From 4401de9b38b4169efa71ad47c81774e74a2fe3da Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 18 Jun 2015 09:58:12 -0700
Subject: Some notes

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+\documentclass[10pt]{article}
+\usepackage[utf8x]{inputenc}
+\usepackage{amsmath,amsfonts,amsthm}
+\usepackage[english]{babel}
+\usepackage[capitalize, noabbrev]{cleveref}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+
+\newenvironment{enumerate*}%
+  {\vspace{-2ex} \begin{enumerate} %
+     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
+  {\end{enumerate}}
+ 
+\newenvironment{itemize*}%
+  {\vspace{-2ex} \begin{itemize} %
+     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
+  {\end{itemize}}
+ 
+\newenvironment{description*}%
+  {\vspace{-2ex} \begin{description} %
+     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
+  {\end{description}}
+
+\DeclareMathOperator{\E}{\mathbb{E}}
+\let\P\relax
+\DeclareMathOperator{\P}{\mathbb{P}}
+\newcommand{\ex}[1]{\E\left[#1\right]}
+\newcommand{\prob}[1]{\P\left[#1\right]}
+\newcommand{\inprod}[1]{\left\langle #1 \right\rangle}
+\newcommand{\R}{\mathbf{R}}
+\newcommand{\N}{\mathbf{N}}
+\newcommand{\C}{\mathcal{C}}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
+\newcommand{\llbracket}{[\![}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\algrenewcommand\algorithmicensure{\textbf{Output:}}
+\algrenewcommand\algorithmicrequire{\textbf{Input:}}
+
+\author{}
+\title{}
+\date{}
+
+\begin{document}
+
+\section{Probabilistic Model}
+During the last decade, a growing body of work has focused on the graph
+inference problem: how can an unknown graph be recovered from the observation
+of cascades propagating along it? (CITE). While our probabilistic model for
+cascades of violence strongly relies on this line of work, we depart from it in
+the following ways:
+\begin{itemize}
+    \item in our case, the graph along which the cascades of violence propagate
+        (the co-offending network) is known, and we are only interested in
+        recovering (and ultimately predicting) the edges of the cascade, that
+        is, identifying who influenced whom.
+    \item the previous models treat multiple cascades as independent
+        observations which occur sequentially. In our case, cascades can die
+        and spawn at all time and temporally overlap. This need to be carefully
+        accounted for in our generative model and the resulting inference
+        algorithms.
+\end{itemize}
+
+\subsection{Pairwise Propagation}
+
+For a pair $(u,v)$ of nodes, we model the propagation of the cascade
+from $u$ to $v$ as the sequence of the following two probabilistic events:
+\begin{enumerate}
+
+    \item \emph{structural propagation:} a Bernouilli variable dictates whether
+        or not a propagation occurs. The parameter of this Bernouilli variable
+        is computed from the edge weights of the co-offending network as
+        follows:
+        \begin{equation}\label{eq:structural}
+            p_{u,v}^s \eqdef \prod_{e\in P(u,v)}
+            \frac{1}{1+e^{-\frac{w_e}{\lambda}}}
+        \end{equation}
+        where $P(u,v)$ denotes the shortest path from $u$ to $v$ and $\lambda$
+        is the \emph{structural parameter}. That is, the weight $w_e$ of each
+        edge $e$ on the  path from $u$ to $v$ is mapped to a probability
+        $p_e\in [0,1]$ through the logistic function:
+        \begin{displaymath}
+            p_e \eqdef \frac{1}{1+e^{-\frac{w_e}{\lambda}}}
+        \end{displaymath}
+        Note that the probability of structural propagation both \emph{(a)}
+        decreases as the distance between $u$ and $v$ in the co-offending
+        network increases, and \emph{(b)} increases as the weights of the edges
+        on the path from $u$ to $v$ increase. The structural parameter
+        $\lambda$ controls how quickly the probability $p_e$ of a given edge
+        converges to one as the weight $w_e$ increases.
+
+    \item \emph{temporal propagation:} if the structural propagation occurs,
+        then the propagation time $\Delta_{u,v}$ between $u$ and $v$ is drawn
+        from a continuous distribution with support in $\R_+$. We denote by
+        $f(t)$ the density function of this probability distribution and by
+        $F(t)$ its cumulative function. Table TODO gives the expression of $f$
+        and $F$ for the different temporal distributions considered in this
+        work.
+
+        In our data, the infection times are only observed with a temporal
+        resolution of one day. As a consequence, for two nodes $u$ and $v$
+        whose \empb{observed} infection times are respectively $t_u$ and $t_v$
+        (days), the \emph{real} propagation time could be anything in the
+    interval $(t_v-t_u-1, t_v-t_u]$. Hence, the probability of observing
+    a propagation time $\Delta_{u,v} = t_v - t_u$ is given by:
+    \begin{equation}\label{eq:temporal}
+        p_{u,v}^t \eqdef \int_{\Delta_{u,v}-1}^{\Delta_{u,v}} f(t)dt
+    \end{equation}
+
+\paragraph{Successful propagation.} For two nodes $u$ and $v$ with observed
+infection times $t_u$ and $t_v$, the probability $p_{u,v}$ that $u$ infected
+$v$ is obtained by multiplying the structural and temporal probabilities given
+by \eqref{eq:structural} and \eqref{eq:temporal}:
+\begin{displaymath}
+    p_{u,v} \eqdef p_{u,v}^s\cdot p_{u,v}^t
+\end{displaymath}
+
+\subsection{Spawning of New Cascades}
+
+\subsection{Cascade Model}
+
+\section{Maximum Likelihood Inference}
+
+\subsection{Finding the Cascades}
+
+\subsection{Finding the Spawning Parameter}
+
+\subsection{Finding the Pairwise Propagation Parameters}
+
+\end{document}
-- 
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