Analysis of Two-Stage Rollout Designs with Clustering for Causal Inference under Network Interference

TL;DR

The paper tackles estimating the Total Treatment Effect under network interference by introducing a two-stage rollout that selects a sub-population in Stage 1 and applies a staged CRD rollout within that sub-population in Stage 2, thereby increasing the effective budget for polynomial interpolation under a $\beta$-order interference model. It derives a bias expression $\tfrac{1}{n}\sum_i\sum_{\mathcal{S}\subseteq\mathcal{N}_i,\mathcal{S}\neq\varnothing} c_{i,\mathcal{S}} [\tfrac{q}{p}\Pr(\mathcal{S}\subseteq\mathcal{U}) - 1]$ and variance bounds that decompose into terms driven by $q$, $p$, cluster structure, and cut edges, plus a specialized, tighter result for $\beta=1$. The work analyzes how clustering in Stage 1 affects bias and variance via $\widehat{\mathrm{Var}}(\bar{L}_\pi)$ and the cut effect $C(\delta(\Pi))$, highlighting a tension between minimizing cross-cluster edges and balancing covariates across clusters. Through simulations on synthetic lattices and real networks (Email, BlogCatalog, Amazon), it demonstrates a bias-variance trade-off: two-stage designs can substantially reduce variance for higher $\beta$ while incurring bias that can be mitigated by carefully chosen clustering strategies, with covariate-informed clustering sometimes outperforming graph-only clustering. Overall, the approach provides a principled way to improve TTE estimation under interference without full network knowledge, guiding practical design choices for clustering and treatment rollout.

Abstract

Estimating causal effects under interference is pertinent to many real-world settings. Recent work with low-order potential outcomes models uses a rollout design to obtain unbiased estimators that require no interference network information. However, the required extrapolation can lead to prohibitively high variance. To address this, we propose a two-stage experiment that selects a sub-population in the first stage and restricts treatment rollout to this sub-population in the second stage. We explore the role of clustering in the first stage by analyzing the bias and variance of a polynomial interpolation-style estimator under this experimental design. Bias increases with the number of edges cut in the clustering of the interference network, but variance depends on qualities of the clustering that relate to homophily and covariate balance. There is a tension between clustering objectives that minimize the number of cut edges versus those that maximize covariate balance across clusters. Through simulations, we explore a bias-variance trade-off and compare the performance of the estimator under different clustering strategies.

Figures (14)

• Figure 1: Visualization of extrapolated polynomials used to estimate $\textrm{TTE}$ across 200 runs of a rollout experiment on a $20\times20$ lattice with $\beta=3$. The left plot uses a one-stage rollout ($p=0.15$), as in cortez2022neurips, while the right plot uses a two-stage rollout ($q=0.375$). The two-stage design incurs bias, but extrapolation in the one-stage design leads to higher variance.
• Figure 2: Performance of different estimators on the Amazon network for various values of $p$. The bold line indicates the mean over 1000 replications. The shading indicates the experimental standard deviation, calculated by taking the square root of the experimental variance over all replications. The 2-Stage estimator uses $q=0.5$ and does not utilize clustering. Note the scaling of the $y$-axes are not the same across $\beta$.
• Figure 3: Mean Squared Error of the Two-Stage TTE estimator for two clusterings of a $100\times100$ lattice graph, compared with no clustering, for a $\beta$-degree potential outcomes model with $\beta=3$. Even with no network knowledge, we see a drastic decrease in MSE even at the cost of incurring bias.
• Figure 4: Mean Squared Error of the Two-Stage TTE estimator for two clusterings (with 250 clusters) of the Amazon network, compared with no clustering, for a $\beta$-degree potential outcomes model with $\beta=3$. Even with no network knowledge, we see a drastic decrease in MSE even at the cost of incurring bias.
• Figure 5: The degree distribution of the Email graph

Theorems & Definitions (22)

• Definition 3.1: Two-Stage Rollout Design
• Example 3.2: Unit CRD Rollout Design
• Example 3.3: Clustered CRD Rollout Design
• Remark 3.4
• Theorem 4.1
• Corollary 4.2
• Corollary 4.3
• Theorem 4.4
• Remark 4.5
• Lemma 4.6