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Identifying the Group to Intervene on to Maximise Effect Under Cross-Group Interference

Xiaojing Du, Jiuyong Li, Lin Liu, Debo Cheng, Jixue Liu, Thuc Duy Le

Abstract

In many networked systems, interventions applied to one group of units can induce substantial causal effects on another group through cross-group interference pathways. Despite its practical importance in domains such as public health, digital marketing, and social policy, the problem of identifying which intervention subset in a source group maximizes the benefit on a target group remains largely unaddressed. We formalize this problem as cross-group causal influence estimation and introduce the core-to-group causal effect (Co2G), a formally defined causal estimand that quantifies the contrast in target-group outcomes under intervention versus non-intervention on a candidate source subset. We establish the nonparametric identifiability of Co2G from observational network data using do-calculus under standard causal assumptions, and develop a graph neural network-based estimator that captures cross-group interference patterns. To navigate the combinatorial search space of candidate subsets, we propose CauMax, an uncertainty-aware causal effect maximization framework with two scalable selection algorithms: (i)CauMax-G, an iterative greedy search with Monte Carlo dropout--based lower confidence bounds, and (ii)CauMax-D, a differentiable gradient-based optimization via Gumbel-Softmax relaxation. Extensive experiments on two real-world social networks demonstrate that CauMax achieves an order-of-magnitude reduction in regret compared with structural heuristics and diffusion-based baselines, and that moderate uncertainty penalization consistently improves subset selection quality.

Identifying the Group to Intervene on to Maximise Effect Under Cross-Group Interference

Abstract

In many networked systems, interventions applied to one group of units can induce substantial causal effects on another group through cross-group interference pathways. Despite its practical importance in domains such as public health, digital marketing, and social policy, the problem of identifying which intervention subset in a source group maximizes the benefit on a target group remains largely unaddressed. We formalize this problem as cross-group causal influence estimation and introduce the core-to-group causal effect (Co2G), a formally defined causal estimand that quantifies the contrast in target-group outcomes under intervention versus non-intervention on a candidate source subset. We establish the nonparametric identifiability of Co2G from observational network data using do-calculus under standard causal assumptions, and develop a graph neural network-based estimator that captures cross-group interference patterns. To navigate the combinatorial search space of candidate subsets, we propose CauMax, an uncertainty-aware causal effect maximization framework with two scalable selection algorithms: (i)CauMax-G, an iterative greedy search with Monte Carlo dropout--based lower confidence bounds, and (ii)CauMax-D, a differentiable gradient-based optimization via Gumbel-Softmax relaxation. Extensive experiments on two real-world social networks demonstrate that CauMax achieves an order-of-magnitude reduction in regret compared with structural heuristics and diffusion-based baselines, and that moderate uncertainty penalization consistently improves subset selection quality.
Paper Structure (34 sections, 2 theorems, 13 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 34 sections, 2 theorems, 13 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Cross-Group Spillovers in Networked Systems
  4. Influence Maximization
  5. Problem Setup and Cross-Group Interference
  6. Notation.
  7. Networked Groups and Interference Structure.
  8. Subset-Level Intervention.
  9. Interventional Target-Group Mean.
  10. Core-to-Group Causal Effect.
  11. Cross-Group Causal Influence Maximization.
  12. Method
  13. Identifiability of the Cross-Group Causal Effect
  14. Causal Effect Estimator
  15. Estimator Overview.
  16. ...and 19 more sections

Key Result

Theorem 4.1

Under Assumptions ass:markov to ass:positivity and the causal graph in Fig. fig:co2g, the interventional target-group mean $\mu_B(t;S)$ is identifiable from the observational distribution $p(X,T_{V_A},Y_B)$ and admits the following representation:

Figures (4)

  • Figure 1: Illustration of the cross-group causal interference problem. The source group $A$ forms a complex interaction structure, and an intervention can be applied to any candidate subset. The goal is to identify the subset that creates the maximal effect on target group $B$. The causal mechanism operates through cross-group interference pathways from $A$ to $B$. For a subset, the counterfactual perspective compares the target group-level outcomes under intervention and non-intervention on a subset. Red nodes denote a candidate intervention subset.
  • Figure 2: Cross-group causal structure and network-to-causal abstraction. (a) Network representation conditioned on a candidate subset $S \subseteq V_A$, with source units partitioned into the intervened subset $S$ and remaining units $V_A \setminus S$. (b) The core-to-group causal effect (Co2G), contrasting the target group outcome $Y_B$ under intervention versus non-intervention on $S$. (c) Corresponding causal graph with subset-level variables: $X_S, X_{V_A\setminus S}, X_{V_B}$ (pre-treatment covariates), $T_S, T_{V_A\setminus S}$ (treatment assignments), and $Y_B$ (target outcomes).
  • Figure 3: Sensitivity analysis of the uncertainty penalty $\lambda$ on $\text{Regret@}K$ across both datasets and all budget levels $K \in \{5,10,15,20,30,50\}$. Solid bars represent BC and hatched bars represent Flickr.
  • Figure 4: Sensitivity analysis of the uncertainty penalty $\lambda$ on RMSE.

Theorems & Definitions (2)

  • Theorem 4.1: Identifiability of the Interventional Target-Group Mean
  • Corollary 4.2: Identifiability of the Core-to-Group Causal Effect