Learning Exposure Mapping Functions for Inferring Heterogeneous Peer Effects

TL;DR

This work tackles estimating heterogeneous peer effects under network interference by learning the exposure mapping function from data. It introduces EgoNetGnn, a graph neural network architecture that constructs ego networks, learns a bounded, informative peer-exposure representation, and integrates it with a counterfactual outcome model trained end-to-end via TARNet/CFR with balance and coverage priors. The method shows superior expressiveness to capture complex local influence mechanisms (e.g., causal network motifs) and delivers more accurate HPE estimates (lower ${\epsilon_{PEHE}}$) across synthetic, semi-synthetic, and real networks, relative to strong baselines. By automating exposure representation learning, EgoNetGnn reduces subjective exposure specification and enhances robustness to unknown peer-influence structures, with implications for policy design and targeted interventions in networked settings.

Abstract

In causal inference, interference refers to the phenomenon in which the actions of peers in a network can influence an individual's outcome. Peer effect refers to the difference in counterfactual outcomes of an individual for different levels of peer exposure, the extent to which an individual is exposed to the treatments, actions, or behaviors of peers. Estimating peer effects requires deciding how to represent peer exposure. Typically, researchers define an exposure mapping function that aggregates peer treatments and outputs peer exposure. Most existing approaches for defining exposure mapping functions assume peer exposure based on the number or fraction of treated peers. Recent studies have investigated more complex functions of peer exposure which capture that different peers can exert different degrees of influence. However, none of these works have explicitly considered the problem of automatically learning the exposure mapping function. In this work, we focus on learning this function for the purpose of estimating heterogeneous peer effects, where heterogeneity refers to the variation in counterfactual outcomes for the same peer exposure but different individual's contexts. We develop EgoNetGNN, a graph neural network (GNN)-based method, to automatically learn the appropriate exposure mapping function allowing for complex peer influence mechanisms that, in addition to peer treatments, can involve the local neighborhood structure and edge attributes. We show that GNN models that use peer exposure based on the number or fraction of treated peers or learn peer exposure naively face difficulty accounting for such influence mechanisms. Our comprehensive evaluation on synthetic and semi-synthetic network data shows that our method is more robust to different unknown underlying influence mechanisms when estimating heterogeneous peer effects when compared to state-of-the-art baselines.

Figures (10)

• Figure 1: Illustration of different possible peer exposure representations for a node (Gaby) in a toy peer network. Red nodes represent peers in the treatment group, and blue nodes represent peers in the control group. Gray star node represents the node that has a fixed treatment.
• Figure 2: An overview of the proposed EgoNetGnn model to learn exposure mapping function for peer effect estimation. EgoNetGnn extracts ego networks, for each node $v_i$, with peer treatments along with feature embedding and its edge attributes as node attributes. Then, node-level aggregations are performed to capture local neighborhood contexts. These contexts are passed through a masked weight layer and encoded by an multi-layer perceptron (MLP) to learn relevant influence mechanisms and summarized with graph-level aggregation. The learned peer exposure embeddings ($\bm{\rho}_i$), along with the feature embeddings (${\mathbf{c}}_i$), and treatment ($t_i$) are passed to a counterfactual outcome model that is used to infer peer effects. The graph transformation ensures expressiveness, while balance, coverage, entropy, and sparsity losses promote the robustness of the peer exposure representation.
• Figure 3: Peer effect estimation error for Barabasi Albert network when true peer exposure depends on mutual connections, clustering coefficient, and attribute similarity. Our method shows robust performance across different underlying peer influence mechanisms and edge densities (low to high).
• Figure 4: Example causal network motifs considered by yuan-www21. Stars represent ego nodes and circles represent their peers. The red circles indicate treated nodes and blue circles indicate control nodes. The gray shapes indicate nodes that could either be treated or control. Here, the characters in red indicate a particular causal network motif (e.g., 3c-2 indicate closed triad with 2 treated peers).
• Figure 5: Examples of higher-order network motifs with four and five nodes. Stars represent ego nodes and circles represent their peers. The gray shapes indicate nodes with any treatment assignment. If the subgraph of a network motif, after removing edges connected to the ego node, forms a tree, then our model is expressive enough to capture the network motif and the corresponding causal network motifs. A network motif is a subgraph without any attributes, whereas a causal network motif is a subgraph that includes peer treatment assignments as attributes.

Theorems & Definitions (14)

• Definition 1: Peer exposure and exposure mapping function
• Proposition 1
• proof
• Definition 2: Subgraph
• Definition 3: Induced subgraph
• Definition 4: Star-shaped pattern
• Definition 5: Connected pattern
• Definition 6: Expressiveness in counting causal network motifs
• Proposition 2: Expressiveness of EgoNetGnn
• proof