From 1ddca2dd5d412326b45d581e633fce473007bff2 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Thu, 17 Sep 2015 10:49:54 -0400
Subject: adding extensions file

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+\documentclass{article}
+\usepackage{fullpage, amsmath, amssymb}
+
+\title{Extensions}
+
+\begin{document}
+
+\section{Gram matrix}
+
+In the ICML paper, we showed that the graph could be recovered as long as the
+expected gram matrix $\mathcal{G}$ of observations had its restricted
+eigenvalues lower bounded away from $0$.  Recall that the gram matrix $G$ is
+given by $X X^T$, where $X$ is an $n\times m$ matrix, where $n$ is the number of
+measurements and $m$ is the number of nodes and if $X_i$ is a column vector
+indicating the nodes infected at step $i$, then the rows of $X$ are given by
+$X_i^T$. It follows that $$\mathcal{G} = \mathbb{E}[\sum_i X_i X_i^T]$$
+
+Note that the indices of the sum  are themselves random variables!  Furthermore,
+recall the definition of a restricted eigenvalue as $\|X_{S^c}\|_1 \leq 3
+\|X_S\|_1$.
+
+\subsection{Voter model}
+
+In the voter model, $\mathbb{P}[X_j^{t+1} = 1 | X^t] = \sum_{i=1}^m A_{i,j}
+X^t_i$, where $A$ is the weighted adjacency matrix of the graph. Furthermore,
+the model runs indefinitely until time $T$, a hyperparameter of the model.
+Therefore, $\mathbb{E}[X_{n+1} | X_n] = A^T X_n$ and $\mathcal{G} = \sum_i A^i
+\mathbb{E}[X_0 X_0^T] (A^T)^i$.  In the case of the single-source model,
+$\mathbb{E}[X_0X_0^T] = \frac{1}{m} I_m$ and it follows that $$\mathcal{G} =
+\frac{1}{m} \sum_{i=1}^T (A A^T)^i$$
+
+\end{document}
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