From e4752124605bfce12b953c06c13425472aacb811 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Thu, 5 Nov 2015 12:14:06 -0500
Subject: Notes for the mid project report, WIP

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+\documentclass[10pt]{article}
+\usepackage[utf8x]{inputenc}
+\usepackage{amsmath, amssymb, amsthm, microtype}
+\usepackage[pagebackref=false,breaklinks=true,colorlinks=true,citecolor=blue]{hyperref}
+\usepackage[capitalize, noabbrev]{cleveref}
+\usepackage{graphicx}
+\usepackage{bbm}
+
+\DeclareMathOperator*{\argmax}{arg\,max}
+\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{\bt}{\boldsymbol{\theta}}
+\newcommand{\bx}{\mathbf{x}}
+\newcommand{\cl}[1]{\text{\textbf{#1}}}
+\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
+
+\newtheorem{theorem}{Theorem}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem*{example}{Example}
+\newtheorem*{remark}{Remark}
+
+\title{Regression Analysis with Network Data}
+\author{Thibaut Horel \and Jean Pouget-Abadie}
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+
+\end{abstract}
+
+\section{Introduction}
+
+Central question: \emph{relating properties of the graph to the difficulty of
+doing graph inference from cascades.}
+
+Two subquestions:
+\begin{itemize}
+    \item which properties of the graph make it impossible or hard to learn
+        (identifiability, convergence rate)?
+    \item which properties of the graph can be exploited to learn better or
+        faster (use a Bayesian prior)?
+\end{itemize}
+
+\section{Model}
+
+Weighted directed graph $G = (V, \Theta)$. $k=|V|$ is the number of nodes.
+$\Theta\in\R_{+}^{k\times k}$. $\Theta$ implicitly defines the edge set $E$ of
+the graph.
+
+Discrete time model, $t\in\N$.  Nodes can be in one of two states:
+\emph{susceptible}, \emph{infected} or \emph{immune}.  $S_t:$ nodes susceptible
+at the beginning of time step $t\in\N$.  $I_t$ nodes infected at time step $t$.
+\begin{displaymath}
+    S_0 = V,\quad S_{t+1} = S_t \setminus I_t
+\end{displaymath}
+Nodes which are no longer susceptible are immune.
+
+The dynamics of $I_t$ is described by a random Markovian process. Let us denote
+by $x_t$ the indicator vector of $I_t$ at time step $t\in\N$. $x_0$ is drawn
+from the \emph{source distribution} $p_s:\{0,1\}^n\to[0,1]$. For $t\geq 1$,
+Markovian process:
+\begin{equation}
+    \label{eq:markov}
+    \forall i\in S_t,\quad
+    \P\big(x_{i}^{t} = 1\,|\, x^{t-1}\big) = f(\bt_i\cdot x^{t-1})
+\end{equation}
+$\bt_i$ is the $i$th column of $\Theta$, $f:\R\to[0,1]$, plus independence for
+$i$. A cascade continues until no more infected nodes. 
+
+
+\section{Identifiability}
+
+A given source distribution $p_s$ and \cref{eq:markov} completely specifies the
+distribution $p$ of a cascade $\mathbf{x} = (x_t)_{t\geq 0}$:
+\begin{displaymath}
+    p(\bx;\Theta) = p_s(x^0)\prod_{t\geq 1}\prod_{i\in S_t}
+    f(\bt_i\cdot x^{t-1})^{x^t_i}\big(1-f(\theta_i\cdot x^{t-1})\big)^{1-x_i^t}
+\end{displaymath}
+
+We see this model as parametrized by $\Theta$. $f$ and $p_s$ are considered
+fixed.
+
+\begin{definition}
+    We say that the model $\mathcal{P}
+    = \big\{p(\cdot\,;\,\Theta)\,|\,\Theta\in\R_+^{k\times k}\big\}$ is
+    identifiable iff:
+    \begin{displaymath}
+        \forall \Theta,\Theta'\in\R_+^{k\times k},\quad
+        p(\cdot\,;\,\Theta) = p(\cdot\, ;\, \Theta') \Rightarrow \Theta = \Theta'
+    \end{displaymath}
+\end{definition}
+
+In our case, identifiability is not guaranteed. In fact there are two factors
+which can lead to unidentifiability: lack of information or ambiguity. We
+illustrate both factors in the subsequent two examples.
+
+\begin{figure}
+\begin{center}
+    \includegraphics[scale=0.8]{example.pdf}
+\end{center}
+\caption{Example of a graph which can be unidentifiable, both because of lack
+    of information or parent ambiguity if the source distribution is chosen
+adversarially.}
+\label{fig:ident}
+\end{figure}
+
+\vspace{.5em}
+
+\begin{example}[Lack of information]
+    Let us consider the graph in \cref{fig:ident} and the source distribution
+    $p_s$ defined by $p_s\big((0, 1, 0)\big) = 1$. In other words, node $2$ is
+    always the only infected node at $t=0$. Then it is clear that all
+    casacades die at $t=1$ at which node 3 becomes infected with probability
+    $f(\theta_{2,3})$. This probability does not depend on $\theta_{1,3}$.
+    Hence all matrices $\Theta$ compatible with the graph and for which
+    $\theta_{2,3}$ is fixed yield the same cascade distribution.
+\end{example}
+
+\vspace{.5em}
+
+\begin{example}[Parent ambiguity]
+Let us consider again the graph in \cref{fig:ident} and let us consider the
+case where the source distribution is such that $p_s\big((1,1,0)\big) = 1$,
+\emph{i.e} nodes 1 and 2 are always jointly infected at time $t=0$. Then it is
+clear that cascades will all die at time $t=1$ where node 3 becomes infected
+with probability $f(\theta_{1,3} + \theta_{2,3})$. This implies that all
+matrices $\Theta$ which are compatible with the graph in \cref{fig:ident}
+(defining the same edge set) and for which $\theta_{1,3} + \theta_{2,3} = c$
+for some $c\in \R_+$ define the same cascade probability distribution.
+\end{example}
+
+\vspace{.5em}
+
+The examples show that for a model to be identifiable, we need conditions on
+the source distribution, in particular in relation to how well it plays with
+the reachability structure of the graph.
+
+We will abuse the notation $p_s$ and for $u\in V$ write:
+\begin{displaymath}
+    p_s(w) \eqdef \sum_{\bx: \bx_w = 1} p_s(\bx)
+\end{displaymath}
+Furthermore we write $u\leadsto v$ if there is a directed path from $u$ to $v$
+in $G$ and $u\leadsto v\leadsto w$ if there is a directed path from $u$ to $w$
+passing through $v$ in $G$.
+
+\begin{proposition}
+Under the assumptions:
+\begin{enumerate}
+    \item $\forall z\in \mathbf{\R_+},\; f(z)< 1$
+    \item for all $S\subseteq V$ with $|S|\geq 2$, $p_s(S)<1$.
+    \item $\forall (u,v)\in V^2$, there exists $w$ such that $p_s(w)\neq 0$ and
+        $w\leadsto u\leadsto v$.
+\end{enumerate}
+then the model is identifiable.
+\end{proposition}
+
+2. prevents parent ambiguity at the source. 1. prevents parent ambiguity from
+   occurring later on in the cascade even when there is no ambiguity at the
+   source (could be relaxed to ``absence of forks''). 3. prevents lack of
+   information.
+
+\begin{proof}
+
+\end{proof}
+
+\begin{remark} The conditions are the weakest under which there is
+    identifiability (we can show that they are necessary and sufficient).
+    Reasonable source distributions which imply the conditions are uniform
+    random, independent Bernouilli.
+\end{remark}
+
+\section{Inference}
+
+\subsection{Maximum-Likelihood Estimation}
+
+Because transitions occur independently for each node $i$, the log-likelihood
+is a sum of $k$ terms, each term $i$ only depending on $\bt_i$. Hence, each
+$\bt_i$ can be learnt separately by solving a node-specific optimization
+problem:
+
+We assume $f$ log-concave to have a concave optimization problem. Depending on
+the model, $\theta_{i,j}$ can either be restricted to live in a compact subset
+of $\R_+$. Otherwise, we need to assume that $\lim_{z\to\infty} f(z) = 1$.
+Then, we can apply standard results on consistency and asymptotic normality of
+MLE estimator.
+
+\subsection{Bayesian Approach}
+
+
+\end{document}
-- 
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