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Exploiting Neighborhood Interference with Low Order Interactions under Unit Randomized Design

Mayleen Cortez-Rodriguez, Matthew Eichhorn, Christina Lee Yu

TL;DR

The paper tackles estimating the total treatment effect $\text{TTE}$ under neighborhood interference in networks with bounded degree using unit Bernoulli randomization. It introduces the Structured Neighborhood Interference Polynomial Estimator (SNIPE), which achieves unbiased estimation under a $\beta$-order polynomial potential outcomes model and yields a variance bound that scales polynomially in network degree and exponentially in $\beta$. A minimax lower bound shows that the $\beta$-order structure captures the intrinsic difficulty of estimation, and a central limit theorem establishes asymptotic normality, enabling confidence intervals. Through simulations, the authors demonstrate that SNIPE can outperform standard estimators in mean squared error when $\beta$ is substantially smaller than the graph degree, highlighting the practical value of balancing model flexibility with statistical complexity in causal network inference.

Abstract

Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we provide an unbiased estimator for the TTE when network interference effects are constrained to low order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. We derive a bound on the variance of our estimator and show in simulated experiments that it performs well compared with standard estimators for the TTE. We also derive a minimax lower bound on the mean squared error of our estimator which suggests that the difficulty of estimation can be characterized by the degree of interactions in the potential outcomes model. We also prove that our estimator is asymptotically normal under boundedness conditions on the network degree and potential outcomes model. Central to our contribution is a new framework for balancing model flexibility and statistical complexity as captured by this low order interactions structure.

Exploiting Neighborhood Interference with Low Order Interactions under Unit Randomized Design

TL;DR

The paper tackles estimating the total treatment effect under neighborhood interference in networks with bounded degree using unit Bernoulli randomization. It introduces the Structured Neighborhood Interference Polynomial Estimator (SNIPE), which achieves unbiased estimation under a -order polynomial potential outcomes model and yields a variance bound that scales polynomially in network degree and exponentially in . A minimax lower bound shows that the -order structure captures the intrinsic difficulty of estimation, and a central limit theorem establishes asymptotic normality, enabling confidence intervals. Through simulations, the authors demonstrate that SNIPE can outperform standard estimators in mean squared error when is substantially smaller than the graph degree, highlighting the practical value of balancing model flexibility with statistical complexity in causal network inference.

Abstract

Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we provide an unbiased estimator for the TTE when network interference effects are constrained to low order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. We derive a bound on the variance of our estimator and show in simulated experiments that it performs well compared with standard estimators for the TTE. We also derive a minimax lower bound on the mean squared error of our estimator which suggests that the difficulty of estimation can be characterized by the degree of interactions in the potential outcomes model. We also prove that our estimator is asymptotically normal under boundedness conditions on the network degree and potential outcomes model. Central to our contribution is a new framework for balancing model flexibility and statistical complexity as captured by this low order interactions structure.
Paper Structure (25 sections, 7 theorems, 95 equations, 2 figures, 2 tables)

This paper contains 25 sections, 7 theorems, 95 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Causal Network
  5. Potential Outcomes Model
  6. Examples of Low Degree Interaction Models
  7. Causal Estimand and Randomized Design
  8. Estimator
  9. Building Intuition in the Linear Setting (beta = 1)
  10. SNIPE in the General Polynomial Setting
  11. Connection to Other Estimator Classes
  12. Properties of Estimator under Bernoulli Design
  13. Minimax Lower Bound on Mean Squared Error
  14. Central Limit Theorem
  15. Variance Estimator
  16. ...and 10 more sections

Key Result

Theorem 1

Under a potential outcomes model satisfying the neighborhood interference assumption with polynomial degree at most $\beta$, the estimator defined in eqn:TTE_PI is unbiased with variance upper bounded by where each $p_i \in [p,1-p]$ and $p>0$.

Figures (2)

  • Figure 1: Plots visualizing the performance of various $\textrm{\normalfont TTE}$ estimators under Bernoulli design on Erdős-Rényi networks for both linear and quadratic potential outcomes models. The height of each line on a plot depicts the experimental relative bias of the estimator and the shaded width depicts the experimental standard deviation. The SNIPE estimator is parametrized by $\beta$, the degree of the potential outcomes model.
  • Figure 2: Plots visualizing the MSE of various $\textrm{\normalfont TTE}$ estimators under Bernoulli design on Erdős-Rényi networks for both linear and quadratic potential outcomes models. The height of each line on a plot depicts the mean squared error when the model is normalized so that the true $\textrm{\normalfont TTE}$ is effectively equal to $1$. Alternatively, we can think of this as the variance of the normalized estimates. Our estimator under a $\beta$-order potential outcomes model is denoted SNIPE$(\beta)$ in the figure.

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Remark 4
  • Remark 5
  • ...and 12 more