From aca5c88f9856c9cdd676e22e44e1955c3e75f676 Mon Sep 17 00:00:00 2001
From: ericbalkanski <ericbalkanski@MACD-01953.local>
Date: Sun, 7 Dec 2014 15:16:18 -0500
Subject: Added examples to report

---
 notes/reportYaron.tex | 73 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 73 insertions(+)

(limited to 'notes/reportYaron.tex')

diff --git a/notes/reportYaron.tex b/notes/reportYaron.tex
index d39d060..1dc0db1 100644
--- a/notes/reportYaron.tex
+++ b/notes/reportYaron.tex
@@ -84,6 +84,42 @@ If the vector $\vec x_i$ is sufficiently sparse, i.e. node $i$ has sufficiently
 
 \subsection{Example}
 
+
+\begin{figure}
+\centering
+\includegraphics[width=0.3\textwidth]{voter.png}
+\caption{A cascade in the voter model with time steps $t = 0,1,2$  over a graph with 5 vertices}
+\end{figure}
+
+
+
+
+To recover the neighbors of $v_1$, we get the following matrix $M$ for the example in Figure 1:
+
+\begin{align*}
+&\hspace{0.35cm} v_2 \hspace{0.2 cm}  v_3 \hspace{0.2 cm} v_4 \hspace{0.2 cm} v_5 &\\
+\vspace{1 cm}
+M  = &  \left( \begin{array}{cccc}
+0 & 0 & 1 & 1 \\
+1 & 1 & 0 & 0 \\
+\end{array} \right)  &  \begin{array}{l} \hspace{ -4.5 cm} 
+\text{time step 0} \\
+ \hspace{ - 4.5cm}  \text{time step 1} \\
+\end{array}
+\end{align*}
+
+and the vector $b_1$:
+
+\begin{align*}
+b_1  = &  \left( \begin{array}{c}
+1 \\
+1 \\
+\end{array} \right) &  \begin{array}{l} \hspace{ - 5cm} 
+\text{time step 1} \\
+ \hspace{ - 5cm}  \text{time step 2} \\
+ \end{array}
+\end{align*}
+
 \section{The Independent Cascade Model}
 
 \subsection{Description}
@@ -127,6 +163,43 @@ e^{M_i \vec \theta_i} = 1 - \vec b_i + \vec \epsilon_i
 
 Note that this is not exactly a sparse recovery model due to the non-linear exponential term. It is of the author's opinion however that if the probabilities $p_{j,i}$ are restrained to a bounded interval $[0, 1- \eta]$, then most of the results which hold for the linear voter model will continue to hold in this case.
 
+\subsection{Example}
+
+\begin{figure}
+\centering
+\includegraphics[width=0.3\textwidth]{icc.png}
+\caption{A cascade in the independent cascade model with time steps $t = 0,1,2$  over a graph with 5 vertices}
+\end{figure}
+
+
+
+To recover the neighbors of $v_5$, we get the following matrix $M$ for the example in Figure 2:
+\begin{align*}
+&v_1 \hspace{0.2 cm}  v_2 \hspace{0.2 cm} v_3 \hspace{0.2 cm} v_4 \\
+\vspace{1 cm}
+M  = &  \left( \begin{array}{cccc}
+1 & 0 & 0 & 0 \\
+0 & 1 & 1 & 0 \\
+\end{array} \right)   \begin{array}{l} \hspace{ 1cm} 
+\text{time step 0} \\
+ \hspace{ 1cm}  \text{time step 1} \\
+ \end{array}
+\end{align*}
+
+and the vector $b_5$:
+
+\begin{align*}
+b_5  = &  \left( \begin{array}{c}
+0 \\
+1 \\
+\end{array} \right)  \begin{array}{l} \hspace{ 1cm} 
+\text{time step 1} \\
+ \hspace{ 1cm}  \text{time step 2} \\
+ \end{array}
+\end{align*}
+
+
+
 \section{Sparse Recovery}
 
 \subsection{Introduction}
-- 
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