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Experimentation on Endogenous Graphs

Wenshuo Wang, Edvard Bakhitov, Dominic Coey

TL;DR

This paper addresses causal inference under endogenous network interference, where the treatment can alter the interference graph itself. It introduces an endogenous bipartite graph model with $r$-driven edges and an anchor-subgraph instrument to identify and estimate the total treatment effect (TTE), proving unbiasedness, consistency, and asymptotic normality under sparsity. The framework extends to endogenous unipartite graphs and is supported by simulations showing reduced bias compared with exposure-based estimators, along with a sharp-null testing procedure. Practically, the method enables valid inference in settings such as social networks and multi-sided platforms where treatment propagates through a dynamic, treatment-influenced graph, expanding the toolbox for network causal inference.

Abstract

We study experimentation under endogenous network interference. Interference patterns are mediated by an endogenous graph, where edges can be formed or eliminated as a result of treatment. We show that conventional estimators are biased in these circumstances, and present a class of unbiased, consistent and asymptotically normal estimators of total treatment effects in the presence of such interference. We show via simulation that our estimator outperforms existing estimators in the literature. Our results apply both to bipartite experimentation, in which the units of analysis and measurement differ, and the standard network experimentation case, in which they are the same.

Experimentation on Endogenous Graphs

TL;DR

This paper addresses causal inference under endogenous network interference, where the treatment can alter the interference graph itself. It introduces an endogenous bipartite graph model with -driven edges and an anchor-subgraph instrument to identify and estimate the total treatment effect (TTE), proving unbiasedness, consistency, and asymptotic normality under sparsity. The framework extends to endogenous unipartite graphs and is supported by simulations showing reduced bias compared with exposure-based estimators, along with a sharp-null testing procedure. Practically, the method enables valid inference in settings such as social networks and multi-sided platforms where treatment propagates through a dynamic, treatment-influenced graph, expanding the toolbox for network causal inference.

Abstract

We study experimentation under endogenous network interference. Interference patterns are mediated by an endogenous graph, where edges can be formed or eliminated as a result of treatment. We show that conventional estimators are biased in these circumstances, and present a class of unbiased, consistent and asymptotically normal estimators of total treatment effects in the presence of such interference. We show via simulation that our estimator outperforms existing estimators in the literature. Our results apply both to bipartite experimentation, in which the units of analysis and measurement differ, and the standard network experimentation case, in which they are the same.

Paper Structure

This paper contains 23 sections, 9 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Our Contributions
  4. Related Work
  5. Outline of the Paper
  6. Setup
  7. Examples of Edge Endogeneity Bias.
  8. Models of Edge Formation
  9. Detecting general departures from edge exogeneity.
  10. Detecting $r$-driven edges.
  11. A Class of Unbiased TTE Estimators
  12. Generalized Linear Exposure-Response Model
  13. Estimation Strategy
  14. Using Anchor Instruments
  15. Statistical Testing under the Sharp Null
  16. ...and 8 more sections

Key Result

Theorem 1

In the setting of Definition def:ebg, suppose $G$ is a known anchor subgraph. If (a) edges are $r$-driven; (b) the treatment assignments are independent Bernoulli random variables with probability $p$; (c) $\{(a,r)\mid u_{ar}\ne0\}\subset G$; (d) $c_{ar}=\mathbb I((a,r)\in G)$; (e) $w_{ar}\ne0$ for is an unbiased estimator of the TTE in equation:gate.

Figures (3)

  • Figure 1: Endogenous graph example. Note that after the treatment is applied, edges (A-1) and (B-4) disappear, while new edges (B-1) and (C-3) emerge.
  • Figure 2: Bias with respect to different parameter values. Error bars are empirical standard deviations of the estimators over 1000 Monte Carlo runs.
  • Figure 3: 95%-CI coverage from 1000 independent Monte Carlo runs with respect to different parameter values. Error bars represent the $95\%$-CI for the true coverage probability.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Example 3
  • Definition 3: Anchor Subgraph
  • Theorem 1
  • ...and 14 more