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\usepackage{graphicx}
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%----------------------------------------------------------------------------------------
%	TITLE SECTION
%----------------------------------------------------------------------------------------

\title{Inferring Graphs from Cascades} % Poster title

\author{Thibaut Horel, Jean Pouget-Abadie} % Author(s)

\institute{Harvard University} % Institution(s)
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\begin{document}
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%----------------------------------------------------------------------------------------
%	INTODUCTION
%----------------------------------------------------------------------------------------


\vspace{- 12.2 cm}
\begin{center}
{\includegraphics[scale=2.5]{../images/SEASLogo_RGB.png}}
\end{center}

\vspace{5 cm}

\begin{block}{Problem}
  \emph{How to recover an unknown network from the observation of contagion
  cascades?}
  \vspace{.5cm}
  \begin{itemize}
  \item {\bf Observe} $X^t_c$ (infections at time $t$ in cascade $c$)
  \item {\bf Objective}: find $\{\theta_{ij}\}$ (graph weight matrix)
\end{itemize}
\end{block}
\vspace{1cm}

\begin{block}{\bf Contagion Model~\cite{}}
\begin{itemize}
  \item Discrete time
  \item Infections drawn indep.~for each node conditioned on previous step
  \item Generalized Linear Model parametrization:
    \begin{framed}
    $$\mathbb{P}(X^{t+1}_j = 1 | X^t) = f(\Theta_j \cdot X^t)$$
    \end{framed}
\end{itemize}
\end{block}

\begin{block}{MLE}
  \begin{itemize}
    \item Log-likelihood is concave for common contagion models (IC model): SGD
      on $\{\theta_{ij}\}$
  \end{itemize}
  \vspace{1cm}
  \begin{equation*}
    \begin{split}
      \hat{\theta}\in \arg\max_\theta \sum_{t}~& y^t\log f(\theta\cdot x^t)
      \\ & + (1-y^t) \log \big(1 - f(\theta\cdot x^t)\big)
  \end{split}
\end{equation*}
\end{block}

\begin{block}{Bayesian Framework}
\begin{figure}
  \centering
  \includegraphics[scale=3]{graphical.pdf}
\end{figure}
\begin{itemize}
  \item $\phi$: fixed source distribution
  \item capture uncertainty on each edge
  \item encode expressive graph priors (tied parameters)
\end{itemize}
\end{block}

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\begin{block}{Active Learning}
  \emph{Can we gain by choosing the source node? If so, how to best choose the
  source node?}
\end{block}

\begin{block}{Heuristic 1 (Baseline)}
\end{block}

\begin{block}{Heuristic 2}
  \begin{itemize}
    \item Choose source to maximize mutual information
    \begin{equation*}
      \begin{split}
        I(\bx ,\Theta | x^0 = i) = &
      E_{\substack{\Theta \sim D_t \\ x |
      \Theta, i \sim D'}}\log p(x|\Theta) \\ - & E_{\substack{\Theta
      \sim D_t \\ x' | \Theta, i \sim D'}} \log p(x')
    \end{split}
    \end{equation*}
  \end{itemize}
\end{block}

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  \begin{block}{Experimental validation}
    \begin{figure}
      \centering
      \includegraphics[scale=1.2]{../images/watts_strogatz.pdf}
    \caption{Watts-Strogatz, $300$ nodes, $4500$ edges, uniform edge weights,
    constant $p_{init}$}
    \end{figure}
  \end{block}


\begin{block}{Conclusion}
  \begin{itemize}
    \item Introduce Generalized Linear Casacade model
    \item Better finite sample guarantees~\cite{Netrapalli:2012, Abrahao:13,
        Daneshmand:2014}
    \item Interpretable conditions on Hessian
    \item Lower bound+approximately sparse case developed in full
      paper~\cite{Pouget:2015}
  \end{itemize}
\end{block}

%-----------------------------------------------------------------------------
%	REFERENCES
%-----------------------------------------------------------------------------

\begin{block}{References}
  {\scriptsize
  \bibliography{../../paper/sparse}
\bibliographystyle{plain}}
\end{block}

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