Time Discounted Infection and Susceptibility

George G. Vega Yon

November 11, 2015

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 \(\mathbf{A}(t)\) 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)