From 4cd49c19e49188afc7bff48ac5877f6f3a2b00a7 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Mon, 18 May 2015 19:42:31 +0200
Subject: fixed typos

---
 paper/sections/model.tex | 13 +++++--------
 1 file changed, 5 insertions(+), 8 deletions(-)

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

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 8d403e1..f1d9585 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -122,14 +122,12 @@ time step $t$, then if $j$ is susceptible at time step $t+1$, we have:
     = 1 - \prod_{i = 1}^m {(1 - p_{i,j})}^{X^t_i}.
 \end{displaymath}
 Defining $\Theta_{i,j} \defeq \log(\frac{1}{1-p_{i,j}})$, this can be rewritten as:
-\begin{equation}\label{eq:ic}
+\begin{align*}\label{eq:ic}
     \tag{IC}
-    \begin{split}
-    \P\big[X^{t+1}_j &= 1\,|\, X^{t}\big]
-    = 1 - \prod_{i = 1}^m e^{-\Theta_{i,j}X^t_i}\\
+    \P\big[X^{t+1}_j = 1\,|\, X^{t}\big]
+    &= 1 - \prod_{i = 1}^m e^{-\Theta_{i,j}X^t_i}\\
     &= 1 - e^{-\inprod{\Theta_j}{X^t}}
-\end{split}
-\end{equation}
+\end{align*}
 
 Therefore, the independent cascade model is a Generalized Linear Cascade model
 with inverse link function $f : z \mapsto 1 - e^{-z}$.
@@ -155,8 +153,7 @@ $|\log (\frac{1}{1 - p}) - \log (\frac{1}{1-p'})| \geq \max(1 - \frac{1-p}{1-p'}
 In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}.
 Without loss of generality, we can suppose that the \emph{blue} nodes are
 contagious. The parameters of the graph are normalized such that $\forall i, \
-\sum_j \Theta_{i,j} = 1$ and we assume that $\Theta_{i,i}$ is always non-zero,
-meaning that all nodes have self-loops.  Each round, every node $j$
+\sum_j \Theta_{i,j} = 1$.  Each round, every node $j$
 independently chooses one of its neighbors with probability $\Theta_{i,j}$ and
 adopts their color. The cascades stops at a fixed horizon time $T$ or if all
 nodes are of the same color.  If we denote by $X^t$ the indicator variable of
-- 
cgit v1.2.3-70-g09d2

