From 190cd5633e7e3237be118c18d374b5542aa38be4 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 5 Feb 2015 23:22:36 -0500
Subject: Polishing

---
 paper/sections/model.tex | 23 ++++++++++++++++-------
 1 file changed, 16 insertions(+), 7 deletions(-)

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

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index dbcbd8d..cf69960 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -52,10 +52,11 @@ A \emph{generalized linear cascade model} is a cascade model such that for each
 susceptible node $j$ in state $s$ at time step $t$, the probability of $j$
 becoming ``contagious'' at time step $t+1$ conditioned on $X^t$ is a Bernoulli
 variable of parameter $f(\theta_j \cdot X^t)$:
-\begin{displaymath}
+\begin{equation}
+    \label{eq:glm}
     \mathbb{P}(X^{t+1}_j = 1|X^t) 
     = f(\theta_j \cdot X^t)
-\end{displaymath}
+\end{equation}
 where $f: \reals \rightarrow [0,1]$
 \end{definition}
 
@@ -128,9 +129,11 @@ with inverse link function $f : z \mapsto 1 - e^z$.
 
 In the Linear Voter Model, nodes can be either \emph{red} or \emph{blue}. Both
 states are symmetric, but without loss of generality, we can suppose that the
-\emph{blue} nodes are contagious. The parameters of the graph are normalized such that $\forall i,
-\ \sum_j \Theta_{i,j} = 1$. Each round, every node $j$ independently chooses
-one of its neighbors with probability $\Theta_{ij}$ and adopts their color. 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 $j$ 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 the set of blue nodes at time
 step $t$, then we have:
@@ -139,7 +142,7 @@ step $t$, then we have:
 \tag{V}
 \end{equation}
 
-Therefore, the independent cascade model is a Generalized Linear Cascade model
+Thus, the independent cascade model is a Generalized Linear Cascade model
 with inverse link function $f: z \mapsto z$.
 
 % \subsection{The Linear Threshold Model}
@@ -204,6 +207,13 @@ where we denote by ${\cal T}_i$ the time steps at which node $i$ is susceptible
     + (1 - x_i^{t+1})\log\big(1-f(\inprod{\theta_i}{x^t})\big)
 \end{multline*}
 
+In the case of the voter model, the measurements include all time steps until
+we reach the time horizon $T$ or the graph coalesces to a single state. In the
+case of the independent cascade model, the measurements include all time steps
+until node $i$ becomes contagious after which its behavior is deterministic.
+Contrary to prior work, we express our results in terms of the number of
+measurements and not the number of cascades.
+
 A sufficient condition for program~\eqref{eq:pre-mle} to be a convex program is
 that $\mathcal{L}_i$ is a concave function. This will be the case if, for
 example, $f$ and $(1-f)$ are both log-concave functions. Remember that
@@ -211,4 +221,3 @@ a twice-differentiable function $f$ is log concave iff. $f''f \leq f'^2$. It is
 easy to verify this property for $f$ and $(1-f)$ in the Independent Cascade
 Model and Voter Model.
 
-Note that in the case of the voter model, TODO horizon time. In the case of the independent cascade model...
-- 
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