The Conflict Graph Design: Estimating Causal Effects under Arbitrary Neighborhood Interference

TL;DR

The paper introduces the Conflict Graph Design (CGD) for estimating causal effects under arbitrary neighborhood interference in networks. It builds a conflict graph from the underlying network and the estimand, uses an importance ordering to resolve incompatible exposures, and employs a modified Horvitz–Thompson estimator whose variance scales with the largest eigenvalue of the conflict graph, $\lambda(\mathcal{H})$, as $O(\lambda(\mathcal{H})/n)$. The authors provide finite-sample and asymptotic variance bounds, a conservative variance estimator, and two types of confidence intervals, along with a spectral-analysis-based comparison to prior dependency-graph approaches. They demonstrate improvements for global, direct, and spill-over effects and validate the framework via numerical simulations on preferential-attachment networks, while acknowledging limitations and open questions about optimality and multi-estimand designs.

Abstract

A fundamental problem in network experiments is selecting an appropriate experimental design in order to precisely estimate a given causal effect of interest. In this work, we propose the Conflict Graph Design, a general approach for constructing experiment designs under network interference with the goal of precisely estimating a pre-specified causal effect. A central aspect of our approach is the notion of a conflict graph, which captures the fundamental unobservability associated with the causal effect and the underlying network. In order to estimate effects, we propose a modified Horvitz--Thompson estimator. We show that its variance under the Conflict Graph Design is bounded as $O(λ(H) / n )$, where $λ(H)$ is the largest eigenvalue of the adjacency matrix of the conflict graph. These rates depend on both the underlying network and the particular causal effect under investigation. Not only does this yield the best known rates of estimation for several well-studied causal effects (e.g. the global and direct effects) but it also provides new methods for effects which have received less attention from the perspective of experiment design (e.g. spill-over effects). Finally, we construct conservative variance estimators which facilitate asymptotically valid confidence intervals for the causal effect of interest.

Figures (23)

• Figure 1: Example conflict graphs for the Direct Treatment Effect (DTE) and the Global Average Treatment Effect (GATE). For each effect, the left panel shows the contrastive exposures, the middle panel shows examples of conflicting units, and the right panel shows the full conflict graph.
• Figure 2: Illustration of the Conflict Graph Design for Direct Effect and GATE. The left panel contains the original graph. The middle panel shows the conflict graph, an importance ordering, and the first step of the algorithm: sampling a realization of the desired exposures $U_i$. The right panel shows the resulting intervention on the original graph, where a dashed circle around unit $i$ indicates that the event $E_{(i, k)}$ occurred.
• Figure 3: Comparison of the variance of various experimental designs. Figure \ref{['fig:gate-variances']} compares RGCR and Conflict Graph Design for estimating the Global Average Treatment Effect. Figure \ref{['fig:dte-variances']} shows the Independent Set Design and Conflict Graph Design for estimating the Direct Treatment Effect.
• Figure 4: Four distinct exposures which all correspond to "50% of neighbors are treated"
• Figure 5: Figure \ref{['fig:illustration-original-network']} depicts a star graph as the original underlying network $G$. Figure \ref{['fig:illustration-conflict-graph']} depicts the conflict graph $\mathcal{H}$ for the direct effect, which in this case is the original graph $G$. Figure \ref{['fig:illustration-for-dep-graph-analysis']} depicts the dependency graph $\mathcal{D}$ which is the complete graph, as the treatment assignment of each node is correlated under the CGD.

Theorems & Definitions (132)

• Definition 1: Conflicting Units
• Definition 2: Importance Ordering
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Corollary 3.4
• Proposition 3.5
• Theorem 4.1
• Lemma 4.2
• proof : Proof Sketch of Theorem \ref{['thm:variance-analysis-finite-sample']}