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\usepackage{amssymb, amsmath, graphicx, amsfonts, color, amsthm}

\newtheorem{proposition}{Proposition}

\title{Learning from Diffusion processes}
\subtitle{What cascades really teach us about networks}
\author{Jean (John) Pouget-Abadie \\ Joint Work with Thibaut (T-bo) Horel}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\frametitle{Introduction}

%notes: Learn what? the network, the parameters of the diffusion process.

\begin{table}
\centering
\begin{tabular}{c | c}
Network & Diffusion process \\[1ex]
\hline
\\
Airports & Infectious diseases (SARS) \\
 & Delays (Eyjafjallajökull) \\[3ex]
Social Network & Infectious diseases (flu) \\
 & Behaviors (Ice Bucket Challenge) \\[3ex]
Internet/WWW & Information diffusion (Memes, Pirated content \dots)
\end{tabular}
\end{table}

\end{frame}

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\begin{frame}
\frametitle{Introduction}

What do we know? What do we want to know?

\begin{itemize}
\item We know the {\bf airport network} structure. We observe delays. Can we learn how delays propagate?
\item  We (sometimes) know the {\bf social network}. We observe behaviors. Can we learn who influences whom?
\item Rarely know {\bf blog network}. We observe discussions. Can we learn who learns from whom?
\end{itemize}

\end{frame}

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\begin{frame}
\frametitle{Independent Cascade Model}

\begin{figure}
\includegraphics[scale=.5]{figures/weighted_graph.png}
\caption{Weighted, directed graph}
\end{figure}

\begin{itemize}
\item Three states: susceptible, {\color{blue} infected}, {\color{red} dead}
\item Each {\color{blue} infected} node $i$ has a probability $p_{i,j}$ of infecting each of his neighbors $j$.
\item A node stays {\color{blue}  infected} for one round, then it {\color{red} dies}
\item At $t=0$, each node is {\color{blue} infected} with probability $p_{\text{init}}$
\end{itemize}

%Notes: Revisit the celebrated independent cascade model -> Influence maximisation is tractable, requires knowledge of weights

\end{frame}


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\begin{frame}
\frametitle{Independent Cascade Model}

\begin{figure}
\includegraphics[scale=.5]{figures/weighted_graph.png}
\caption{Weighted, directed graph}
\end{figure}

\begin{itemize}
\item If $3$ and $4$ are {\color{blue} infected} at $t$, what is the probability that node $0$ is infected at $t+1$?
$$1 - \mathbb{P}(\text{not infected}) = 1 - (1 - .45)(1-.04)$$
\item In general, $X_t$ {\color{blue} infected} nodes at t:
$$\mathbb{P}(j \text{ becomes infected at t+1}|X_{t}) = 1 - \prod_{i \in {\cal N}(j) \cap X_{t}} (1 - p_{i,j})$$
\end{itemize}


\end{frame}

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\begin{frame}
\frametitle{Independent Cascade Model}
\begin{proposition}
The ICC, conditioned on previous time step, can be cast as a {\bf generalized linear model}
$$\mathbb{P}(j \in X_{t+1} | X_t) = f(X_t \cdot \theta_j)$$
\end{proposition}

\begin{proof}
\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{proof}

\end{frame}

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\begin{frame}
\frametitle{Independent Cascade Model}
\begin{block}{Decomposability}
\begin{itemize}
\item Conditioned on $X_t$, the state of each node is sampled independently
\item We can focus on learning vector $\theta_{j}$ for each node
\end{itemize}
\end{block}

\begin{block}{Sparsity}
\begin{itemize}
\item $\theta_{i,j} = 0 \Leftrightarrow p_{i,j} = 0$
\item If graph is ``sparse'', then $\theta_j$ is sparse.
\end{itemize}
\end{block}
\end{frame}



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\begin{frame}
\frametitle{Sparse Recovery}
\begin{figure}
\includegraphics[scale=.6]{../images/sparse_recovery_illustration.pdf}
\caption{$f(X_t\cdot \theta) = \mathbb{P}(j \in X_{t+1}| X_t)$}
\end{figure}
\end{frame}

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\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 parameterized by $\left((\theta_{i,j})_{i,j}, f, p_{\text{init}}\right)$. 
\item Suppose we {\bf only} observe $(X_t)$ from the diffusion process.
\item Under what conditions can we learn $\theta_{i,j}$ for all $(i,j)$? How many $(X_t)$ are necessary?
\end{itemize}
\end{block}
\end{frame}

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\begin{frame}
\frametitle{Learning from Diffusion Processes}

\begin{figure}
\includegraphics[scale=.4]{../images/sparse_recovery_illustration.pdf}
\caption{Generalized Linear Model for node $i$}
\end{figure}

\begin{block}{Likelihood Function}
$${\cal L}(\theta| X_1, \dots X_N) = \frac{1}{{\cal T}_i}  \sum_{t \in {\cal T}_i} x^{t+1}_i \log f(\theta_i \cdot x^t) + (1 - x^{t+1}_i) \log(1 - f(\theta_i \cdot x^t))$$
\end{block}

\end{frame}

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\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 bounded gradient
\end{itemize}
\end{block}

\begin{block}{On $(X_t)$}
\begin{itemize}
\item Want ${\cal H}$ be the hessian of ${\cal L}$ with respect to $\theta$ to be ``inversible''
\item $ n < dim(\theta) \implies {\cal H}$ is degenerate.
\item {\bf Restricted Eigenvalue condition} = ``almost invertible'' on sparse vectors. 
\end{itemize}
\end{block}

\end{frame}

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\begin{frame}
\frametitle{Restricted Eigenvalue Condition}

\begin{definition}
Let $S$ be the set of parents of node $i$.
$${\cal C} := \{ \Delta : \|\Delta\|_2 = 1, \|\Delta_{\bar S}\| \leq 3 \| \Delta_S\|_1 \}$$
${\cal H}$ verifies the $(S, \gamma)$-RE condition if:
$$\forall X \in {\cal C}, \Delta {\cal H} \Delta \geq \gamma$$
\end{definition}
\end{frame}
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\begin{frame}
\frametitle{Main Result}
Adapting a result from \cite{Negahban:2009}, we have the following theorem:

\begin{theorem}
Assume 
\begin{itemize}
\item the Hessian verifies the $(S,\gamma)$-RE condition
\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}

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\begin{frame}
Comments: Correlated Measurements
TODO: still need to mention somewhere that we are doing penalized log likelihood
condition on $X_t$ is not great, we would like to have condition on the parameters $\theta, p_{\text{init}}$ $->$ Slides about expected hessian
TODO: slide about matrice de gram!
\end{frame}

\bibliography{../../paper/sparse}
\bibliographystyle{apalike}

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