\sum_{ij} B_{ij} \mathbf{1}[C(i) = C(j)].\end{aligned}$$ Then under independent assignment, $$\begin{aligned}
\tau^\mathrm{ind}_\mathrm{ITR}(1,0) = \frac{1}{N} \sum_{i} B_{ii}.\end{aligned}$$ Because $B_{ij} \geq 0$, together this implies that $\tau(1,0) - \tau^\mathrm{gcr}_\mathrm{ITR}(1,0)  \leq 
\tau(1,0)-\tau^\mathrm{ind}_\mathrm{ITR}(1,0)$, where monotonicity dictates that each side of this inequality is positive.

This comparison allows seeing how, at least in this linear model, the magnitude of bias reduction from graph cluster randomization depends on the “strength” of the interactions within clusters. That is, this clarifies the intuition that using clusters formed from more distant vertices will not generally reduce bias as much as clusters formed from closer vertices, as is the aim of using graph partitioning methods such as $\epsilon$-net partitioning or community detection methods.[^12] It also highlights that when there are mainly non-zero $B_{ij}$’s, *ceteris paribus* large clusters result in more bias reduction; of course, there are corresponding costs to precision.

To clarify this further, let’s consider the relative bias defined by $$\begin{aligned}
\tau^\mathrm{gcr}_\mathrm{ITR}(1,0)/\tau(1,0) - 1 = \frac{\sum_{ij} B_{ij} \mathbf{1}[C(i) = C(j)]}{\sum_{ij} B_{ij}} - 1.\end{aligned}$$ Assume that there are ${\ensuremath{\mathcal{O}}}(N)$ clusters of size ${\ensuremath{\mathcal{O}}}(1)$ used for the graph cluster randomization.[^13] Under this condition, the numerator has ${\ensuremath{\mathcal{O}}}(N)$ terms and the denominator has ${\ensuremath{\mathcal{O}}}(N^2)$ terms. So unless there is a judicious choice of 