From eb2e948cb7918999f723393cdc79ac7b512794c5 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sat, 26 Sep 2015 16:13:49 -0400
Subject: Adapt Section 3.1 to be consistent with the main paper

---
 supplements/main.tex | 29 ++++++++++++++++++++---------
 1 file changed, 20 insertions(+), 9 deletions(-)

(limited to 'supplements/main.tex')

diff --git a/supplements/main.tex b/supplements/main.tex
index 5e68542..1f3e69f 100644
--- a/supplements/main.tex
+++ b/supplements/main.tex
@@ -14,7 +14,7 @@
 \newtheorem{theorem}{Theorem}
 \newtheorem{proposition}[theorem]{Proposition}
 
-\title{Hawkes contagion model}
+\title{Hawkes Contagion model}
 \author{Ben Green \and Thibaut Horel \and Andrew Papachristos}
 \date{September 2015}
 
@@ -224,7 +224,7 @@ obtain the following form for the exogenous intensity:
 \end{equation}
 where $\mu_0 = \frac{A}{|V|}$.
 
-\subsection{Kernel Function Parameters}
+\subsection{Exciting Function Parameters}
 
 Using the exogenous intensity from Section~\ref{sec:background}, the
 log-likelihood now depends on three parameters $(\mu_0, \alpha, \beta)$, and
@@ -295,7 +295,15 @@ the optimum:
 
 \section{Inferring the Patterns of Infections}
 
-\subsection{Methodology}
+\subsection{Methodology and Results}
+\label{sec:meth}
+
+\begin{figure}[h]
+\centering
+\includegraphics[width=.8\textwidth]{cascade-distribution}
+\caption{The distribution of cascade sizes follows a power-law distribution.}
+\label{fig:cascade-sizes}
+\end{figure}
 
 Given fitted values of the parameters of the Hawkes contagion model, it is then
 possible to determine whether an infection event $(t, v)$ was primarily caused
@@ -324,13 +332,15 @@ the infection events attributed to exogenous intensity, and tree nodes are
 events attributed to peer contagion and are connected to their cause.  We note
 that cycles are impossible since edges are directed forward in time.
 
-\begin{figure}
-\centering
-\includegraphics[width=.8\textwidth]{cascade-distribution}
-\caption{The distribution of cascade sizes follows a power-law distribution.}
-\label{fig:cascade-sizes}
-\end{figure}
+Following the terminology found in the literature on information diffusion in
+social networks, we call \emph{cascade} a single tree in the forest of
+infections. The distribution of the cascade sizes extracted from our dataset
+can be found in Figure~\ref{fig:cascade-sizes}. Consistently with previous
+findings in other domains of application (CITE) this distributions follows
+a power-law of exponent XX. Figure SX shows three cascades extracted from the
+forest.
 
+\begin{comment}
 \subsection{Experiments with synthetic data}
 
 \subsubsection{Generating networks}
@@ -343,6 +353,7 @@ We also simulated contagions on the co-offending network. Since we are most inte
 \subsubsection{Simulating contagions}
 
 \subsubsection{Results}
+\end{comment}
 
 \subsection{Comments on Causality}
 
-- 
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