(\cdot): \mathbb{Z} \times \mathbb{Y}^{k_i + 1} \times \mathbb{U}^N \rightarrow \mathbb{Y},$$ where $k_i$ is the degree of vertex $i$ and $Y_{\cdot,0}$ is initialized by some prior process. That is, $h_{i,t}(\cdot)$ is the nonparametric structural equation (NPSE) for $Y_{i,t}$.

Together with the graph $G$, the function $h_{i,t}(\cdot)$ determine the treatment response function $f_i(\cdot)$. Thus, this outcome generating process implies some CTR assumptions. After the first time step (i.e., at time 1), the effective treatment for a vertex, the function $g_i(\cdot)$ considered earlier, maps to the space of the vertex’s treatment. After the second time step, it maps to the space of the vertex’s treatment and its neighbors treatment. After the third time step (i.e., at time 3), the effective treatment is no finer than the treatment of all vertices within distance 2. At time step $t$, the effective treatment is no finer than the treatment of all vertices within distance $t - 1$. We see here that under such a dynamic outcome generating process, Manski’s notion of effective treatment, conceived of to limit the scope of dependence, quickly expands to encompass the full graph.[^11]

### Utility linear-in-means {#section:utility_linear-in-means}

Many familiar models are included in the above outcome generating process. To make this 