\label{eq:bias_in_effective_contribution}
{\mathrm{E}}^d[Y_i - Y_i(Z = z) {\, | \,}g_i(Z) = g_i(z)],\end{aligned}$$ where $g_i(\cdot)$ is the potentially incorrect (i.e., too coarse) specification of effective treatments for vertex $i$.

Considering two or more specifications of effective treatments can allow us to elaborate on the intuition that using a finer specification of effective treatments will reduce bias by comparing only vertices that are in conditions “closer” to the global treatments of interest. For example, the NTR assumption corresponds to finer effective treatments than the ITR assumption. We also relax the NTR assumption to a fractional $\lambda$-neighborhood treatment in which a vertex is considered effectively in global treated if a fraction $\lambda$ of its neighbors are treated (and the same for control) [@ugander_graph_2013].

Here we analyze functions $g_i(\cdot)$ such that $g_i(Z) = g_i(z)$ just implies that for some subset of vertices $J_i$ we have that $\sum_{j \in J_i} 1\{Z_j = z_j\} \geq l_i$ and that $Z_i = z_i$. These are conditions such that some subset of size $l_i$ of a set of vertices $J_i$ has treatment assignment matching that in $z$, the global treatment vector of interest. The fractional neighborhood treatment response (FNTR) assumption corresponds to such a function with $J_i = \delta(i)$ and 