the intuition that graph cluster randomization puts vertices in conditions “closer” to the global treatments of interest and thereby reduces bias in estimates of average treatment effects, even if a vertex’s outcome depends on the global treatment vector. The result below uses a linear outcome model that has as a special case the linear-in-means model, as made clear at the end of this subsection.

\[prop:linear\_bias\_reduction\_design\] Assume we have a linear outcome model for all vertices $i \in V$ such that $$\begin{aligned}
{\mathrm{E}}_U[Y_i(z, U)] = a_i + \sum_{j \in V} B_{ij} z_j\end{aligned}$$ and further assume that $Y_i(z, u)$ is monotonically increasing in $z$ for every $u \in \mathbb{U}^N$ and vertex $i$ such that $B_{ij} \geq 0$.

Then for some mapping of vertices to clusters, the absolute bias of $\tau^{d}_\mathrm{ITR}(1,0)$ when the design $d$ is graph cluster randomization is less than or equal to the absolute bias when $d$ is independent assignment, with a fixed treatment probability $p$.

Using the linear model for $Y_i$ and the definition of $\tau$, we have that the true ATE $\tau$ is given by $$\begin{aligned}
\label{eq:linear_outcome_model}
\tau(1,0) = \mu(1) - \mu(0) = \frac{1}{N} \sum_{ij} B_{ij}\end{aligned}$$ for this outcome model. Under graph cluster randomization, $$\begin{aligned}
\tau^\mathrm{gcr}_\mathrm{ITR}(1,0) = \frac{1}{N} 