From f95792d89fff45e9a3630486805a7c516778db84 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Tue, 3 Feb 2015 00:35:47 -0500
Subject: changes to model section

---
 paper/sections/model.tex | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

(limited to 'paper/sections/model.tex')

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 5e55c7a..cd7485d 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -51,7 +51,7 @@ In other words, each generalized linear cascade provides a series for each node
 % $f$ can be interpreted as the inverse link function of our generalized linear
 % cascade model.
 
-\subsection{examples}
+\subsection{Examples}
 
 \subsubsection{Independent Cascade Model}
 
@@ -78,11 +78,11 @@ Defining $\Theta_{i,j} \defeq \log(1-p_{i,j})$, this can be rewritten as:
     = 1 - e^{\inprod{\theta_j}{X^t}}
 \end{equation}
 
-Therefore, the independent cascade model is a Generalized Linear Cascade model with link function $f : z \mapsto 1 - e^z$
+Therefore, the independent cascade model is a Generalized Linear Cascade model with canonical link function $f : z \mapsto 1 - e^z$
 
 \subsubsection{The Linear Voter Model}
 
-In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}. Both states verify the properties of a contagious state, but for ease of presentation, we will suppose that the contagious nodes are the \emph{blue}. The parameters of the graph are normalized such that $\forall i,
+In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}. Both states are symetric, but without loss of generality, we can suppose that the \emph{blue} nodes are contagious and the \emph{red} nodes are susceptible. The parameters of the graph are normalized such that $\forall i,
 \ \sum_j \Theta_{i,j} = 1$. Each round, every node $j$ independently chooses
 one of its neighbors with probability $\Theta_{ij}$ and adopts their color. The
 cascades stops at a fixed horizon time T or if all nodes are of the same color.
@@ -93,7 +93,7 @@ step $t$, then we have:
 \tag{V}
 \end{equation}
 
-Therefore, the independent cascade model is a Generalized Linear Cascade model with link function $f: z \mapsto z$
+Therefore, the independent cascade model is a Generalized Linear Cascade model with canonical link function $f: z \mapsto z$
 
 % \subsection{The Linear Threshold Model}
 
@@ -137,7 +137,7 @@ $\Theta$ via Maximum Likelihood Estimation (MLE). Denoting by $\mathcal{L}$ the
 log-likelihood function, we consider the following $\ell_1$-regularized MLE
 problem:
 \begin{displaymath}
-    \hat{\Theta} \in \argmax_{\Theta} \mathcal{L}(\Theta\,|\,x^1,\ldots,x^n)
+    \hat{\Theta} \in \argmax_{\Theta} \frac{1}{n} \mathcal{L}(\Theta\,|\,x^1,\ldots,x^n)
     + \lambda\|\Theta\|_1
 \end{displaymath}
 where $\lambda$ is the regularization factor which helps at preventing
@@ -149,16 +149,16 @@ of $m$ terms, each term $i\in\{1,\ldots,m\}$ only depending on $\theta_i$.
 Since this is equally true for $\|\Theta\|_1$, each column $\theta_i$ of
 $\Theta$ can be estimated by a separate optimization program:
 \begin{equation}\label{eq:pre-mle}
-    \hat{\theta}_i \in \argmax_{\theta}\frac{1}{n_i}\mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n)
+    \hat{\theta}_i \in \argmax_{\theta} \frac{1}{n}\mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n)
     + \lambda\|\theta_i\|_1
 \end{equation}
-where we denote by $n_i$ the first step at which node $i$ becomes contagious and
-where:
+where we denote by ${\cal T}_i$ the time steps at which node $i$ is susceptible and:
 \begin{multline}
-    \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{n_i}
-    \sum_{t=1}^{n_i-1}\log\big(1-f(\inprod{\theta_i}{x^t})\big)\\
-+\log f(\inprod{\theta_i}{x^{n_i}})+ \lambda\|\theta_i\|_1
+    \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{|{\cal T}_i|} \sum_{t\in {\cal T}_i } x_i^{t+1}\log f(\inprod{\theta_i}{x^{t}}) + \dots \nonumber \\
+    (1 - x_i^{t+1})\log\big(1-f(\inprod{\theta_i}{x^t})\big)  + \lambda\|\theta_i\|_1  \nonumber
 \end{multline}
 
+Note that in the case of the voter model, TODO horizon time. In the case of the independent cascade model...
+
 TODO: discuss conditions on $f$ under which this program is convex. For LC and
 VM it is convex.
-- 
cgit v1.2.3-70-g09d2

