism when possible, compare fixed-effects specifications when the needed variation exists, and report sensitivity across defensible models.

What the SRM Sampler Is Trying to Learn

The observed network has to inform four connected pieces:

• Associations with the measured variables, $\boldsymbol\beta$
• One adjusted cooperation effect for every organization, $\mathbf a$
• How much those organization effects vary, $\sigma_a^2$
• How much pair-level variation remains, $\sigma_\epsilon^2$

$$p(\boldsymbol\beta,\mathbf a,\sigma_a^2,\sigma_\epsilon^2\mid Y,X) \propto \underbrace{p(Y\mid X,\boldsymbol\beta,\mathbf a,\sigma_\epsilon^2)}_{\text{fit to the observed partnerships}} \times \underbrace{p(\boldsymbol\beta,\mathbf a,\sigma_a^2,\sigma_\epsilon^2)}_{\text{regularization and pooling}}$$

How MCMC Explores That Target

One sweep updates connected blocks:

1. Draw coefficients conditional on the current actor effects.
2. Draw organization effects conditional on their partnerships and the current coefficients.
3. Draw the actor-effect and remaining-error variances.
4. Repeat, discard burn-in, and summarize the retained draws.

The sampler does not maximize the posterior and stop at one value. Its retained draws represent the posterior distribution.

One Switch Separates the Benchmark and SRM

# Independent pairs
ame(..., symmetric = TRUE, nvar = FALSE)

# Persistent organization effects
ame(..., symmetric = TRUE, nvar = TRUE)

Read Point Estimates and Intervals Separately

• Greater ideological distance has a negative point estimate, but its SRM interval crosses zero.
• A larger difference in estimated fighters has a negative estimate, and its SRM interval stays below zero.
• Shared operating location has a positive point estimate, but its interval crosses zero.
• Shared state sponsorship remains uncertain.

The Actor Effects Describe Named Organizations

The model gives Al-Nusrah Front (ANF) and Ahrar al-Sham Islamic Movement (ASIM) the two highest adjusted cooperation scores.

The SRM Is Already a Latent-Variable Model

The effect $a_i$ is inferred from the network rather than observed directly.

Posterior Predictive Checks Show the SRM Boundary

Where These Methods Come From

• Back and Kenny (2010): the general SRM decomposition
• Hoff (2005): network regression with additive actor effects
• Gade et al. (2019): ideology and tactical cooperation among Syrian armed organizations
• Minhas et al. (2022): dyadic dependence and omitted-variable limits
• Boschee et al. (2015): ICEWS event coding and source-target records
• Olivella, Pratt, and Imai (2022): mixed-membership models for repeated networks

From Actor Effects to Discrete Roles

The SRM gives every organization one adjusted score for how broadly it cooperates. The blockmodel asks a different question: was cooperation spread evenly, divided into separate camps, or organized around a small set of organizations that worked across a much less connected field?

This analysis uses the 31 organizations with at least one recorded cooperation tie. The SRM used square-root counts for the 30 organizations with complete covariates.

Blocks Are Not Necessarily Communities

A Role Is a Compressed Partner List

Imagine describing the network with only three numbers:

• How often core organizations cooperate with one another
• How often core and peripheral organizations cooperate
• How often peripheral organizations cooperate with one another

We also need one role assignment for each organization. If those few quantities reconstruct the partner lists well, they provide a useful summary of the network.

The Blockmodel Names the Pieces

$$Y_{ij}\mid z_i,z_j\sim\operatorname{Bernoulli}(\theta_{z_i z_j})$$

• $z_i$ is organization $i$’s relational role.
• $\theta_{gh}$ is the probability of a recorded tie between roles $g$ and $h$.
• The full $\boldsymbol\theta$ matrix contains every within-role and across-role probability.

What the Blockmodel Likelihood Rewards

$$L(\mathbf z,\boldsymbol\theta;Y) =\prod_{i<j} \theta_{z_i z_j}^{Y_{ij}} (1-\theta_{z_i z_j})^{1-Y_{ij}}$$

• A recorded tie rewards assignments with a large relevant $\theta_{z_i z_j}$.
• A recorded non-tie rewards assignments with a small relevant $\theta_{z_i z_j}$.
• The preferred roles make entire partner lists unsurprising, not merely actor degrees.

How Variational EM Fits the Blockmodel

1. Begin with tentative role probabilities for every organization.
2. Update each organization’s probabilities using its ties, non-ties, and the current role-pair rates.
3. Update each role-pair rate as a membership-weighted proportion of observed ties.
4. Alternate until the evidence lower bound stops improving.
5. Repeat from several starts because the objective has local optima.

$$\operatorname{ELBO} =E_q[\log p(Y,\mathbf z\mid\boldsymbol\theta)] -E_q[\log q(\mathbf z)]$$

Two Roles Best Balance Simplicity and Fit

The integrated completed likelihood, or ICL, rewards fit and penalizes unnecessary roles. Higher is better here. It prefers $K=2$ roles, but the $K=3$ solution is close enough to inspect. Here $K$ is the number of roles.

The Stable Core Is ANF and ASIM

Across the two- and three-block solutions, Al-Nusrah Front (ANF) and Ahrar al-Sham Islamic Movement (ASIM) remain together in the core.

The other organizations occupy one broad periphery or two peripheral roles with different access to the core.

Read the Cooperation Probabilities as Actions

Observed ties and approximate fitted probabilities:

• Core with core: 1 of 1 observed, fitted 0.96
• Core with periphery: 44 of 58 observed, fitted 0.73
• Periphery with periphery: 41 of 406 observed, fitted 0.10

Compare the Observed Network With the Role Model

Simulate Networks From the Fitted Roles

• Spread in the number of partners: 5.43 observed, 4.72 in an average fitted simulation
• Three-organization cooperation triangles: 99 observed, about 61 in an average fitted simulation

Two Longitudinal Questions Need Different Panels

The outcome equals one when an ICEWS source-target dyad records more than 20 materially conflictual events in a year.

Polity scores run from -10 for strongly autocratic regimes to 10 for strongly democratic regimes. We divide the absolute gap by 20, so a 10-point difference enters the model as 0.5.

The Binary Longitudinal SRM

$$\Pr(Y_{ijt}=1)=\Phi\!\left(\alpha_t+\mathbf{x}_{ijt}^{\mathsf T}\boldsymbol\beta+a_{it}+b_{jt}\right)$$

• Year intercepts absorb changes in overall above-threshold network density.
• $\Phi$ turns the fitted probit score into a probability between zero and one.
• $\alpha_t$ gives each year its own baseline rate, while $\mathbf{x}_{ijt}^{\mathsf T}\boldsymbol\beta$ contains the measured dyad-year predictors.
• $a_{it}$ records how often state $i$ appears as the coded source in year $t$ beyond those predictors.
• $b_{jt}$ records how often state $j$ appears as the coded target in year $t$ beyond those predictors.
• dynamic_ab = TRUE links adjacent annual scores with an AR(1)-style smoothing penalty.

What Dynamic ALS Is Trying to Fit

lame(...,
  family = "binary",
  method = "als",
  dynamic_ab = TRUE,
  bootstrap = 100,
  bootstrap_type = "parametric")

The binary outcome is converted into a temporary probit working score. The estimator finds coefficients and annual source and target effects that reconstruct those scores while keeping adjacent annual paths reasonably smooth.

What Dynamic ALS Minimizes

$$\sum_{t,i\ne j}w_{ijt} \left[z_{ijt}-(\mathbf x_{ijt}^{\mathsf T}\boldsymbol\beta+a_{it}+b_{jt})\right]^2 +\lambda_a\sum_{t>1}\|\mathbf a_t-\rho\mathbf a_{t-1}\|^2 +\lambda_b\sum_{t>1}\|\mathbf b_t-\rho\mathbf b_{t-1}\|^2$$

1. Update the probit working responses $z$ and weights $w$.
2. Solve for the measured coefficients.
3. Update the annual source and target paths.
4. Repeat until the penalized objective stops changing.

Measured Associations and Bootstrap Intervals

Both associations are positive in this selected panel. The coefficients move a latent probit score, so they are not percentage-point changes.

Events Coded From and Toward Each State

Syria’s Adjusted Source and Target Scores Changed

The fitted score for events coded from Syria rises:

• 2002: -1.34, with 95% bootstrap interval [-2.46, -0.85]
• 2012: 0.09, with interval [-0.53, 0.49]
• 2014: 0.29, with interval [-0.38, 0.79]

The fitted score for events coded toward Syria rises more sharply:

• 2002: -1.51, with interval [-2.18, -0.39]
• 2012: 0.71, with interval [0.11, 1.04]
• 2014: 0.72, with interval [0.33, 1.27]

The Other Focus States

• The United States stays clearly above zero for events coded both from it and toward it throughout 2002 to 2014.
• Iran is clearly above zero for events coded toward it from 2006 onward, with especially high values from 2010 through 2013.
• Russia is clearly above zero for events coded toward it from 2012 through 2014.

What the Bootstrap Ribbons Mean

For each of 100 bootstrap replications:

1. Generate a new 13-year panel from the fitted dynamic SRM.
2. Refit the same model to that generated panel.
3. Save the annual trajectories for events coded from and toward each state.
4. Use the spread across refits to build the ribbons.

What This Dynamic SRM Does Not Show

The AR(1)-style penalty smooths adjacent annual actor effects. It does not identify the process that produced a rise or fall.

The model does not give every state pair its own long-run baseline, include a lagged outcome, or show that a source-side change caused a target-side change.

Did Post-Cold War Defense Partners Follow the Same Path?

Did Poland, Romania, and Russia develop similar sets of defense partners after the Cold War, or did their treaty networks take different paths? Which states connected partner systems that would otherwise remain separate?

The panel contains 50 states from 1991 through 2000. A tie means the pair had a recorded defense commitment in that year.

What NetMix Learns

1. A table showing how often every pair of roles contains an alliance
2. Three weights for each state-year showing how much its partner list resembles each role

The three weights add to one. A state-year can mix roles rather than receiving one permanent label.

Turn the Notation Into One Sentence

$$\Pr(Y_{ijt}=1) =\sum_g\sum_h\pi_{it,g}\pi_{jt,h}\theta_{gh} =\boldsymbol\pi_{it}^{\mathsf T}\boldsymbol\Theta\boldsymbol\pi_{jt}$$

• $\pi_{it,g}$: how much state $i$ in year $t$ resembles role $g$
• $\pi_{jt,h}$: how much state $j$ in that year resembles role $h$
• $\theta_{gh}$: how often role $g$ and role $h$ are allied

How One State-Year Informs Its Role Mixture

Take Poland in 1997. The estimator examines:

• Every state with which Poland recorded a defense commitment
• Every state with which it did not
• The current alliance rates among the three candidate roles

It asks which mixture of regional, dense-multilateral, and sparse partner patterns makes that entire treaty list least surprising.

How Variational Inference Fits NetMix

1. Begin with rough role weights for every state-year.
2. Update approximate role assignments for the endpoints of every dyad.
3. Update state-year membership vectors and role-pair alliance log-odds.
4. Continue until the evidence lower bound stops improving.

$$\operatorname{ELBO} =E_q[\log p(Y,\text{roles},\boldsymbol\pi\mid\boldsymbol\Theta)] -E_q[\log q(\text{roles},\boldsymbol\pi)]$$

One Hidden State Means No Transition Model

Every state-year receives three role weights, and all ten years share one role-to-role alliance table.

With n.hmmstates = 1, there is no transition matrix, smoothing, or estimated switching process.

The Fitted Alliance Pattern

Why Mixed Membership Matters

• The United States mixes the regional and dense multilateral patterns early in the panel, so forcing it into only one role would discard part of its alliance portfolio.
• Poland and Romania both have few ties into the two connected systems early in the decade.
• From 1997 onward, Poland’s partners are mainly states already tied into the dense multilateral system, while Romania’s partner list does not make the same move.

What the Profiles Show

How Much Does It Add Beyond Degree?

Dense-role membership correlates about 0.90 with the number of alliances a state-year has, so much of the result is a density ranking.

Partner identities still add something. Poland and Romania begin with similarly short partner lists, then separate when Poland forms defense commitments with states already connected to the dense multilateral system and Romania does not.

Does the Model Recreate the Alliances We Gave It?

The Brier score is the average squared gap between an estimated alliance probability and the observed zero or one. Lower is better.

Boundaries on the Interpretation

• Role weights summarize recorded treaty partners, not treaty credibility or compliance.
• The role labels are descriptions, not natural kinds.
• Three roles were set before estimation.
• One hidden state means no estimated transition process.
• Role numbers must be aligned before comparing separate fits.
• The model describes this 50-state panel, not every state in the system.

Three Ways to Represent the Missing Pattern

What Happened Among the Syrian Organizations?

The data count claimed joint tactical operations between pairs of armed organizations from July 2012 through June 2015. Pairs closer in estimated fighter strength carried out more of these operations after accounting for ideology, operating location, state sponsorship, and whether each organization cooperated broadly with many partners.

Al-Nusrah Front and Ahrar al-Sham Islamic Movement carried out more claimed joint operations across their many partnerships than the measured variables predicted. In the binary network, these two organizations connected to 44 of 58 possible pairs involving the rest of the network, while the other organizations connected to only 41 of 406 possible pairs among themselves.

What Happened in the ICEWS Example?

The outcome is whether ICEWS recorded more than 20 materially conflictual events from one state toward another in a year. The United States crossed that cutoff with many states in both directions from 2002 through 2014.

Syria crossed the cutoff with relatively few states through 2010. After 2011, Syria appeared in many more above-threshold relationships, especially as the state toward which events were directed. The bootstrap intervals show a clear change from its earlier years.

What Happened in the Alliance Example?

The treaty records show that Poland, Romania, and Russia did not follow one common post-Cold War path.

From 1997 onward, Poland’s defense partners are mainly states already tied into a dense multilateral system. Romania remains connected to relatively few states in this panel, while Russia’s partners concentrate in a different regional system.

What to Carry Into Day 10

1. Relationships that share an actor are connected, even when each pair appears only once.
2. A missing process can pull a coefficient up, pull it down, or leave it centered while still making the usual uncertainty calculation wrong.
3. A random-effects SRM assumes the remaining actor effects are conditionally unrelated to the predictors; Mundlak terms can relax that assumption in repeated data.
4. SRM actor effects show which actors repeatedly sit above or below the measured prediction. They do not create a causal design.
5. The SRM reproduces broad actor heterogeneity here but misses third-order dependence.
6. Blockmodels find recurring relationship patterns, not automatically substantive communities.
7. Dynamic SRM trajectories describe adjusted source and target prominence in ICEWS; NetMix memberships compare defense-alliance partner profiles.