$$\begin{aligned}
{\mathrm{E}}^U[Y_{t}(z)] = (\gamma {\mathbf{D}}^{-1} {\mathbf{A}})^t {\mathrm{E}}^U[Y_0] + \sum_{q=0}^{t-1} (\gamma  {\mathbf{D}}^{-1} {\mathbf{A}})^q (\alpha + \beta z)\end{aligned}$$ where ${\mathbf{D}}^{-1}$ is the diagonal matrix of inverse degrees, ${\mathbf{A}}$ is the adjacency matrix, and $Y_0$ is the vector of initial states. This is a linear outcome model with $a_i = \alpha (1 - \gamma^t) / (1 - \gamma) + ((\gamma {\mathbf{D}}^{-1} {\mathbf{A}})^t {\mathrm{E}}^U[Y_0])_i$ and $B_{ij} = \beta \sum_{q = 0}^{t-1} (\gamma {\mathbf{D}}^{-1} {\mathbf{A}})^q_{ij}$.

### Bias reduction through analysis

Definitions of effective treatments other than ITR correspond to different estimands. In particular, we can incorporate assumptions about effective treatments into Equation \[tau\_is\_ate\]. Let $$\begin{aligned}
\label{mu_g}
\mu^d_g(z) = \frac{1}{N} \sum_i {\mathrm{E}}^d[Y_i {\, | \,}g_i(Z) = g_i(z)]\end{aligned}$$ be the mean outcome for the global treatment $z$ when $g$ specifies the effective treatments and $d$ is the experimental design. Then we have $$\begin{aligned}
\tau^d_g(z_1, z_0) &=  \mu^d_g(z_1) - \mu^d_g(z_0)
\label{effective_treatment_tau}\end{aligned}$$ as our revised estimand for the ATE.[^14]

If the effective treatment assumption corresponding to this estimator is satisfied, then it is unbiased. As with the ITR assumption, we can again describe the bias that occurs when effective treatments are incorrectly specified. For some global treatment vector $z$, vertex $i$ contributes to the bias of $\mu^d_{g}(z)$ through $$\begin{aligned}
