\times \mathbb{U}^N \rightarrow \mathbb{Y}.$$ One way for this assumption to hold is if the vertices do not interact.[^8] This specification of $f_i(\cdot)$ corresponds to the assumption that a vertex’s outcome is invariant to changes in other vertices’ assignments. That is, for any two global assignments $z_0, z_1 \in \mathbb{Z}^N$ and any stochastic component $U \in \mathbb{U}^N$, $$z_{1,i} = z_{0,i} \Rightarrow f_i(z_{1}, U) = f_i(z_{0}, U).$$ ITR is a particular version of the more general notion of *constant treatment response* (CTR) assumptions [@manski_identification_2013]. More generally, a CTR assumption involves establishing equivalence classes of treatment vectors by defining a function $g_i(\cdot): \mathbb{Z}^N \rightarrow \mathbb{G}_i$ that maps global treatment vectors to the space $\mathbb{G}_i$ of *effective treatments* for vertex $i$ [@manski_identification_2013] such that $$g_i(z_1) = g_i(z_0) \Rightarrow f_i(z_{1}, U) = f_i(z_{0}, U)$$ for any two global assignments $z_0, z_1 \in \mathbb{Z}^N$ and any stochastic component $U \in \mathbb{U}^N$. Specifying the functions $g_i$ is then a general way to specify a CTR assumption. Such assumptions can be described as constituting an *exposure model* [@aronow_estimating_2011; @ugander_graph_2013].

Other CTR assumptions have been proposed that allow for some interference. @aronow_estimating_2011 simply posit different restrictions on this function, such as that a vertex’s outcome only depends on its 