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+\documentclass[10pt]{beamer}
+
+\usepackage{amssymb, amsmath, graphicx, amsfonts, color, amsthm, wasysym, framed}
+
+\newtheorem{proposition}{Proposition}
+
+\title{Learning from Diffusion processes}
+\subtitle{What cascades really teach us about networks}
+\author{Jean Pouget-Abadie, Thibaut Horel}
+\institute[]{\includegraphics[scale=.35]{figures/SEASLogo_RGB.png}}
+\begin{document}
+
+\begin{frame}
+\titlepage
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Introduction}
+
+\begin{itemize}
+\item If we observe who catches the flu and when over several years, can we guess {\bf who is friends with whom?}
+\item If we observe behaviors spreading on Facebook (\emph{Ice bucket challenge, Je suis Charlie}), can we know {\bf who is most likely to influence you?}
+\item If we observe memes/ideas on internet, can we tell {\bf which blogs/commmunities take their information from which source?}
+\end{itemize}
+
+\end{frame}
+
+%% Intuition says yes --> this problem can be formalized and is known as the
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Introduction}
+
+\begin{definition}{{\bf Network Inference Problem} {\small \cite{GomezRodriguez:2010}} }
+
+
+If \emph{only} the diffusion process is observed, but the network is \emph{unknown}: 
+\begin{itemize}
+\item Can we learn the network?
+\item For which types of diffusion process?
+\item After how many observations?
+\end{itemize}
+\end{definition}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Prior work}
+\begin{itemize}
+\item {\small \cite{GomezRodriguez:2010, gomezbalduzzi:2011, Abrahao:13, Daneshmand:2014}} : Continuous-time Independent Cascade-like diffusion model
+\item {\small \cite{Netrapalli:2012}}: Discrete-time Independent Cascade Model (formalized in {\small \cite{Kempe:03}}) with correlation decay
+\item {\small \cite{?}}
+\end{itemize}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Discrete-time Independent Cascade Model}
+
+\begin{figure}
+\includegraphics[scale=.3]{figures/ic_illustration.pdf}
+\caption{Weighted, directed graph}
+\end{figure}
+
+\begin{itemize}
+
+\item At $t=0$, each node is {\color{blue} infected} with probability $p_{\text{init}}$ and {\bf susceptible} with probability $1-p_{\text{init}}$
+
+\item Each {\color{blue} infected} node $i$ has a ``one-shot'' probability $p_{i,j}$ of infecting each of its susceptible neighbors $j$ at $t+1$.
+
+\item A node stays {\color{blue}  infected} for one round, then it {\color{red} dies}
+
+\item Process continues until random time $T$ when no more {\bf susceptible} nodes are left
+\end{itemize}
+
+%Notes: Revisit the celebrated independent cascade model -> Influence maximisation is tractable, requires knowledge of weights
+\end{frame}
+
+% %%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Discrete-time Independent Cascade Model}
+
+% \begin{figure}
+% \includegraphics[scale=.5]{figures/weighted_graph.png}
+% \caption{Weighted, directed graph}
+% \end{figure}
+
+% \begin{block}{Example}
+% \begin{itemize}
+% \item At $t=0$, the {\color{orange} orange} node is infected, and the two other nodes are susceptible. $X_0 = $\{{\color{orange} orange}\}
+% \item At $t=1$, the {\color{orange}} node infects the {\color{blue} blue} node and fails to infect the {\color{green} green} node. The {\color{orange} orange} node dies. $X_1 = $\{{\color{blue} blue}\}
+% \item At $t=2$, {\color{blue} blue} dies. $X_2 = \emptyset$
+% \end{itemize}
+% \end{block}
+
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Discrete-time Independent Cascade Model}
+
+\begin{figure}
+\includegraphics[scale=.5]{figures/weighted_graph.png}
+\caption{Weighted, directed graph}
+\end{figure}
+
+\begin{itemize}
+\item If the {\color{orange} orange} node and the {\color{green} green} node are infected at $t=0$, what is the probability that the {\color{blue} blue} node is infected at $t=1$?
+$$1 - \mathbb{P}(\text{not infected}) = 1 - (1 - .72)(1-.04)$$
+\end{itemize}
+
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Independent Cascade Model}
+% \begin{figure}
+% \includegraphics[scale=.5]{figures/weighted_graph.png}
+% \caption{Weighted, directed graph}
+% \end{figure}
+
+% \begin{itemize}
+% \item In general, for each susceptible node $j$:
+% $$\mathbb{P}(j \text{ becomes infected at t+1}|X_{t}) = 1 - \prod_{i \in {\cal N}(j) \cap X_{t}} (1 - p_{i,j})$$
+% \end{itemize}
+
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Independent Cascade Model}
+
+\begin{itemize}
+\item $X^t$: set of {\color{red} infected} nodes
+
+\item Probability that node $j$ gets infected at $t+1$:
+\begin{framed}
+      \begin{align*}
+        \tag{IC}
+        \mathbb{P}\big[X^{t+1}_j  = 1\,|\, X^{t}\big]
+            & = 1 - \prod_{i = 1}^m {(1 - p_{i,j})}^{X^t_i} \\
+            & = 1 - \prod_{i = 1}^m e^{\Theta_{i,j}X^t_i} \\
+            & = 1 - e^{\Theta_j \cdot X^t}
+      \end{align*}
+      \end{framed}
+% \item $f: z \mapsto 1 - e^z$
+% \item $(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$
+\item $\Theta_{i,j} \equiv \log(1-p_{i,j})$ 
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Voter Model}
+\begin{figure}
+\includegraphics[scale=.3]{figures/vt_illustration.pdf}
+\end{figure}
+\begin{itemize}
+\item Probability that node $j$ is infected at $t+1$:
+\begin{equation*}
+        \tag{VT}
+      \mathbb{P}\big[X^{t+1}_j = 1\,|\, X^{t}\big]
+    = \sum_{i \in X^t} p_{i,j} = p_j \cdot X^t
+\end{equation*}
+\end{itemize}
+
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Generalized Linear Cascades}
+
+{\bf Generalized Linear Cascade Model}
+\begin{itemize}
+  \item $f: \mathbb{R} \rightarrow [0,1]$: inverse link function
+  \item Probability depends on $f$-transform of {\bf scalar product}:
+    \begin{framed}
+    $$\mathbb{P}(X^{t+1}_j = 1 | X^t) = f(\Theta_j \cdot X^t)$$
+    \end{framed}
+    \item {\bf Decomposable} node by node conditionally on infected nodes
+    \item Examples:
+    \begin{itemize}
+    \item IC model : $f: z \mapsto 1- e^z$
+    \item VT model : $f: z \mapsto z$
+    \item Discretized version of continuous diffusion model $f: z \mapsto 1-e^{-z}$
+    \end{itemize}
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+\begin{frame}
+\frametitle{Setup}
+\begin{figure}
+\centering
+\includegraphics[scale=.6]{../images/drawing.pdf}
+\caption{$\mathbb{P}(j \in X_{t+1}| X_t) = f(X_t\cdot \theta)$}
+\end{figure}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}
+\frametitle{Sparse Recovery}
+\begin{itemize}
+\item Solving for $Ax = b$ when $A$ is non-degenerate is possible if:
+\begin{itemize}
+\item A is {\bf almost invertible}
+\item x is {\bf sparse}
+\end{itemize}
+\item If $x$ is solution to $\min L(x)$ where $L$ is convex, then {\small \cite{Negahban:2009}} solving for:
+\begin{framed}
+\begin{equation*}
+\min_x L(x) + \lambda \| x \|
+\end{equation*}
+\end{framed}
+gives finite-sample guarantees under certain assumptions.
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Result}
+{\bf Assumptions}
+\begin{itemize}
+\item $f$ and $1-f$ are $(1)$ log-concave and have $(2)$ log-gradient bounded by $\frac{1}{\alpha}$
+\item $\nabla^2{\cal L}$ verifies the $(S, \gamma)-({\bf RE})$ condition
+\end{itemize}
+\vspace{1cm}
+{\bf Algorithm}
+\begin{itemize}
+\item Solve $MLE$ program with $\lambda = 2 \sqrt{\frac{\log m}{\alpha n}}$:
+\begin{framed}
+\begin{equation*}
+          \hat \theta_i \in \arg \max_{\theta} {\cal L}_i(\theta_i | x^1,
+          \dots x^n) - \lambda \|\hat \theta_i\|_1
+\end{equation*}
+\end{framed}
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Theorem}
+If previous assumptions are met, with high probability,
+    \begin{framed}
+    \begin{equation*}
+    \|\hat \theta - \theta^*\|_2 \leq \frac{6}{\gamma} \sqrt{\frac{s \log
+      m}{\alpha n}}
+    \end{equation*}
+  \end{framed}
+    where 
+    \begin{itemize}
+    \item $s$ is degree of node, 
+    \item $m$ is number of nodes, 
+    \item $n$ is the number of observations, 
+    \item $\alpha$ is the gradient bound,
+    \item $\gamma$ is the $({\bf RE})$-constant
+    \end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Restricted Eigenvalue Condition}
+\begin{block}{Almost sparse vectors}
+\begin{itemize}
+\item ${\cal C} := \{ X : \|X\|_2 = 1, \|X_{\bar S}\|_1 \leq 3 \| X_S\|_1 \}$
+\end{itemize}
+\end{block}
+\begin{definition}
+$A$ verifies the $(S, \gamma)$-({\bf RE}) condition \cite{bickel2009simultaneous} if:
+$$\forall X \in {\cal C}, X^T A X \geq \gamma$$
+\end{definition}
+\begin{block}{Properties}
+\begin{itemize}
+\item If $\mathbb{E}[A]$ verifies the $(S, \gamma)$-{(\bf RE)} condition,
+      then $A$ verifies the $(S, \gamma/2)$-{(\bf RE)}
+     condition~{\small \cite{vandegeer:2009}}
+\end{itemize}
+\end{block}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Restricted Eigenvalue Condition}
+\begin{itemize}
+\item If $f$ and $1-f$ are $c$-strictly log-concave, then if the {\bf Gram matrix} verifies the $\gamma$-({\bf RE})-condition, then the Hessian verifies the $c\gamma$-({\bf RE})-condition.
+\end{itemize}
+
+\begin{align*}
+\mathbb{E}[X^T X] = \left( \begin{array}{ccc} a_1 & b_{1, 2} & b_{1, m} \\
+\dots & \dots & \dots \\
+b_{1, m} & \dots & a_m \end{array}\right)
+\end{align*}
+where 
+\begin{itemize}
+\item $a_i \equiv$ ``average time node $i$ is infected''
+\item $b_{i,j} \equiv$ ``average time node $i$ and node $j$ are infected \emph{together}'' 
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}
+\frametitle{Conclusion}
+\begin{itemize}
+\item Introduced Generalized Linear Cascades
+\item Better finite sample guarantees
+\item Interpretable conditions
+\item Lower bound+approx. sparse case developped in full paper~\cite{Pouget:2015}
+\end{itemize}
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Independent Cascade Model}
+% For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable:
+% $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
+% \begin{itemize}
+% \item $\theta_{i,j} := \log(1 - p_{i,j})$
+% \item $\theta_j := (0, 0, 0, \theta_{4,j}, 0 \dots, \theta_{k,j}, \dots)$
+% \item $f : x \mapsto 1 - e^x$
+% \begin{align*}
+% \mathbb{P}(j\in X_{t+1}|X_{t}) & = 1 - \prod_{i \in {\cal N}(j) \cap X_{t}} (1 - p_{i,j}) \\
+% & = 1 - \exp \left[ \sum_{i \in {\cal N}(j) \cap X_{t}} \log(1 - p_{i,j}) \right] \\
+% & = 1 - \exp \left[ X_{t} \cdot \theta_{j}\right]
+% \end{align*}
+% \end{itemize}
+% \end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Independent Cascade Model}
+% For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable:
+% $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
+
+% \begin{block}{Decomposability}
+% \begin{itemize}
+% \item Conditioned on $X_t$, the state of the nodes at the next time step are mutually independent
+% \item We can learn the parents of each node independently
+% \end{itemize}
+% \end{block}
+
+% \begin{block}{Sparsity}
+% \begin{itemize}
+% \item $\theta_{i,j} = 0 \Leftrightarrow \log(1 - p_{i,j}) = 0 \Leftrightarrow p_{i,j} = 0$
+% \item If graph is ``sparse'', then $p_{j}$ is sparse, then $\theta_j$ is sparse.
+% \end{itemize}
+% \end{block}
+% \end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Learning from Diffusion Processes}
+% \begin{block}{Problem Statement}
+% \begin{itemize}
+% \item We are given a graph ${\cal G}$, and a diffusion process $f$ parameterized by $\left((\theta_j)_j, p_{\text{init}}\right)$. 
+% \item Suppose we {\bf only} observe $(X_t)$ from the diffusion process.
+% \item Under what conditions can we learn $\theta_{i,j}$ for all $(i,j)$?
+% \end{itemize}
+% \end{block}
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Sparse Recovery}
+% \begin{figure}
+% \centering
+% \includegraphics[scale=.6]{../images/drawing.pdf}
+% \caption{$\mathbb{P}(j \in X_{t+1}| X_t) = f(X_t\cdot \theta)$}
+% \end{figure}
+% \end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Learning from Diffusion Processes}
+
+% % \begin{figure}
+% % \includegraphics[scale=.4]{../images/sparse_recovery_illustration.pdf}
+% % \caption{Generalized Cascade Model for node $i$}
+% % \end{figure}
+
+% \begin{block}{Likelihood Function}
+% \begin{align*}
+% {\cal L}(\theta_1, \dots, \theta_m| X_1, \dots  X_n) = \sum_{i=1}^m \sum_{t} & X_{t+1}^i \log f(\theta_i \cdot X_t) + \\
+% & (1 - X_{t+1}^i) \log(1 - f(\theta_i \cdot X_t))
+% \end{align*}
+% \end{block}
+
+% \begin{block}{Penalized log-likelihood}
+% For each node $i$, $$\theta_i \in \arg \max \frac{1}{n_i}{\cal {L}}_i(\theta_i | X_1, X_2, \dots, X_{n_i}) - \lambda \|\theta_i\|_1$$
+% \end{block}
+
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Conditions}
+% \begin{block}{On $f$}
+% \begin{itemize}
+% \item $\log f$ and $\log (1-f)$ have to be concave
+% \item $\log f$ and $\log (1-f)$ have to have bounded gradient
+% \end{itemize}
+% \end{block}
+
+% \begin{block}{On $(X_t)$}
+% \begin{itemize}
+% \item Want ${\cal H}$, the hessian of ${\cal L}$ with respect to $\theta$, to be well-conditioned.
+% \item $ n < dim(\theta) \implies {\cal H}$ is degenerate.
+% \item {\bf Restricted Eigenvalue condition} = invertible on ``almost sparse'' vectors. 
+% \end{itemize}
+% \end{block}
+
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+% %%%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Main Result}
+% Adapting a result from \cite{Negahban:2009}, we have the following theorem:
+
+% \begin{theorem}
+% For node $i$, assume
+% \begin{itemize}
+% \item the Hessian verifies the $(S,\gamma)$-RE condition where $S$ is the set of parents of node $i$ (support of $\theta_i$)
+% \item $f$ and $1-f$ are log-concave
+% \item $|(\log f)'| < \frac{1}{\alpha}$ and $|(\log 1- f)'| < \frac{1}{\alpha}$ 
+% \end{itemize} then with high probability:
+% $$\| \theta^*_i - \hat \theta_i \|_2 \leq \frac{6}{\gamma}\sqrt{\frac{s\log m}{\alpha n}}$$
+% \end{theorem}
+
+% \begin{corollary}
+% By thresholding $\hat \theta_i$, if $n > C' s \log m$, we recover the support of $\theta^*$ and therefore the edges of ${\cal G}$
+% \end{corollary}
+
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%
+
+% \begin{frame}
+% \frametitle{Voter Model}
+% \begin{itemize}
+
+% \item {\color{red} Red} and {\color{blue} Blue} nodes. At every step, each node $i$ chooses one of its neighbors $j$ with probability $p_{j,i}$ and adopts that color at $t+1$
+
+% \item If {\color{blue} Blue} is `contagious' state:
+
+% \begin{equation}
+% \nonumber
+% \mathbb{P}(i \in X^{t+1}|X^t) = \sum_{j \in {\cal N}(i)\cap X^t} p_{ji} = X^t \cdot \theta_i
+% \end{equation}
+% \end{itemize}
+% \end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Future Work}
+\begin{itemize}
+\item Lower bound restricted eigenvalues of expected gram matrix
+\item Confidence Intervals
+\item Show that $n > C' s \log m$ measurements are necessary w.r.t. expected hessian.
+\item Linear Threshold model $\rightarrow$ 1-bit compressed sensing formulation
+\item Better lower bounds
+\item Active Learning
+\end{itemize}
+\end{frame}
+
+
+%%%%%%%%%%%%%%%%%
+{\scriptsize
+\bibliography{../../paper/sparse}
+\bibliographystyle{apalike}
+}
+%%%%%%%%%%%%%%%%%%%
+
+
+\begin{frame}
+\frametitle{Analysis}
+
+\begin{block}{Guarantees}
+\begin{itemize}
+\item Positive result despite correlated measurements \smiley
+\item Several measurements per cascade
+\item Good finite-sample guarantee
+\end{itemize}
+\end{block}
+
+\begin{block}{Assumptions}
+\begin{itemize}
+\item The Hessian must verify the $(S,\gamma)$-RE condition \frownie
+\item Can we make a conditional statement on $\Theta$ and not $X_t$?
+\end{itemize}
+\end{block}
+
+\end{frame}
+
+
+
+%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+
+\begin{frame}
+\frametitle{Restricted Eigenvalue Condition}
+
+\begin{block}{From Hessian to Gram Matrix}
+\begin{itemize}
+\item If $f$ and $1 -f$ are strictly log-concave with constant $c$, then if $(S, \gamma)$-RE holds for the gram matrix $\frac{1}{n}X X^T$ , then $(S, c \gamma)$-RE holds for ${\cal H}$
+\end{itemize}
+\end{block}
+
+\begin{block}{From Gram Matrix to Expected Gram Matrix}
+\begin{itemize}
+\item If $n > C' s^2 \log m$ and $(S, \gamma)$-RE holds for $\mathbb{E}(\frac{1}{n}XX^T)$, then $(S, \gamma/2)$-RE holds for $\frac{1}{n}XX^T$ w.h.p
+\item $\mathbb{E}(\frac{1}{n}XX^T)$ only depends on $p_{\text{init}}$ and $(\theta_j)_j$
+\end{itemize}
+\end{block}
+
+\begin{block}{Expected Gram Matrix}
+\begin{itemize}
+\item Diagonal : average number of times node is infected
+\item Outer-diagonal : average number of times pair of nodes is infected {\emph together}
+\end{itemize}
+\end{block}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Approximate Sparsity}
+\begin{itemize}
+\item $\theta^*_{\lceil s \rceil}$ best s-sparse approximation to $\theta^*$
+\item $\|\theta^* - \theta^*_{\lceil s \rceil} \|_1$: `tail' of $\theta^*$
+\end{itemize}
+\begin{theorem}
+Under similar conditions on $f$ and a relaxed RE condition, there $\exists C_1, C_2>0$ such that with high probability:
+\begin{equation}
+\|\hat \theta_i - \theta^*_i\|_2 \leq C_1 \sqrt{\frac{s\log m}{n}} + C_2 \sqrt[4]{\frac{s\log m}{n}}\|\theta^* - \theta^*_{\lceil s \rceil} \|_1
+\end{equation}
+\end{theorem}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Lower Bound}
+\begin{itemize}
+\item Under correlation decay assumption for the IC model, ${\Omega}(s \log N/s)$ cascades necessary for graph reconstruction (Netrapalli et Sanghavi SIGMETRICS'12)
+\item Adapting (Price \& Woodruff STOC'12), in the approximately sparse case, any algorithm for any generalized linear cascade model such that:
+$$\|\hat \theta - \theta^*\|_2 \leq C \|\theta^* - \theta^*_{\lfloor s \rfloor}\|_2$$
+requires ${\cal O}(s \log (n/s)/\log C)$ measurement.
+\end{itemize}
+\end{frame}
+
+\end{document}
\ No newline at end of file
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diff --git a/presentation/WWW15/figures/voter.png b/presentation/WWW15/figures/voter.png
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index 0000000..c1325cb
--- /dev/null
+++ b/presentation/WWW15/figures/voter.png
diff --git a/presentation/WWW15/figures/weighted_graph.png b/presentation/WWW15/figures/weighted_graph.png
new file mode 100644
index 0000000..7deccc3
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+++ b/presentation/WWW15/figures/weighted_graph.png
diff --git a/presentation/WWW15/sparse.bib b/presentation/WWW15/sparse.bib
new file mode 100644
index 0000000..3efaf85
--- /dev/null
+++ b/presentation/WWW15/sparse.bib
@@ -0,0 +1,510 @@
+@article {CandesRomberTao:2006,
+author = {Candès, Emmanuel J. and Romberg, Justin K. and Tao, Terence},
+title = {Stable signal recovery from incomplete and inaccurate measurements},
+journal = {Communications on Pure and Applied Mathematics},
+volume = {59},
+number = {8},
+publisher = {Wiley Subscription Services, Inc., A Wiley Company},
+issn = {1097-0312},
+pages = {1207--1223},
+year = {2006},
+}
+
+
+@inproceedings{GomezRodriguez:2010,
+ author = {Gomez Rodriguez, Manuel and Leskovec, Jure and Krause, Andreas},
+ title = {Inferring Networks of Diffusion and Influence},
+ booktitle = {Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining},
+ series = {KDD '10},
+ year = {2010},
+ isbn = {978-1-4503-0055-1},
+ location = {Washington, DC, USA},
+ pages = {1019--1028},
+ numpages = {10},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+} 
+
+
+@article{Netrapalli:2012,
+ author = {Netrapalli, Praneeth and Sanghavi, Sujay},
+ title = {Learning the Graph of Epidemic Cascades},
+ journal = {SIGMETRICS Perform. Eval. Rev.},
+ volume = {40},
+ number = {1},
+ month = {June},
+ year = {2012},
+ issn = {0163-5999},
+ pages = {211--222},
+ numpages = {12},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+ keywords = {cascades, epidemics, graph structure learning},
+}
+
+@article{Negahban:2009,
+ author = {Negahban, Sahand N. and Ravikumar, Pradeep and Wrainwright, Martin J. and Yu, Bin},
+ title = {A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers},
+ Journal = {Statistical Science},
+ year = {2012},
+ month = {December},
+ volume = {27},
+ number = {4},
+ pages = {538--557},
+}
+
+@article{Zhao:2006,
+ author = {Zhao, Peng and Yu, Bin},
+ title = {On Model Selection Consistency of Lasso},
+ journal = {J. Mach. Learn. Res.},
+ issue_date = {12/1/2006},
+ volume = {7},
+ month = dec,
+ year = {2006},
+ issn = {1532-4435},
+ pages = {2541--2563},
+ numpages = {23},
+ url = {http://dl.acm.org/citation.cfm?id=1248547.1248637},
+ acmid = {1248637},
+ publisher = {JMLR.org},
+} 
+
+@inproceedings{Daneshmand:2014,
+  author    = {Hadi Daneshmand and
+               Manuel Gomez{-}Rodriguez and
+               Le Song and
+               Bernhard Sch{\"{o}}lkopf},
+  title     = {Estimating Diffusion Network Structures: Recovery Conditions, Sample
+               Complexity {\&} Soft-thresholding Algorithm},
+  booktitle = {Proceedings of the 31th International Conference on Machine Learning,
+               {ICML} 2014, Beijing, China, 21-26 June 2014},
+  pages     = {793--801},
+  year      = {2014},
+  url       = {http://jmlr.org/proceedings/papers/v32/daneshmand14.html},
+  timestamp = {Fri, 07 Nov 2014 20:42:30 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/icml/DaneshmandGSS14},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{Kempe:03,
+  author    = {David Kempe and
+               Jon M. Kleinberg and
+               {\'{E}}va Tardos},
+  title     = {Maximizing the spread of influence through a social network},
+  booktitle = {Proceedings of the Ninth {ACM} {SIGKDD} International Conference on
+               Knowledge Discovery and Data Mining, Washington, DC, USA, August 24
+               - 27, 2003},
+  pages     = {137--146},
+  year      = {2003},
+  url       = {http://doi.acm.org/10.1145/956750.956769},
+  doi       = {10.1145/956750.956769},
+  timestamp = {Mon, 13 Feb 2006 15:34:20 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/kdd/KempeKT03},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{Abrahao:13,
+  author    = {Bruno D. Abrahao and
+               Flavio Chierichetti and
+               Robert Kleinberg and
+               Alessandro Panconesi},
+  title     = {Trace complexity of network inference},
+  booktitle = {The 19th {ACM} {SIGKDD} International Conference on Knowledge Discovery
+               and Data Mining, {KDD} 2013, Chicago, IL, USA, August 11-14, 2013},
+  pages     = {491--499},
+  year      = {2013},
+  url       = {http://doi.acm.org/10.1145/2487575.2487664},
+  doi       = {10.1145/2487575.2487664},
+  timestamp = {Tue, 10 Sep 2013 10:11:57 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/kdd/AbrahaoCKP13},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+
+@article{vandegeer:2009,
+author = "van de Geer, Sara A. and B{\"u}hlmann, Peter",
+doi = "10.1214/09-EJS506",
+fjournal = "Electronic Journal of Statistics",
+journal = "Electron. J. Statist.",
+pages = "1360--1392",
+publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
+title = "On the conditions used to prove oracle results for the Lasso",
+url = "http://dx.doi.org/10.1214/09-EJS506",
+volume = "3",
+year = "2009"
+}
+
+@article{vandegeer:2011,
+author = "van de Geer, Sara and Bühlmann, Peter and Zhou, Shuheng",
+doi = "10.1214/11-EJS624",
+fjournal = "Electronic Journal of Statistics",
+journal = "Electron. J. Statist.",
+pages = "688--749",
+publisher = "The Institute of Mathematical Statistics and the Bernoulli Society",
+title = "The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso)",
+url = "http://dx.doi.org/10.1214/11-EJS624",
+volume = "5",
+year = "2011"
+}
+
+@article{Zou:2006,
+author = {Zou, Hui},
+title = {The Adaptive Lasso and Its Oracle Properties},
+journal = {Journal of the American Statistical Association},
+volume = {101},
+number = {476},
+pages = {1418-1429},
+year = {2006},
+doi = {10.1198/016214506000000735},
+URL = {http://dx.doi.org/10.1198/016214506000000735},
+}
+
+@article{Jacques:2013,
+  author    = {Laurent Jacques and
+               Jason N. Laska and
+               Petros T. Boufounos and
+               Richard G. Baraniuk},
+  title     = {Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse
+               Vectors},
+  journal   = {{IEEE} Transactions on Information Theory},
+  volume    = {59},
+  number    = {4},
+  pages     = {2082--2102},
+  year      = {2013},
+  url       = {http://dx.doi.org/10.1109/TIT.2012.2234823},
+  doi       = {10.1109/TIT.2012.2234823},
+  timestamp = {Tue, 09 Apr 2013 19:57:48 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/tit/JacquesLBB13},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{Boufounos:2008,
+  author    = {Petros Boufounos and
+               Richard G. Baraniuk},
+  title     = {1-Bit compressive sensing},
+  booktitle = {42nd Annual Conference on Information Sciences and Systems, {CISS}
+               2008, Princeton, NJ, USA, 19-21 March 2008},
+  pages     = {16--21},
+  year      = {2008},
+  url       = {http://dx.doi.org/10.1109/CISS.2008.4558487},
+  doi       = {10.1109/CISS.2008.4558487},
+  timestamp = {Wed, 15 Oct 2014 17:04:27 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/ciss/BoufounosB08},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{Gupta:2010,
+  author    = {Ankit Gupta and
+               Robert Nowak and
+               Benjamin Recht},
+  title     = {Sample complexity for 1-bit compressed sensing and sparse classification},
+  booktitle = {{IEEE} International Symposium on Information Theory, {ISIT} 2010,
+               June 13-18, 2010, Austin, Texas, USA, Proceedings},
+  pages     = {1553--1557},
+  year      = {2010},
+  url       = {http://dx.doi.org/10.1109/ISIT.2010.5513510},
+  doi       = {10.1109/ISIT.2010.5513510},
+  timestamp = {Thu, 15 Jan 2015 17:11:50 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/isit/GuptaNR10},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{Plan:2014,
+  author    = {Yaniv Plan and
+               Roman Vershynin},
+  title     = {Dimension Reduction by Random Hyperplane Tessellations},
+  journal   = {Discrete {\&} Computational Geometry},
+  volume    = {51},
+  number    = {2},
+  pages     = {438--461},
+  year      = {2014},
+  url       = {http://dx.doi.org/10.1007/s00454-013-9561-6},
+  doi       = {10.1007/s00454-013-9561-6},
+  timestamp = {Tue, 11 Feb 2014 13:48:56 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/dcg/PlanV14},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{bickel:2009,
+author = "Bickel, Peter J. and Ritov, Ya’acov and Tsybakov, Alexandre B.",
+doi = "10.1214/08-AOS620",
+fjournal = "The Annals of Statistics",
+journal = "Ann. Statist.",
+month = "08",
+number = "4",
+pages = "1705--1732",
+publisher = "The Institute of Mathematical Statistics",
+title = "Simultaneous analysis of Lasso and Dantzig selector",
+url = "http://dx.doi.org/10.1214/08-AOS620",
+volume = "37",
+year = "2009"
+}
+
+@article{raskutti:10,
+  author    = {Garvesh Raskutti and
+               Martin J. Wainwright and
+               Bin Yu},
+  title     = {Restricted Eigenvalue Properties for Correlated Gaussian Designs},
+  journal   = {Journal of Machine Learning Research},
+  volume    = {11},
+  pages     = {2241--2259},
+  year      = {2010},
+  url       = {http://portal.acm.org/citation.cfm?id=1859929},
+  timestamp = {Wed, 15 Oct 2014 17:04:32 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/jmlr/RaskuttiWY10},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{rudelson:13,
+  author    = {Mark Rudelson and
+               Shuheng Zhou},
+  title     = {Reconstruction From Anisotropic Random Measurements},
+  journal   = {{IEEE} Transactions on Information Theory},
+  volume    = {59},
+  number    = {6},
+  pages     = {3434--3447},
+  year      = {2013},
+  url       = {http://dx.doi.org/10.1109/TIT.2013.2243201},
+  doi       = {10.1109/TIT.2013.2243201},
+  timestamp = {Tue, 21 May 2013 14:15:50 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/tit/RudelsonZ13},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{bipw11,
+  author    = {Khanh Do Ba and
+               Piotr Indyk and
+               Eric Price and
+               David P. Woodruff},
+  title     = {Lower Bounds for Sparse Recovery},
+  journal   = {CoRR},
+  volume    = {abs/1106.0365},
+  year      = {2011},
+  url       = {http://arxiv.org/abs/1106.0365},
+  timestamp = {Mon, 05 Dec 2011 18:04:39 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/corr/abs-1106-0365},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{pw11,
+  author    = {Eric Price and
+               David P. Woodruff},
+  title     = {{(1} + eps)-Approximate Sparse Recovery},
+  booktitle = {{IEEE} 52nd Annual Symposium on Foundations of Computer Science, {FOCS}
+               2011, Palm Springs, CA, USA, October 22-25, 2011},
+  pages     = {295--304},
+  year      = {2011},
+  crossref  = {DBLP:conf/focs/2011},
+  url       = {http://dx.doi.org/10.1109/FOCS.2011.92},
+  doi       = {10.1109/FOCS.2011.92},
+  timestamp = {Tue, 16 Dec 2014 09:57:24 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/focs/PriceW11},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@proceedings{DBLP:conf/focs/2011,
+  editor    = {Rafail Ostrovsky},
+  title     = {{IEEE} 52nd Annual Symposium on Foundations of Computer Science, {FOCS}
+               2011, Palm Springs, CA, USA, October 22-25, 2011},
+  publisher = {{IEEE} Computer Society},
+  year      = {2011},
+  url       = {http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6108120},
+  isbn      = {978-1-4577-1843-4},
+  timestamp = {Mon, 15 Dec 2014 18:48:45 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/focs/2011},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{pw12,
+  author    = {Eric Price and
+               David P. Woodruff},
+  title     = {Applications of the Shannon-Hartley theorem to data streams and sparse
+               recovery},
+  booktitle = {Proceedings of the 2012 {IEEE} International Symposium on Information
+               Theory, {ISIT} 2012, Cambridge, MA, USA, July 1-6, 2012},
+  pages     = {2446--2450},
+  year      = {2012},
+  crossref  = {DBLP:conf/isit/2012},
+  url       = {http://dx.doi.org/10.1109/ISIT.2012.6283954},
+  doi       = {10.1109/ISIT.2012.6283954},
+  timestamp = {Mon, 01 Oct 2012 17:34:07 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/isit/PriceW12},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@proceedings{DBLP:conf/isit/2012,
+  title     = {Proceedings of the 2012 {IEEE} International Symposium on Information
+               Theory, {ISIT} 2012, Cambridge, MA, USA, July 1-6, 2012},
+  publisher = {{IEEE}},
+  year      = {2012},
+  url       = {http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6268627},
+  isbn      = {978-1-4673-2580-6},
+  timestamp = {Mon, 01 Oct 2012 17:33:45 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/isit/2012},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{Leskovec:2010,
+  author    = {Jure Leskovec and
+               Deepayan Chakrabarti and
+               Jon M. Kleinberg and
+               Christos Faloutsos and
+               Zoubin Ghahramani},
+  title     = {Kronecker Graphs: An Approach to Modeling Networks},
+  journal   = {Journal of Machine Learning Research},
+  volume    = {11},
+  pages     = {985--1042},
+  year      = {2010},
+  url       = {http://doi.acm.org/10.1145/1756006.1756039},
+  doi       = {10.1145/1756006.1756039},
+  timestamp = {Thu, 22 Apr 2010 13:26:26 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/jmlr/LeskovecCKFG10},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{Holme:2002,
+  author= {Petter Holme and Beom Jun Kim},
+  title = {Growing scale-free networks with tunable clustering},
+  journal = {Physical review E},
+  volume = {65},
+  issue = {2},
+  pages = {026--107},
+  year = {2002}
+}
+
+
+@article{watts:1998,
+  Annote = {10.1038/30918},
+  Author = {Watts, Duncan J. and Strogatz, Steven H.},
+  Date = {1998/06/04/print},
+  Isbn = {0028-0836},
+  Journal = {Nature},
+  Number = {6684},
+  Pages = {440--442},
+  Read = {0},
+  Title = {Collective dynamics of `small-world' networks},
+  Url = {http://dx.doi.org/10.1038/30918},
+  Volume = {393},
+  Year = {1998},
+}
+
+@article{barabasi:2001,
+  author    = {R{\'{e}}ka Albert and
+               Albert{-}L{\'{a}}szl{\'{o}} Barab{\'{a}}si},
+  title     = {Statistical mechanics of complex networks},
+  journal   = {CoRR},
+  volume    = {cond-mat/0106096},
+  year      = {2001},
+  url       = {http://arxiv.org/abs/cond-mat/0106096},
+  timestamp = {Mon, 05 Dec 2011 18:05:15 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/corr/cond-mat-0106096},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+
+@article{gomezbalduzzi:2011,
+  author    = {Manuel Gomez{-}Rodriguez and
+               David Balduzzi and
+               Bernhard Sch{\"{o}}lkopf},
+  title     = {Uncovering the Temporal Dynamics of Diffusion Networks},
+  journal   = {CoRR},
+  volume    = {abs/1105.0697},
+  year      = {2011},
+  url       = {http://arxiv.org/abs/1105.0697},
+  timestamp = {Mon, 05 Dec 2011 18:05:23 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/journals/corr/abs-1105-0697},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{Nowell08,
+  author = {Liben-Nowell, David and Kleinberg, Jon},
+  biburl = {http://www.bibsonomy.org/bibtex/250b9b1ca1849fa9cb8bb92d6d9031436/mkroell},
+  doi = {10.1073/pnas.0708471105},
+  eprint = {http://www.pnas.org/content/105/12/4633.full.pdf+html},
+  journal = {Proceedings of the National Academy of Sciences},
+  keywords = {SNA graph networks},
+  number = 12,
+  pages = {4633-4638},
+  timestamp = {2008-10-09T10:32:56.000+0200},
+  title = {{Tracing information flow on a global scale using Internet chain-letter data}},
+  url = {http://www.pnas.org/content/105/12/4633.abstract},
+  volume = 105,
+  year = 2008
+}
+
+@inproceedings{Leskovec07,
+  author    = {Jure Leskovec and
+               Mary McGlohon and
+               Christos Faloutsos and
+               Natalie S. Glance and
+               Matthew Hurst},
+  title     = {Patterns of Cascading Behavior in Large Blog Graphs},
+  booktitle = {Proceedings of the Seventh {SIAM} International Conference on Data
+               Mining, April 26-28, 2007, Minneapolis, Minnesota, {USA}},
+  pages     = {551--556},
+  year      = {2007},
+  url       = {http://dx.doi.org/10.1137/1.9781611972771.60},
+  doi       = {10.1137/1.9781611972771.60},
+  timestamp = {Wed, 12 Feb 2014 17:08:15 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/sdm/LeskovecMFGH07},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+
+@inproceedings{AdarA05,
+  author    = {Eytan Adar and
+               Lada A. Adamic},
+  title     = {Tracking Information Epidemics in Blogspace},
+  booktitle = {2005 {IEEE} / {WIC} / {ACM} International Conference on Web Intelligence
+               {(WI} 2005), 19-22 September 2005, Compiegne, France},
+  pages     = {207--214},
+  year      = {2005},
+  url       = {http://dx.doi.org/10.1109/WI.2005.151},
+  doi       = {10.1109/WI.2005.151},
+  timestamp = {Tue, 12 Aug 2014 16:59:16 +0200},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/webi/AdarA05},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@inproceedings{Pouget:2015,
+   title={Inferring graphs from cascades: A Sparse Recovery Framework},
+   author={Pouget-Abadie, Jean and Horel, Thibaut},
+   series={ICML'15},
+   year={2015}
+}
+
+@inproceedings{Kleinberg:00,
+  author    = {Jon M. Kleinberg},
+  title     = {The small-world phenomenon: an algorithm perspective},
+  booktitle = {Proceedings of the Thirty-Second Annual {ACM} Symposium on Theory
+               of Computing, May 21-23, 2000, Portland, OR, {USA}},
+  pages     = {163--170},
+  year      = {2000},
+  url       = {http://doi.acm.org/10.1145/335305.335325},
+  doi       = {10.1145/335305.335325},
+  timestamp = {Thu, 16 Feb 2012 12:06:08 +0100},
+  biburl    = {http://dblp.uni-trier.de/rec/bib/conf/stoc/Kleinberg00},
+  bibsource = {dblp computer science bibliography, http://dblp.org}
+}
+
+@article{zhang2014,
+  title={Confidence intervals for low dimensional parameters in high dimensional linear models},
+  author={Zhang, Cun-Hui and Zhang, Stephanie S},
+  journal={Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
+  volume={76},
+  number={1},
+  pages={217--242},
+  year={2014},
+  publisher={Wiley Online Library}
+}
+
+@article{javanmard2014,
+  title={Confidence intervals and hypothesis testing for high-dimensional regression},
+  author={Javanmard, Adel and Montanari, Andrea},
+  journal={The Journal of Machine Learning Research},
+  volume={15},
+  number={1},
+  pages={2869--2909},
+  year={2014},
+  publisher={JMLR. org}
+}