In Myers (2000), susceptibility and infection is defined for a given time period and as a constant throughout the network–so only varies on t. In order to include effects from previous/coming time periods, it adds up through the of the rioting, which in our case would be strength of tie, hence a dichotomous variable, whenever the event occurred a week within t, furthermore, he then introduces a discount factor in order to account for decay of the influence of the event. Finally, he obtains
$$ V_{(t)} = \sum_{a\in \mathbf{A}(t)} \frac{S_{(a)}m_{T(a), T\leq t-T(a)}}{t- T(a)} $$
where A(t) is the set of all riots that occurred by time t, S(a) is the severity of the riot a, T(a) is the time period by when the riot a accurred and m is an indicator function.
In order to include this notion in our equations, I modify these by also adding whether a link existed between i and j at the corresponding time period. Furthermore, in a more general way, the time windown is now a function of the number of time periods to include, K, this way, instead of looking at time periods t and t + 1 for infection, we look at the time range between t and t + K.
Following the paper’s notation, a more generalized formula for infectiousness is
$$\label{eq:infect-dec} \left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j(t+k)}}{k} \right)\left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j([t+k;T])}}{k} \right)^{-1} $$
Where $\frac{1}{k}$ would be the equivalent of $\frac{1}{t - T(a)}$ in mayers. Alternatively, we can include a discount factor as follows
$$\label{eq:infect-exp} \left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j(t+k)}}{(1+r)^{k-1}} \right)\left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j([t+k;T])}}{(1+r)^{k-1}} \right)^{-1} $$
Observe that when K = 1, this formula turns out to be the same as the paper.
Likewise, a more generalized formula of susceptibility is
$$\label{eq:suscept-dec} \left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j(t-k)}}{k} \right)\left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j([1;t-k])}}{k} \right)^{-1} $$
Which can also may include an alternative discount factor
$$\label{eq:suscept-exp} \left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j(t-k)}}{(1+r)^{k-1}} \right)\left( \sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j([1;t-k])}}{(1+r)^{k-1}} \right)^{-1} $$
Also equal to the original equation when K = 1. Furthermore, the resulting statistic will lie between 0 and 1, been the later whenever i acquired the innovation lastly and right after j acquired it, been j its only alter.
(PENDING: Normalization of the stats)