of friends, and so on out through the network. In an extreme case, treating one individual could alter the behavior of everyone in the network.

To see the challenges this introduces, consider what is, in many cases, a primary quantity of interest for experiments in networks — the average treatment effect (ATE) of applying a treatment to all units compared with applying a different (control) treatment to all units.[^1] Let $Z$ be a vector of length $N$ giving each unit’s treatment assignment, so that $Y_i(Z = z)$ is the potential outcome of interest for unit $i$ when $Z$ is set to $z$. Then the ATE is a contrast between two such treatment vectors, $$\label{tau_is_ate}
\tau(z_1, z_0) = \frac{1}{N} \sum_i {\mathrm{E}}[Y_i(Z = z_1) - Y_i(Z = z_0)],$$ where $N$ is the number of units and $z_1$ and $z_0$ are two treatment assignments vectors; the prototypical case has $z_1 = 1$ and $z_0 = 0$, the vectors of all ones and of all zeros. Note that each unit’s potential outcome is a function of the global treatment assignment vector $Z$, not just its own treatment $Z_i$. Additional assumptions will thus be required for $\tau$ to be identifiable.[^2]

The standard approach is to 