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\begin{document}

\section{Probabilistic Model}
During the last decade, a growing body of work has focused on the graph
inference problem: how can an unknown graph be recovered from the observation
of cascades propagating along it? (CITE). While our probabilistic model for
cascades of violence strongly relies on this line of work, we depart from it in
the following ways:
\begin{itemize}
    \item in our case, the graph along which the cascades of violence propagate
        (the co-offending network) is known, and we are only interested in
        recovering (and ultimately predicting) the edges of the cascade, that
        is, identifying who influenced whom.
    \item the previous models treat multiple cascades as independent
        observations which occur sequentially. In our case, cascades can die
        and spawn at all time and temporally overlap. This need to be carefully
        accounted for in our generative model and the resulting inference
        algorithms.
\end{itemize}

\subsection{Pairwise Propagation}

For a pair $(u,v)$ of nodes, we model the propagation of the cascade
from $u$ to $v$ as the sequence of the following two probabilistic events:
\begin{enumerate}

    \item \emph{structural propagation:} a Bernouilli variable dictates whether
        or not a propagation occurs. The parameter of this Bernouilli variable
        is computed from the edge weights of the co-offending network as
        follows:
        \begin{equation}\label{eq:structural}
            p_{u,v}^s \eqdef \prod_{e\in P(u,v)}
            \frac{1}{1+e^{-\frac{w_e}{\lambda}}}
        \end{equation}
        where $P(u,v)$ denotes the shortest path from $u$ to $v$ and $\lambda$
        is the \emph{structural parameter}. That is, the weight $w_e$ of each
        edge $e$ on the  path from $u$ to $v$ is mapped to a probability
        $p_e\in [0,1]$ through the logistic function:
        \begin{displaymath}
            p_e \eqdef \frac{1}{1+e^{-\frac{w_e}{\lambda}}}
        \end{displaymath}
        Note that the probability of structural propagation both \emph{(a)}
        decreases as the distance between $u$ and $v$ in the co-offending
        network increases, and \emph{(b)} increases as the weights of the edges
        on the path from $u$ to $v$ increase. The structural parameter
        $\lambda$ controls how quickly the probability $p_e$ of a given edge
        converges to one as the weight $w_e$ increases.

    \item \emph{temporal propagation:} if the structural propagation occurs,
        then the propagation time $\Delta_{u,v}$ between $u$ and $v$ is drawn
        from a continuous distribution with support in $\R_+$. We denote by
        $f(t)$ the density function of this probability distribution and by
        $F(t)$ its cumulative function. Table TODO gives the expression of $f$
        and $F$ for the different temporal distributions considered in this
        work.

        In our data, the infection times are only observed with a temporal
        resolution of one day. As a consequence, for two nodes $u$ and $v$
        whose \empb{observed} infection times are respectively $t_u$ and $t_v$
        (days), the \emph{real} propagation time could be anything in the
    interval $(t_v-t_u-1, t_v-t_u]$. Hence, the probability of observing
    a propagation time $\Delta_{u,v} = t_v - t_u$ is given by:
    \begin{equation}\label{eq:temporal}
        p_{u,v}^t \eqdef \int_{\Delta_{u,v}-1}^{\Delta_{u,v}} f(t)dt
    \end{equation}

\paragraph{Successful propagation.} For two nodes $u$ and $v$ with observed
infection times $t_u$ and $t_v$, the probability $p_{u,v}$ that $u$ infected
$v$ is obtained by multiplying the structural and temporal probabilities given
by \eqref{eq:structural} and \eqref{eq:temporal}:
\begin{displaymath}
    p_{u,v} \eqdef p_{u,v}^s\cdot p_{u,v}^t
\end{displaymath}

\subsection{Spawning of New Cascades}

\subsection{Cascade Model}

\section{Maximum Likelihood Inference}

\subsection{Finding the Cascades}

\subsection{Finding the Spawning Parameter}

\subsection{Finding the Pairwise Propagation Parameters}

\end{document}
