Data-Adaptive Exposure Thresholds under Network Interference

We thank Reviewer Y8AE for the thoughtful questions. We appreciate it, and would like to address the reviewer’s questions and concerns below, and hope that our response provides more clarity on the focus of our work.

Response to Concern 1 / Question 1:

We are not assuming a linear potential outcome model. Our model is much more general (please refer to eq. 1). We are simply using linear regression for a first-order approximation to the potential outcomes model for bias estimation. Only in the special case of a true linear model is this exact. We would also like to draw the reviewer’s attention to Baird et al.(2018) , where the authors leverage the slope of spillovers with respect to treatment saturation . Similar to our approach, this work uses the linear model, i.e. the slope, of the spillovers for an approximation without assuming it as the generating model. This is discussed in lines 151–157 of the paper.

In Corollary 4.6, we provide error bounds for models of the form presented on line 112. To demonstrate the method’s robustness beyond linear settings, we report simulation results for nonlinear potential outcomes models in Section A.7.4. Finally, we extend our method to incorporate local regression approaches in Section A.7.6, and apply these to nonlinear outcome models as well. This is discussed in lines 301–313 of the paper.

Question 2: “Is it possible to estimate the effect of the treatment itself, excluding the influence of the exposure?”.. This is a great question. Extending our approach to other estimands, such as the direct effect, is indeed possible, though the characterization of bias and variance would be somewhat different. For example, to estimate the direct effect, one might focus on treated units whose neighbors are all assigned to control, yielding an unbiased estimate under certain conditions. In this case, rather than defining extremes as $(z_i=1,e_i \geq h)$ and $(z_i=0, e_i \leq 1−h)$, one might instead consider $(z_i=1,e_i \in x_h)$ and $(z_i=0,e_i \in x_h)$ for some interval $x_h=[x−h,x+h]$, where $x_h$ forms part of a cover of $[0,1]$. While the underlying principles remain similar, the bias–variance trade-offs do not translate in a straightforward manner between estimands, and would need to be re-analyzed in each case.

We hope our response has provided more clarity on the motivations and the focus of our work. We thank the reviewer once again for the thoughtful questions, and we hope that this discussion has enhanced your perspective on our paper.

We thank Reviewer mQMx very much for the encouraging feedback and the thoughtful comments and questions. We would like to address them below.

Response to Question 1: This corresponds to a cluster-level randomization design, where clusters are first defined, and then randomization is performed at the cluster level, i.e., if a cluster is assigned to treatment (resp. control), all units within that cluster receive treatment (resp. control).

For Figures 2, 3, and 4: The node colors in Figure 2 reflect the clustering used in our experiments for both the graph generated from the stochastic block model (SBM) and the real-world network data. For the SBM, we used the underlying block memberships (from which the graph was generated) as the clusters. For the real dataset, we applied $\epsilon$-net clustering with $\epsilon$ = 1. In $\epsilon$-net clustering, nodes in the net are at least ε apart (in shortest path distance in the graph), and the union of their ε-radius neighborhoods (balls) covers the entire graph. These were described briefly in the captions of the figures. We apologize if this was unclear, and we are happy to improve on this clarity in the revision. We also refer the reviewer to Ugander et al. (2013) for a more detailed discussion of this clustering method. For the k-th power cycle graphs, clusters are formed by grouping consecutive $2k + 1$ nodes. This specific clustering choice follows the design in Ugander et al. (2013), where it is shown to achieve minimum variance. We discuss this in lines 533–534 in the Appendix.

Regarding Questions 2 and 3: Thank you very much for pointing this out. This is correct, and we will make this assumption more explicit in Theorem 4.5. Corollary 4.6 does assume the slightly more restrictive form of $\psi(z,e) = g(z) + f(e)$. We will also make that clearer in the revision. Apologies for any confusion caused. Additionally, regarding clarity in Figure 1, we are happy to improve this in the revised version.

Thank you very much again for all of your insightful questions and feedback.

We thank Reviewer rfKt very much for the encouraging feedback, and the thoughtful and detailed comments and questions. We would like to address them below:

Response to Question 1: Thank you for pointing this out. We will be clarifying and improving the notation there.

Response to Question 2: Thank you for pointing this out. We will fix the typo in the proof.

Response to Question 3: Thank you for this feedback. We agree that the notion of the “bias slope” would benefit from an earlier introduction and clearer motivation, and we will incorporate this in the revision. To clarify the intuition: as you point out, the regression coefficient estimates the average change in the mean outcome per unit change in exposure. In our context, this also reflects the average change in bias as we move away from the boundary exposures, i.e., from 0 for control or 1 for treated, since bias arises (depending on the outcome function) when exposures deviate from these extremes. This interpretation motivates the use of the weights $e_i- 0 = e_i$ and $1− e_i$ in Equation (6), corresponding to deviations from the control and treated boundaries, respectively. Equation (6) defines a Horvitz-Thompson-style estimator for the bias, which integrates this regression slope over the empirical exposure distribution to yield an average (absolute) bias estimate at a given threshold.

Response to Question 4: This comes from the $n(\Delta_n - \tilde{\delta})$ terms in the bounds in Corollary 4.6. Indeed, when these terms are larger, the $\exp(- (n(\Delta_n - \tilde{\delta}))$ terms are small. We hope this is helpful, and we are happy to add this explanation in the revision.

Response to Question 5: Thank you for pointing this out. The issue you observed was due to this particular illustrative plot having been generated using an earlier version of the code, with significantly fewer Monte Carlo trials. This was an oversight; all other plots and results in the paper had already been updated to use 1,000 Monte Carlo trials for bias computation and 10,000 for exposure probability estimation. We have now regenerated this specific plot using the updated setup. As expected, the bias at threshold 1 is numerically close to zero across all $\gamma/\beta$ ratios, consistent with your intuition, and bias remains near zero for $\gamma/\beta = 0$. We will replace the outdated figure in the revision to ensure consistency with the rest of the results.

Response to Question 6: Thank you for pointing this out. Upon reviewing the code used to generate this plot, we discovered an inconsistency: the ATE used to normalize the RMSE was inadvertently computed using a $\gamma/\beta$ ratio of $10^6$, while the RMSE values themselves were computed using the correct ratio of $10^4$. This mismatch led to an artificially low normalized RMSE for the final data point. The value $\gamma/\beta = 10^6$ was used elsewhere in the cluster-randomized setting to illustrate the effects of stronger interference, as cluster randomization helps control interference bias. Unfortunately, that parameter setting was mistakenly carried over when computing the ATE for this plot. We have since corrected the figure, and the updated version behaves as expected: the normalized RMSE no longer drops sharply for the final ratio and reflects a consistent trend across all $\gamma/\beta$ ratios. We apologize for this oversight and will include the corrected figure in the revision. We have verified that this issue is isolated to this plot and does not affect any other results in the paper.

Once again, we very much appreciate the reviewer's thoughtful and detailed feedback.

We thank Reviewer Szy4 for the thoughtful review and questions. We would like to address the questions and concerns the reviewer has posed below.

Response to Question 1: The Horvitz-Thompson estimator is used heavily in practice! Motivated by this, we sought to address our framework through the setting of the Horvitz-Thompson estimator. Our framework, of course, could be adapted for the Hajek, Difference-in-Means, AIPW etc. estimators. We formulate the adaptively-thresholded estimator for the Difference-in-Means estimator, which is a special case of the Hajek estimator, in the Appendix (Section A.7.5.). A similar adaptation to the Hajek estimator could be used. We note however, that the Hajek estimator is only approximately unbiased at the true threshold. Therefore, characterizing the exact bias-variance tradeoff formally would be more complex compared to the Horvitz-Thompson estimator. We discuss this in lines 74-80 of the paper.

Response to Question 2: We appreciate the curiosity of how the performance of our approach compares to direct modelling approaches. This, however, is not the focus of our paper. Instead, our work focuses on an adaptive optimal threshold for a specific estimator class decided upon by the practitioner a priori. In particular, our goal is to show that there is something to be gained when using an adaptive estimator, instead of a fixed estimator like HT(0) or HT(1), as in our approach. The only other viable adaptive approach that exists for that purpose is Lepski’s method which we had adapted to our settings. Therefore, we believe that the useful benchmarks for comparisons are the estimators adapted from Lepski’s method, and the fixed Horvitz-Thompson estimators.

We would also like to emphasize that, while we use linear regression to obtain a first-order approximation of how interference bias varies with exposures, which is similar in spirit to Baird et al. (2018), we go further by pairing this with a Horvitz-Thompson-style bias estimator. This approach uses the estimated regression coefficients in conjunction with exposures, enabling the bias estimator to account for the interaction between regression structure and exposure distribution present in the data.

We hope that our response has provided more clarity on the motivations and the focus of our work. We thank the reviewer once again for the thoughtful feedback, and we hope that this discussion has enhanced your perspective on our paper.