Connections as treatment: causal inference with edge interventions in networks

TL;DR

This work addresses the causal effect of connections between units in networks by formalizing edge interventions. It develops a design-based framework with local interference and stochastic interventions, pairing inverse probability weighting with constrained exponential random graph models to estimate edge-treatment probabilities. The authors prove identification, consistency, and asymptotic normality of the estimators under suitable dependence conditions and implement a dependent wild bootstrap for variance, validating the approach through simulations. They apply the method to China’s inter-city rail network, finding suggestive but imprecise evidence that increasing local rail connectivity may modestly boost regional economic development, illustrating both feasibility and limitations of edge-level causal inference in real-world networks. This framework provides a principled route for analyzing how network structure and connections causally influence outcomes, with potential applications to transportation, social, and organizational networks.

Abstract

Causal inference has traditionally focused on interventions at the unit level. In many applications, however, the central question concerns the causal effects of connections between units, such as transportation links, social relationships, or collaborative ties. We develop a causal framework for edge interventions in networks, where treatments correspond to the presence or absence of edges. Our framework defines causal estimands under stochastic interventions on the network structure and introduces an inverse probability weighting estimator under an unconfoundedness assumption on edge assignment. We estimate edge probabilities using exponential random graph models, a widely used class of network models. We establish consistency and asymptotic normality of the proposed estimator. Finally, we apply our methodology to China's transportation network to estimate the causal impact of railroad connections on regional economic development.

Figures (10)

• Figure 1: Bias and RMSE of the Hájek estimator $\hat{\theta}^{\delta}_{2}$. The first row displays bias, and the second row displays RMSE. Each column represents a different exclusion neighborhood size $l \in \{3, 5, 7\}$.
• Figure 2: Comparison of estimated and true standard errors (SEs) for $\hat{\theta}^{\delta}_{2}$. Each panel shows standard errors with different values of $\lambda_\delta$. Columns correspond to sample sizes ($n = 300, 1000, 5000$), and rows correspond to exclusion neighborhood sizes ($l = 3, 5, 7$). Solid lines indicate the estimated standard errors. Dashed lines represent the true standard errors accounting for ERGM estimation uncertainty. Dotted lines represent the true standard errors assuming the fitted ERGM is the true model.
• Figure 3: Panel (a) shows the degree distribution of the Chinese railway network. Panel (b) overlays the network on a geographic base map, where nodes represent prefecture-level cities and edges indicate rail connections with travel times under two hours. Node size is proportional to the number of direct connections.
• Figure 4: The impact of China's railway network on the tertiary industry value added under a 2-hour travel-time cutoff. Each panel displays results on for a fixed exclusion neighborhood size ($l \in {3, 4, 5, 6}$). The lines show the estimates of $\theta^\delta-\theta^0$ across varying levels of intervention intensity ($\lambda_\delta$), and the shaded regions indicate 95% pointwise confidence intervals.
• Figure S1: Bias and RMSE of $\hat{\theta}^{\delta}_{1}$. The first row displays bias, and the second row displays RMSE. Each column represents a different exclusion neighborhood size $l \in \{3, 5, 7\}$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition S1
• Lemma S1
• Lemma S2
• Definition S2: Local dependence