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Causal inference in connected populations with contagion

This paper addresses the gap in understanding how contagion impacts causal inference by deriving closed-form expressions that reveal the complex interplay between interventions, spillovers, and contagion, while highlighting the resulting statistical biases in standard estimators and proposing potential remedies.

Original authors: Subhankar Bhadra, Michael Schweinberger

Published 2026-06-02

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Subhankar Bhadra, Michael Schweinberger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out if a new fertilizer makes plants grow taller. In a normal garden, you might just compare the plants you treated with the ones you didn't. But now, imagine your garden is a connected community where the plants are actually people holding hands, and they can "catch" feelings or behaviors from each other.

This paper, written by S. Bhadra and M. Schweinberger, tackles the messy reality of figuring out cause-and-effect in these connected groups when contagion (the spread of behavior or outcomes from one person to another) is happening.

Here is the breakdown of their findings using simple analogies:

1. The Problem: The "Ripple Effect" vs. The "Echo"

The authors distinguish between two ways things spread in a group:

Spillover (The Ripple): If you tell your friend about a cool new song (the intervention), they hear it and buy the album. This is a direct ripple from you to them.
Contagion (The Echo): If you buy the album and start dancing to it, your friend sees you dancing, gets excited, and also buys the album. This is contagion. The outcome (buying the album) spreads because of the result of the first person's action, not just the action itself.

The Big Issue: Most standard statistical tools assume that one person's result doesn't depend on their neighbor's result. But in a connected world, if your neighbor gets happy, you might get happy too. If you ignore this "contagion," your math breaks.

2. The Discovery: A Tangled Knot

The authors created new mathematical formulas (closed-form expressions) to untangle three different effects that usually get mixed up:

The Direct Effect: How much did the intervention help the person who got it?
The Spillover Effect: How much did it help the people who didn't get it but knew someone who did?
The Contagion Effect: How much did the results of the first group spread to others?

The Surprising Twist: They found that these three effects are intertwined like a knot. You can't just pull one thread without affecting the others.

Contagion can amplify the effect (making the intervention look super powerful).
Contagion can also dampen the effect (making it look weaker than it is).
Even in the simplest scenarios (like a pair of friends), the math shows that the "direct" effect you think you are measuring is actually a mix of the direct effect, the spillover, and the contagion.

3. The Examples: Households and Communities

To prove their point, they looked at two scenarios:

The Household Scenario: Imagine a world where people only interact with their roommates. If you get a happy boost, your roommate might catch that happiness. The authors show that if you ignore this "catching," you might think your intervention worked differently than it actually did. Sometimes, the happiness of one person bounces back and forth between roommates, making the total effect larger or smaller depending on how strong that "bounce" is.
The Community Scenario: Imagine a large city with different neighborhoods. They found that larger communities contribute more to the total effect than smaller ones. If a big neighborhood catches the "vibe," it changes the overall result more than a small neighborhood would.

4. The Warning: Why Old Tools Fail

The paper warns that if you use standard statistical models (like the ones used in many studies) that ignore this contagion, your results will be biased.

The "Blind Spot": It's like trying to measure the wind speed while standing in a hurricane, but your anemometer (wind gauge) assumes the air is still. The gauge will give you a wrong number.
The "Design" Problem: There are other methods designed to handle connected groups (like assuming people only feel the influence of their immediate neighbors). The authors show that unrestricted contagion breaks these rules too. If a rumor (or a behavior) can travel across the whole network, not just to the next-door neighbor, these "neighborhood" assumptions are violated, and the math fails again.

The Bottom Line

The authors conclude that contagion is a double-edged sword:

The Challenge: It makes calculating the true effect of an intervention incredibly difficult because the effects are tangled together.
The Opportunity: If you understand how contagion works, you can use it. For example, policymakers could identify "influencers" or build specific communities to amplify the positive effects of an intervention, rather than just hoping it spreads naturally.

In short: In a connected world, you can't just look at the person who got the treatment; you have to understand the whole web of connections and how the results "infect" the rest of the group. If you ignore the infection, your math is wrong.

Technical Summary: Causal Inference in Connected Populations with Contagion

Problem Statement
Causal inference in connected populations is complicated by dependence among outcomes induced by real-world processes such as spillover and contagion. While existing literature (e.g., Toulis & Kao, 2013; Ugander et al., 2013; Ogburn et al., 2020) has developed methods to estimate causal effects under interference, there is a significant gap in understanding how contagion—the process where an outcome $Y_i$ influences another outcome $Y_j$ —specifically impacts causal effects and statistical inference. Unlike spillover, where an intervention $X_i$ affects $Y_j$ , contagion involves the transmission of outcomes themselves. The authors note that little is known about how this mechanism alters the magnitude or direction of causal effects, nor how it biases standard estimators.

Methodology and Framework
The authors develop a theoretical framework to derive closed-form expressions for causal effects under contagion. They define the total causal effect $\tau_T$ as the sum of the ego effect ( $\tau_E$ ), representing the direct impact of an intervention on the treated unit, and the alter effect ( $\tau_A$ ), representing the impact of an intervention on other units.

The analysis relies on two primary conditions:

Ignorability (Condition 1): Standard assumptions regarding the independence of interventions and potential outcomes given the network structure.
Linear-in-Means with Contagion (Condition 2): A data-generating process where the conditional expectation of an outcome $Y_i$ $Y_{i}$ depends on:
- The unit's own intervention ( $\beta_i x_i$ ).
- The interventions of neighbors (spillover, $\gamma_i$ ).
- The outcomes of neighbors (contagion, $\delta_i$ ).
- Scaling constants $c_{N,i}(\mathbf{z})$ .

Unlike standard linear-in-means models, this framework allows for dependent residuals due to contagion and does not restrict scaling constants to $1/N_i$ . The authors utilize matrix algebra involving the adjacency matrix $\mathbf{Z}$ and the matrix $\mathbf{M}(\delta, \mathbf{Z}) = (\mathbf{I}_N - \delta \mathbf{C}_N(\mathbf{Z}) \mathbf{Z})^{-1}$ to derive these effects.

Key Contributions and Results

Closed-Form Expressions:
The paper provides the first closed-form expressions for $\tau_E$ , $\tau_A$ , and $\tau_T$ in the presence of both spillover and contagion (Theorems 1 and 2). These expressions reveal that the effects of interventions ( $\beta$ ), spillover ( $\gamma$ ), and contagion ( $\delta$ ) are intertwined, even in simple settings.
- Theorem 1 establishes the identification of these effects using the trace and spectral properties of $\mathbf{M}(\delta, \mathbf{Z})$ .
- Theorem 2 provides simplified expressions when the expected network structure $\mathbf{P} = E(\mathbf{Z})$ approximates the realized network.
Impact of Contagion on Causal Effects:
Through specific examples, the authors demonstrate that contagion can either decrease or increase causal effects, challenging the assumption that interference is purely additive or negligible.
- Restricted Contagion (Households): In a model where connections are restricted to households of size 2, the ego effect $\tau_E$ becomes a convex combination of the direct effect $\beta$ and a term involving $\beta, \gamma,$ and $\delta$ . Specifically, $\tau_E = (1-\chi)\beta + \chi \frac{\beta + \gamma\delta}{1-\delta^2}$ . This shows that contagion ( $\delta$ ) modifies the direct effect, potentially amplifying or diminishing it depending on the sign and magnitude of $\delta$ .
- Unrestricted Contagion (Random Graphs): In sparse and dense interference graphs (including stochastic block models), the authors show that while the ego effect $\tau_E$ may remain $\beta$ under certain growth conditions, the total effect $\tau_T$ is scaled by a factor dependent on $\delta$ . Crucially, contagion cannot cancel or reverse the sign of the total effect if the base effect ( $\beta + \gamma$ ) is non-zero, but it can significantly alter its magnitude.
- Community Size: In degree-corrected block models, the authors find that larger communities contribute more to the total causal effect than smaller ones.
Statistical Implications and Bias:
The paper analyzes the consequences of ignoring contagion in estimation:
- Model-Based Estimators: Ordinary Least Squares (OLS) estimators that ignore outcome dependence (setting $\delta=0$ ) suffer from asymptotic bias. The authors prove that the expected value of these estimators converges to a complex function of $\beta, \gamma, \delta,$ and the network structure, rather than the true causal parameters.
- Design-Based Estimators: While design-based estimators (e.g., Horvitz-Thompson) avoid model misspecification, they rely on "neighborhood exposure" assumptions. The authors demonstrate that unrestricted contagion violates these assumptions because an intervention can affect an outcome through a chain of contagion even if the units are not direct neighbors (Figure 3). Consequently, existing design-based estimators (like the PC balancing estimator) may exhibit non-vanishing bias in the presence of unrestricted contagion.

Significance and Claims
The authors claim that their work provides critical theoretical insight into how contagion fundamentally alters causal inference. The primary significance lies in demonstrating that:

Interventions, spillover, and contagion are inextricably linked; one cannot isolate the effect of an intervention without accounting for how outcomes propagate.
Ignoring contagion leads to biased inference, whether using model-based approaches (due to unmodeled dependence) or design-based approaches (due to violated exposure assumptions).
Contagion is not merely a nuisance parameter; it can actively amplify or dampen the effects of interventions.

The paper concludes that while contagion poses challenges for standard causal inference, it also offers opportunities. For instance, if contagion exists, policymakers could potentially recruit influencers or incentivize community building to amplify intervention effects. However, the authors remain modest regarding practical solutions, noting that remedies for the identified biases (such as restricting contagion or developing new bias-variance reduction methods) are necessary areas for future work.