added continuous time+figure

1 files changed, 30 insertions, 6 deletions

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index fbcedf3..4241f93 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -148,8 +148,30 @@ step $t$, then we have:
 Thus, the linear voter model is a Generalized Linear Cascade model
 with inverse link function $f: z \mapsto z$.
 
-{\color{red} \subsubsection{Discretization of Continous Model}
-TODO}
+\subsubsection{Discretization of Continous Model}
+
+Consider the continuous-time independent cascade model with exponential
+transmission function (CICE) of~\cite{GomezRodriguez:2010, Abrahao:13,
+Daneshmand:2014} where time is binned into intervals of length $\epsilon$. Let
+$X^k$ be the set of nodes `infected' before or during the $k^{th}$ time
+interval. Note that contrary to the discrete-time independent cascade model,
+$X^k_i = 1 \implies X^{k+1}_i = 1$.  Let $\exp(p)$ be an
+exponentially-distributed random variable of parameter $p$.  By the memoryless
+property of the exponential, if we consider that $\forall i,\Theta_{i,j} = 0 if
+(i,j) \notin E$, then if $X^k_j \neq 1$:
+\begin{align*}
+  \mathbb{P}(X^{k+1}_j = 1 | X^k) & = \mathbb{P}(\min_{i \in {\cal N}(j)}
+  \exp(p_{i,j}) \leq \epsilon) \\
+  & = \mathbb{P}(\exp( \sum_{i=1}^m \Theta_{i,j} X^t_i) \leq \epsilon) \\
+  & = 1 - e^{- \epsilon \cdot \theta_j \cdot X^t}
+\end{align*}
+This formulation is consistent when $X^k_j = 1$ by considering the dummy
+variables $\forall i, \Theta_{i,i} = +\infty$.  Therefore, the
+$\epsilon-$binned-process of the continuous-time model with exponential
+transmission function is a Generalized Linear Cascade model with inverse link
+function $f:z\mapsto 1-e^{-\epsilon\cdot z}$.
+
+\subsubsection{Logistic Cascades}
 
 % \subsection{The Linear Threshold Model}
 
@@ -174,13 +196,15 @@ TODO}
 % \begin{equation} \label{eq:lt} \tag{LT} \mathbb{P} \left[X^{t+1}_j = 1 |
 % X^t\right] = \text{sign} \left(\inprod{\theta_j}{X^t} - t_j \right)
 % \end{equation} where ``sign'' is the function $\mathbbm{1}_{\cdot > 0}$.
-
-{\color{red} Add drawing of math problem as in Edo's presentation}
-
-
 \subsection{Maximum Likelihood Estimation}
 \label{sec:mle}
 
+\begin{figure}
+  \includegraphics[scale=.4]{figures/drawing.pdf}
+  \caption{Illustration of the sparse-recovery approach}
+\end{figure}
+
+
 Inferring the model parameter $\Theta$ from observed influence cascades is the
 central question of the present work. Recovering the edges in $E$ from observed
 influence cascades is a well-identified problem known as the \emph{Graph