In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ from the interval $[0,1]$. Furthermore, there is a weight $\Theta_{i,j}\in[0,1]$ for each edge $(i,j)$, such that the sum of incoming weights is less than $1$: $\forall j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$. Nodes remain active until the end of the cascade, which is reached when no new susceptible nodes become active.

Nodes can be either susceptible or active. At each time step, each susceptible
node $j$ becomes active if the sum of the incoming weights from active parents is greater than $j$'s threshold $t_j$. 

We formalize the model in the following way: let $X^t$ be the indicator variable of the set of active nodes at time step $t-1$, then:
\begin{equation}
    X^{t+1}_j = \left[\sum_{i=1}^m \Theta_{i,j}X^t_i > t_j\right]
    = \left[\inprod{\theta_j}{X^t} > t_j \right]
\end{equation}
where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$ and $[\cdot]$ is the boolean evaluator. In other words, for every step in the linear threshold model, we observe the following signal:

\begin{equation}
    \label{eq:lt}
    \tag{LT}
    y_{t} = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
\end{equation}

The linear threshold model can therefore be cast as a variant of the recently introduced 1-bit compressed sensing model \cite{Boufounos:2008}. Several recent papers study the theoretical guarantees obtained for 1-bit compressed sensing with specific measurements \cite{Gupta:2010}, \cite{Plan:2014}. Others study adaptive mechanisms \cite{Jacques:2013}.

Support recovery results in the 1-bit compressed sensing model rely on the fact that well-chosen random hyperplanes segment a compact set into small cells. Knowing which side of the hyperplanes you are on situates you in one of these cells of bounded size. Whilst the obtained bounds are also of the order ${\cal O}(n \log \frac{n}{s}$), no current theory exists for recovering positive bounded signal from bernoulli hyperplanes. We leave this research direction to future work.