SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb {E} [ \operatorname {s g n} (X) \operatorname {s g n} (Y) ] = \mathbb {P} (\operatorname {s g n} (X) \operatorname {s g n} (Y) > 0) - \mathbb {P} (\operatorname {s g n} (X) \operatorname {s g n} (Y) <   0) \\ = 2 \mathbb {P} (\operatorname {s g n} (X) \operatorname {s g n} (Y) > 0) - 1 \\ = 2 \left(\frac {1}{\pi} (\pi - \operatorname {a r c c o s} (\langle x, y \rangle)) - 1 \right. \tag {47} \\ = 1 - \frac {2}{\pi} \arccos (\langle x, y \rangle) \\ = \frac {2}{\pi} \arcsin (\langle x, y \rangle) \\ \end{array}
$$
which gives the desired outcome.
B Simulation Details
B.1 Methods
We consider following linear potential outcome model,
$$
Y _ {i} (\boldsymbol {z}) = \alpha + \beta \cdot z _ {i} + c \cdot \frac {d _ {i}}{\bar {d}} + \sigma \cdot \epsilon_ {i} + \gamma \frac {\sum_ {j \in N _ {i}} z _ {j}}{d _ {i}} \tag {48}
$$
$$
Y _ {i} (\boldsymbol {z}) = \left(\alpha + \sigma \cdot \epsilon_ {i}\right) \cdot \frac {d _ {i}}{\bar {d}} \cdot \left(1 + \beta z _ {i} + \gamma \frac {\sum_ {j \in N _ {i}} z _ {j}}{d _ {i}}\right) \tag {49}
$$
We set the clusters as given by Louvain algorithm with fixed random seed and resolution parameter as 2, 5, 10.
We consider social network FB-Stanford3[32], and FB-Cornell5[32]. These two social networks provide the network topology, and we generate potential outcome for each unit with mentioned potential outcome models.
Besides our optimized covariance design (OCD), we implement following randomization schemes. We first consider two baselines, independent Bernoulli randomization (Ber) and complete randomization (CR), both of which are cluster-level. We also implement two adaptive schemes, rerandomized-adaptive randomization (ReAR) and pairwise-sequential randomization (PSR) [25, 31], which balance heuristic covariates adaptively and act as competitive baselines, since the average degree is considered as a covariate in these two methods and exactly appear in both two of our models, explicitly. Then we implement the independent block randomization (IBR) [5] and heuristic version IBR-p that creates blocks with size 2.
In the methods mentioned above, ReAR is the only one concerned with hyperparameters setting. We set  $(q,B,\alpha) = (0.85,400,0.1)$ , which corresponds to the recommendation in the original paper.
We estimate GATE with standard HT estimator and difference-in-means (DIM) estimator, where the latter refers to
$$
\hat {\tau} _ {D I M} = \sum_ {i \in [ n ]} \left(\left(\frac {z _ {i}}{\sum_ {j \in [ n ]} z _ {j}} - \frac {(1 - z _ {i})}{\sum_ {j \in [ n ]} (1 - z _ {j})}\right) Y _ {i} (z)\right) \tag {50}
$$
To summarize, we provide the bias, standard deviation and MSE of two estimators under two potential outcome models, three  $\gamma$  levels and two datasets. All of these three metrics are calculated by repetition of Monte Carlo simulation of randomization, 10,000 times. In the main paper we've presented the results of HT estimator on FB-Stanford3, and we'll present the detailed results in this section.
B.2 Discussion on Estimators
Here we also provide discussion on another popular estimator that's considered in existing literature, which is the HT estimator with exposure indicator. We consider the exposure condition that's fully treated or fully controlled in 1-hop neighborhood here, and the corresponding indicator is defined as  $\delta_{i}(z_{0}) = \mathbb{I}\{\sum_{j\in N_{i}}z_{j} = d_{i}z_{0}\}$ . The estimator is
$$
\hat {r} ^ {\prime} = \frac {1}{n} \sum_ {i \in [ n ]} \left(\left(\frac {\delta_ {i} (1)}{\mathbb {E} \left[ \delta_ {i} (1) \right]} - \frac {\delta_ {i} (0)}{\mathbb {E} \left[ \delta_ {i} (0) \right]}\right) Y _ {i} (\boldsymbol {z})\right) \tag {51}
$$
We don't consider this estimator not only because of it's high variance resulted from low effective sample size, and but also its high calculation cost. For calculating it in our repeated simulations, we need to estimate  $\mathbb{E}[\delta_i(1)]$ , the probability of fully treated, for each unit  $i$ . We denote the number of clusters a unit  $i$  connects to by  $c_{i}$ . For a unit  $i$  in the exterior of its cluster, namely,  $c_{i} > 1$ , such a quantity can very high in a social network, which is also decided by the resolution of clustering, we present an instance in figure 1.
Unfortunately, we must estimate such generalized propensity score [10] by Monte Carlo simulation for most of randomization schemes, even if it's just a little bit more complex than independent Bernoulli randomization, where such quantity can be calculated directly,  $(1/2)^{c_i}$ . For every simulation, we should visit every node and query the treatment assignment of its 1-hop neighborhood, whose time complexity is  $O(|E|)$ . Roughly, we need  $2^{c_i}$  repetitions of randomization to achieve effective estimation on  $\delta_i(1), \delta_i(0)$ , which is too time-consuming to be acceptable for many units.
$^{4}$ Network topology data can be found in https://networkrepository.com/socfb-Stanford3.php  
 $^{5}$ https://networkrepository.com/socfb-Cornell5.php
We stress the computational issue here since the resulted variance can be reduced by self-normalization, which corresponds to the Hajek estimator.
$$
\hat {\tau} ^ {\prime} = \sum_ {i \in [ n ]} \left(\left(\frac {\frac {\delta_ {i} (1)}{\mathbb {E} \left[ \delta_ {i} (1) \right]}}{\sum_ {j \in [ n ]} \frac {\delta_ {j} (1)}{\mathbb {E} \left[ \delta_ {j} (1) \right]}}\right) - \left(\frac {\frac {\delta_ {i} (0)}{\mathbb {E} \left[ \delta_ {i} (0) \right]}}{\sum_ {j \in [ n ]} \frac {\delta_ {j} (0)}{\mathbb {E} \left[ \delta_ {j} (0) \right]}}\right)\right) Y _ {i} (\boldsymbol {z}) \tag {52}
$$
However, we argue that the computational cost can't be bypassed without further modification and restricts the practicality of such HT estimator with exposure indicator. [25] proposes the so-called cluster-adjusted estimator that assign the units on the exterior of the same cluster the same modified propensity score, which reduces the computation complexity at the cost of unpredictable distortion of estimator, which can be viewed as heuristics.
Therefore, we choose to consider the standard HT estimator and DIM estimator in our simulation.
B.3 Detailed Simulation Results
We present the results according to following sequence: HT-DIM estimator, linear-multiplicative model, 2-5-10 clustering resolution. In every table, we report the average bias, standard deviation (SD) and MSE of randomization schemes with three  $\gamma$  levels. There are 24 tables in total, which are organized to demonstrate the advantage and robustness of our proposed method, optimized covariance design (OCD).