Day 14: Which Model, When?

The Question That Outlasts the Course

Days 8 Through 10 Built the Relational Toolkit

Days 11 Through 13 Added Structure, Change, and Design

Start by Naming the Target

Possible targets include:

• An observed covariate association with uncertainty that allows shared actors
• Actor sending and receiving propensities
• Discrete relational types
• A latent relational surface
• An explicit graph configuration such as closure
• Influence among actors

Every Lens Must Earn Its Place

For each model, finish four sentences:

1. The substantive question is…
2. This model helps because…
3. It is distinct from the previous lens because…
4. After fitting it, I can say…

The Same Network Becomes Different Evidence

One Dataset Keeps the Models Comparable

The rebel-cooperation network supplies:

• 30 armed organizations active in the Syrian civil war
• An undirected tie for at least one claimed tactical joint operation, July 2012 through June 2015
• Organization ideology and size
• Pair-level ideological distance, power difference, shared location, and shared sponsor
• Strong differences in how many cooperation partners organizations have

Lens 0: Start With the Dyadic GLM

What Dyadic Cluster-Robust Standard Errors Do

The rows in a dyadic regression overlap:

• ANF with ASIM and ANF with another organization both contain ANF.
• Unmeasured changes involving ANF can therefore affect both regression errors.

Dyadic cluster-robust standard errors, or DCRSEs, allow errors to be correlated when two dyads share either organization.

What DCR Does Not Fix

• It does not repair an omitted actor trait, a wrong functional form, or a missing network process.
• It still treats dyads with no shared organization as independent. Broader shocks or network dependence can violate that assumption.
• Its justification is asymptotic in the number of actors. With only 30 organizations, the intervals can be unstable.
• It does not turn a conditional association into a causal effect.
• A corrected standard error can be larger or smaller because the estimated shared-actor covariances can have either sign.

Lens 1: SRM Locates Organization-Level Dependence

$$\eta_{ij}=x_{ij}^{\mathsf T}\beta+a_i+a_j$$

Because this cooperation network is undirected, the SRM asks:

• Which organizations cooperate with more partners than the measured covariates predict?
• How much of the network’s hub structure is organization-level heterogeneity?
• Which covariate results change after every organization receives an additive effect?

Lens 2: Blockmodels Ask Whether Actors Come in Kinds

$$\Pr(Y_{ij}=1\mid z_i,z_j)=\theta_{z_i z_j}$$

Lens 3: AME Adds a Relational Surface

$$\eta_{ij}=x_{ij}^{\mathsf T}\beta+a_i+a_j+u_i^{\mathsf T}\Lambda u_j$$

The SRM gives each organization one overall cooperation effect. AME adds a pair-specific score:

• A high score raises fitted cooperation for that particular pair.
• A low score lowers fitted cooperation for that particular pair.
• The scores summarize recurring partner patterns that the measured variables and overall organization effects missed.

Lens 4: ERGM Makes Configurations Explicit

$$\Pr(Y=y)\propto\exp\{\theta^{\mathsf T}g(y)\}$$

Use change statistics and simulation to interpret:

• Edges and covariates
• Actor sociality and degree-related terms
• Geometrically weighted closure

After Naming the Target, Preserve the Data Structure

Choose a Model in Three Steps

1. Say the question in ordinary language. Example: Which organizations cooperate unusually widely?
2. Choose the model feature that answers it. An SRM gives each organization its own overall cooperation tendency.
3. Name the check that could change your mind. If the SRM still misses important partner patterns, add a blockmodel, AME surface, or explicit ERGM configuration depending on the miss.

Carry Six Questions From the Causal Lecture

1. What changes? Name the treatment or exposure precisely.
2. Whose outcome may change, and when?
3. Compared with what? Name the units or periods supplying the comparison.
4. Why might treated and untreated cases already differ?
5. Can treatment reach a unit through its network neighbors?
6. Which result would still be useful if a causal claim is not credible?

Turn the Questions Into a Workable Analysis

Example: Do sanctions reduce later trade?

• Define the treatment as a sanction imposed before the trade outcome.
• Compare sanctioned and unsanctioned pair-periods with similar pre-treatment histories.
• Measure pre-sanction trade, security relations, regime ties, and economic conditions.
• If allies’ sanctions may alter the focal pair’s trade, define that indirect exposure.
• Use pre-treatment outcomes or negative controls to probe hidden selection.
• Model repeated states and dyads, but keep that dependence problem separate from causal identification.

A Negative Control Starts as a Deliberate Non-Effect

Egami and Tchetgen Tchetgen (2024) ask whether students do better academically when their friends have higher GPAs.

Main exposure: friends’ average baseline GPA
Outcome: the student’s later GPA
Confounding worry: friends may share motivation, family resources, courses, stress, or school conditions before any influence occurs

Friends’ baseline headaches may reflect that hidden background, but should not directly change the student’s later GPA. As a basic placebo check, an association between friends’ headaches and later GPA would warn that hidden selection may remain.

What Node2vec Does With the Network

1. Reserve ties and non-ties for testing, then remove the test ties from the graph.
2. Start at one organization and follow a few observed ties.
3. Repeat these short network trips from every organization.
4. Give every organization a short row of coordinates. Organizations reached in similar network neighborhoods receive similar rows.
5. Combine two organizations’ profiles, fit a model on visible pairs, and test it on hidden pairs.

Three Methods, Three Definitions of “Similar”

NetMF and GraphSAGE Solve Two Practical Problems

Question 1: Can We Predict a Hidden Tie?

Syrian cooperation example

1. Before learning coordinates, reserve some cooperation ties and an appropriate sample of non-ties.
2. Remove the reserved ties, then learn the walk-based coordinates from the reduced graph.
3. Turn each candidate pair into predictors using the two organizations’ coordinates.
4. Fit on the remaining pairs and predict only the pairs hidden in step 1.
5. Compare with degree and shared-partner baselines. In a full application, also add observed pair characteristics.

Question 2: Can Network Position Predict an Actor Label?

Brown et al. (2021) use linking patterns to help predict ideological labels for Twitter users and internet domains.

• The outcome belongs to an actor, not a pair.
• The predictors summarize which other actors or domains link to it.
• Keep test actors in the graph, but hide their labels from the classifier.
• Compare network features with a simple label-frequency baseline and with useful non-network attributes.
• Check whether performance survives across different samples, platforms, or time periods.

Question 3: Can We Build Features for a New Actor?

Ordinary node2vec learns a separate coordinate row for every actor in the training graph. A genuinely new actor has no stored row.

GraphSAGE instead learns a rule:

1. Start with the new actor’s observed attributes.
2. summarize the attributes of its observed neighbors;
3. combine those summaries using the learned rule;
4. produce coordinates for the new actor;
5. use those coordinates in the prediction task.

Node2vec and Day 10 Are Cousins

Both approaches reduce complicated relational patterns to a few numbers per actor. They do that for different reasons.

SIR Asks How Conflict Patterns Carry Forward

Minhas and Hoff (2026) study monthly material-conflict events among countries. These include physical attacks, destruction of property, and other coercive actions recorded by ICEWS.

The model separates two questions:

• Direct dyadic history: Does state $i$ direct more conflict toward state $j$ after earlier conflict within that same pair?
• Network influence: Does conflict elsewhere in the system help predict which states $i$ targets next and which states $j$ is targeted by next?

Direct Effects and Influence Paths Are Different

Direct part of the model

• The pair’s earlier conflict and conflict in the reverse direction
• Geographic distance, joint democracy, alliance, trade, and verbal cooperation

Influence part of the model

• Sender side: Does state $i$ target countries that its allies or verbal-cooperation partners targeted previously?
• Receiver side: Are geographically nearby states targeted by similar sets of countries?

What the Published SIR Application Finds

Alliances organize sender-side influence

States tend to direct more material-conflict events toward countries that their allies fought in the previous month.

Verbal cooperation also organizes sender-side influence

States tend to target some of the same countries as their cooperative diplomatic partners.

Geography organizes receiver-side influence

Nearby states tend to receive conflict from similar sets of senders.

The Same Modeling Move Can Structure Different Objects

Multilayer Networks Keep Several Relations Visible

For the same source $i$ and target $j$,

$$Y_{ij\ell}$$

keeps relation type $\ell$ visible. With repeated years, the object becomes $Y_{ij\ell t}$.

Examples include:

• Military threats, sanctions, trade, and alliances among the same states
• Verbal cooperation, material cooperation, verbal conflict, and material conflict
• Advice, friendship, and collaboration among the same people

Theory Can Make the Layers Interdependent

With repeated observations, $Y_{ij\ell t}$ can represent several relations evolving together.

• International relations: Conflict can reduce trade, while trade dependence may discourage later conflict.
• American politics: Legislative cosponsorship can build voting coalitions, while repeated voting agreement can make future cosponsorship easier.
• Education: Friendship can create opportunities for study help, while repeated help can strengthen a friendship.

What netify Can Do With the Layers

One Network Can Contain Four Different Stories

Do Not Flatten Away the Question

Collapsing verbal cooperation, material cooperation, verbal conflict, and material conflict into “any interaction” changes the outcome.

Fitting four separate networks preserves the outcomes but cannot estimate:

• A source profile shared across relations
• A target profile shared across relations
• Which layers express the same relational component
• Shared cross-layer mean structure

Follow One Shared Component Across the Layers

For component $r$, the contribution to one source-target-layer cell is

$$u_{ir}\times v_{jr}\times w_{\ell r}.$$

1. $u_{ir}$ says how strongly source $i$ expresses the source side of this profile.
2. $v_{jr}$ says how strongly target $j$ expresses the target side.
3. Their product identifies source-target pairs matching the profile.
4. $w_{\ell r}$ says whether that same pair pattern is strong, weak, reversed, or absent in layer $\ell$.

What the Tensor Estimator Optimizes

This teaching fit uses the same broad ALS logic as lame. For the transformed Gaussian outcome, it minimizes

$$\min \sum_{i\ne j,\ell} \left(Y^*_{ij\ell}-\eta_{ij\ell}\right)^2.$$

1. Remove the current layer-specific source and target effects.
2. Hold $V,W$ fixed and solve an ordinary least-squares update for $U$.
3. Cycle through analogous updates for $V$ and $W$, then refresh the additive effects.
4. Repeat to convergence. This example compares eight starts; research should use more starts and compare ranks out of sample.

Overlap Needs a Density Benchmark

At the one-event threshold:

• Jaccard 0.385: 38.5% of dyads appearing in either conflict layer appear in both.
• Lift 4.91: a dyad appears in both layers 4.91 times as often as independent layers with these densities would predict.

What Should We Do After Finding Layer Overlap?

1. Check measurement choices. Repeat the comparison across meaningful event thresholds and coding rules.
2. Use a stronger null. Ask whether overlap remains after preserving actor activity, time patterns, or other obvious sources of co-occurrence.
3. Match the model to the claim. Use a joint factor model for shared profiles; use repeated layers and lags for cross-layer prediction.
4. Check what remains unexplained. Compare held-out prediction and residual degree, triad, and cross-layer overlap patterns.

Do the Same State Pairs Stand Out Across Relations?

First remove each state’s broad source and target activity within every layer. Then ask whether a few recurring pair patterns explain what remains.

Pattern 1

After removing the United States’ overall source and target activity, particular United States dyads remain unusually active across several relations.

Pattern 2

Pairs involving Iran, Israel, Lebanon, Egypt, and Iraq repeatedly stand out.

Both patterns appear in verbal cooperation, material cooperation, verbal conflict, and material conflict. Together they account for about 23% of the transformed variation left after the broad actor effects.

How Should We Check the Joint Fit?

• Compare rank 1, rank 2, and higher ranks on held-out actor-pair-layer cells.
• Repeat ALS from several starts and retain solutions reaching the best objective.
• Add defensible covariates and see whether the shared profiles remain.
• Check residual source activity, target exposure, triads, and cross-layer overlap.
• Use a binary or count likelihood when that matches the outcome, and use bootstrap or Bayesian fitting when uncertainty is required.

Write the Model Choice as an Argument

Use this paragraph structure:

1. Question: Name the estimand.
2. Data: State the network, time, and eligibility structure.
3. Primary model: Explain where it represents dependence.
4. Runner-up: Name the strongest alternative representation.
5. Flip condition: Give a specific diagnostic that would change the choice.
6. Claim: State what the fitted result would and would not establish.

Published Example: With Whom Do Syrian Rebels Cooperate?

Gade et al. (2019) ask why armed organizations chose some partners rather than others during the Syrian civil war.

Data: Claims of tactical joint operations from July 2012 through June 2015, combined with measures of ideology, power, state sponsorship, and shared location.

Competing explanations: Rebels may choose ideologically similar partners, partners of similar strength, or partners backed by the same state.

The Map, Filled In

After the Models Run, Write the Substantive Paragraph

Include the question, model choice, substantive result, diagnostic, limit, and the finding that would change the model choice.

What to Take With You

1. Define actors, ties, eligibility, missingness, and time before modeling.
2. Identify where the important variation and dependence live.
3. Match the model to the estimand rather than to a preferred software package.
4. Diagnose the feature that would change the substantive conclusion.
5. Treat latent quantities, graph statistics, temporal mechanisms, and robust standard errors as different quantities that require different interpretations.