From 10e206159d4d5b803ff789eae5f029456cfc2603 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Fri, 6 Nov 2015 16:54:39 -0500
Subject: Pass over model section

---
 finale/mid_report.tex | 40 +++++++++++++++++++++++++++-------------
 1 file changed, 27 insertions(+), 13 deletions(-)

(limited to 'finale/mid_report.tex')

diff --git a/finale/mid_report.tex b/finale/mid_report.tex
index d22a557..75f5f4a 100644
--- a/finale/mid_report.tex
+++ b/finale/mid_report.tex
@@ -155,29 +155,43 @@ authors' knowledge, novel in this context.
 
 \section{Model}
 
-Weighted directed graph $G = (V, \Theta)$. $k=|V|$ is the number of nodes.
-$\Theta\in\R_{+}^{k\times k}$. $\Theta$ implicitly defines the edge set $E$ of
-the graph.
+The GLC model is described over a directed graph $G = (V, \Theta)$. Denoting by
+$k=|V|$ the number of nodes in the graph, $\Theta\in\R_{+}^{k\times k}$ is the
+matrix of edge weights. Note that $\Theta$ implicitly defines the edge set $E$ of
+the graph through the following equivalence:
+\begin{displaymath}
+    (u,v)\in E\Leftrightarrow \Theta_{u,v} > 0,\quad
+    (u,v)\in V^2
+\end{displaymath}
 
-Discrete time model, $t\in\N$.  Nodes can be in one of two states:
-\emph{susceptible}, \emph{infected} or \emph{immune}.  $S_t:$ nodes susceptible
-at the beginning of time step $t\in\N$.  $I_t$ nodes infected at time step $t$.
+The time is discretized and indexed by a variable $t\in\N$. The nodes can be in
+one of three states: \emph{susceptible}, \emph{infected} or \emph{immune}.
+Let us denote by $S_t$ the set of nodes susceptible at the beginning of time
+step $t\in\N$ and by $I_t$ the set of nodes who become infected at this time
+step. The following dynamics:
 \begin{displaymath}
     S_0 = V,\quad S_{t+1} = S_t \setminus I_t
 \end{displaymath}
-Nodes which are no longer susceptible are immune.
+expresses that the nodes infected at a time step are no longer susceptible
+starting from the next time step (they become part of the immune nodes).
 
-The dynamics of $I_t$ is described by a random Markovian process. Let us denote
-by $x_t$ the indicator vector of $I_t$ at time step $t\in\N$. $x_0$ is drawn
-from the \emph{source distribution} $p_s:\{0,1\}^n\to[0,1]$. For $t\geq 1$,
-Markovian process:
+The dynamics of $I_t$ are described by a random Markovian process. Let us
+denote by $x_t$ the indicator vector of $I_t$ at time step $t\in\N$. $x_0$ is
+drawn from a \emph{source distribution} $p_s:\{0,1\}^n\to[0,1]$. For $t\geq 1$,
+we have:
 \begin{equation}
     \label{eq:markov}
     \forall i\in S_t,\quad
     \P\big(x_{i}^{t} = 1\,|\, x^{t-1}\big) = f(\bt_i\cdot x^{t-1})
 \end{equation}
-$\bt_i$ is the $i$th column of $\Theta$, $f:\R\to[0,1]$, plus independence for
-$i$. A cascade continues until no more infected nodes. 
+where $\bt_i$ is the $i$th column of $\Theta$. The function $f:\R\to[0,1]$ can
+be interpreted as the inverse link function of the model. Finally, the
+transitions in \cref{eq:markov} occur independently for each $i$. A cascade
+continues until no infected nodes remains.
+
+We refer the reader to \cite{pouget} for a more complete description of the
+model and examples of common contagion models which can be interpreted as
+specific instances of the GLC model.
 
 \section{Identifiability}
 
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