From 8c2d3d070a3db1b469e3e32e3c20ef67bc274b3b Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 21 Sep 2015 16:15:13 -0400
Subject: Supplements, section 3.1

---
 supplements/main.tex | 33 +++++++++++++++++++++++++++++----
 1 file changed, 29 insertions(+), 4 deletions(-)

(limited to 'supplements/main.tex')

diff --git a/supplements/main.tex b/supplements/main.tex
index 8ba30bd..71bfcc4 100644
--- a/supplements/main.tex
+++ b/supplements/main.tex
@@ -295,9 +295,34 @@ the optimum:
 
 \section{Inferring the Patterns of Infections}
 
-We can estimate if a person was primarily infected via peer contagion by comparing the contributions from the background rate and from his or her peers.
-
-We take this approach one step further to determine the person most responsible for infecting each of these 7,016 individuals infected by social contagion.
+\subsection{Methodology}
+
+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
+by the exogenous process or endogenous peer contagion. For this, using formula
+\eqref{eq:hawkes}, we simply compare the value of the exogenous intensity and 
+the sum of endogenous exciting functions at the time $t$ of infection, and
+attribute the infection event to the largest of the two quantities. In other
+words, for our contagion model we compare:
+\begin{enumerate}
+    \item $\mu(t) = \mu(t) = \mu_0\left[1 + 0.43 \sin\left(\frac{2\pi}{365.24}
+    t + 4.36\right)\right]$ and
+\item  $\sum_{i:t_i< t} g_\alpha(u_i, v)\betae^{-\beta(t-t_i)}$
+\end{enumerate}
+
+It is possible to go one step further: for an infection event attributed to
+peer contagion, we can single out a single past infection event as its cause.
+This is achieved by choosing the event $i$ for which the value of the summand
+in 2. is maximum. That is, the event $i$ for which the value of $g_\alpha(u_i, v)\beta
+e^{-\beta(t-t_i)}$ is largest.
+
+We thus uncover the patterns of infections: each infection event is attributed
+to either the exogenous intensity or a single past infection event.  Drawing an
+edge between infection events $(t_1, u_1)$ and $(t_2, u_2)$ if $(t_1, u_1)$ is
+the cause of $(t_2, u_2)$, we obtain the forest of infections: tree roots are
+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
@@ -319,6 +344,6 @@ We also simulated contagions on the co-offending network. Since we are most inte
 
 \subsubsection{Results}
 
-\section{Regarding causality [THIBAUT WRITE THIS SECTION]}
+\subsection{Comments on Causality}
 
 \end{document}
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