From 752ca266a44613a2b505f4f81cfed38cfd8fa5d4 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Thu, 29 Jan 2015 17:33:32 -0500
Subject: linear threshold section

---
 paper/sections/model.tex | 30 ++++++------------------------
 1 file changed, 6 insertions(+), 24 deletions(-)

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

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 98712d7..25a0a47 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -26,16 +26,16 @@ problems.
 
 \subsection{Independent Cascade Model}
 
-In the independent cascade model, nodes can be either uninfected, active or
-infected. All nodes start either as uninfected or active. At each time step
-$t$, for each edge $(i,j)$ where $j$ is uninfected and $i$ is active, $i$
+In the independent cascade model, nodes can be either susceptible, active or
+infected. All nodes start either as susceptible or active. At each time step
+$t$, for each edge $(i,j)$ where $j$ is susceptible and $i$ is active, $i$
 attempts to infect $j$ with probability $p_{i,j}\in[0,1]$. If $i$ succeeds, $j$ will
 become active at time step $t+1$. Regardless of $i$'s success, node $i$ will be
 infected at time $t+1$: nodes stay active for only one time step. The cascade
 continues until no active nodes remain.
 
 If we denote by $X^t$ the indicator variable of the set of active nodes at time
-step $t-1$, then if $j$ is uninfected at time step $t-1$, we have:
+step $t-1$, then if $j$ is susceptible at time step $t-1$, we have:
 \begin{displaymath}
     \P\big[X^{t+1}_j = 1\,|\, X^{t}\big]
     = 1 - \prod_{i = 1}^m (1 - p_{i,j})^{X^t_i}.
@@ -50,25 +50,7 @@ as:
 \end{equation}
 where we defined $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$.
 
-\subsection{Linear Threshold Model}
-
-In the Linear Threshold Model, each node $j\in V$ has a threshold $t_j$ drawn
-uniformly from the interval $[0,1]$. Furthermore, there is a weight
-$\Theta_{i,j}\in[0,1]$ for each edge $(i,j)$. We assume that the weights are
-such that for each node $j\in V$, $\sum_{i=1}^m \Theta_{i,j} \leq 1$.
-
-Nodes can be either uninfected or active. At each time step, each uninfected
-node $j$ becomes active if the sum of the weights $\Theta_{i,j}$ for $i$ an
-active parent of $j$ is larger than $j$'s threshold $t_j$. Denoting by $X^t$
-the indicator variable of the set of active nodes at time step $t-1$, if
-$j\in V$ is uninfected at time step $t-1$, then:
-\begin{equation}
-    \label{eq:lt}
-    \tag{LT}
-    \P\big[X^{t+1}_j = 1\,|\, X^{t}\big] = \sum_{i=1}^m \Theta_{i,j}X^t_i
-    = \inprod{\theta_j}{X^t}
-\end{equation}
-where we defined again $\theta_j\defeq (\Theta_{1,j},\ldots,\Theta_{m,j})$.
+\subsection{Generalized Linear Models}
 
 \subsection{Maximum Likelihood Estimation}
 
@@ -104,7 +86,7 @@ a separate optimization program:
 \end{equation}
 
 Furthermore, the state evolution of a node $j\in V$ has the same structure in
-both models: the transition from uninfected to active at time step $t+1$ is
+both models: the transition from susceptible to active at time step $t+1$ is
 controlled by a Bernoulli variable whose parameter can be written
 $f(\inprod{\theta_j}{x^t})$ for some function $f$. Hence, if we denote by $n_j$
 the first step at which $j$ becomes active, we can rewrite the  MLE program
-- 
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