more concrete, and for our subsequent simulations, we consider a model in which a vertex’s behavior is a stochastic function of the mean of neighbors’ prior behaviors, so that behavior at some new time step $t$ is generated as: $$\begin{aligned}
\label{probit_model}
Y_{i, t}^* &= \alpha +  \beta Z_i+ \gamma \frac{{A}_i^{'} Y_{t - 1}}{k_i}  + U_{i,t} \\ 
\label{probit_model_link}
Y_{i, t} &= a\big( Y_{i, t}^* \big)\end{aligned}$$ where $A_i$ is a row of the adjacency matrix and $k_i$ is the degree of vertex $i$. In the case of a binary behavior, we work with $a(x) = 1\{x > 0\}$ and $U_{i,t} \sim \mathcal{N}(0, 1)$, which is a probit model. We initialize behaviors with $Y_{i, 0} = 0$. Here $\alpha$ is the baseline, where a negative $\alpha$ determines the threshold that must be crossed for $Y_{i, t}^*$ to be positive. Setting $\beta$ determines the strength of the direct effect of the treatment, while $\gamma$ is the slope for peer behavior, and therefore determines the strength of the peer effects. This process is then run up to a maximum time $T$. As described above, with a small value of $T$, this implies CTR assumptions.

This can be interpreted as a *noisy best response* or *best 