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%----------------------------------------------------------------------------------------
%	TITLE SECTION 
%----------------------------------------------------------------------------------------

\title{Sparse Recovery for Graph Reconstruction } % Poster title

\author{Eric Balkanski, Jean Pouget-Abadie} % Author(s)

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


%\vspace{- 15.2 cm}
%\begin{center}
%{\includegraphics[height=7em]{logo.png}} % First university/lab logo on the left
%\end{center}

%\vspace{4.6 cm}

\begin{block}{Problem Statement}
\begin{center}
\bf{How can we reconstruct a graph on which observed cascades spread?}
\end{center} 
\end{block}


%{\bf Graph Reconstruction}:

%\begin{itemize}
%\item \{${\cal G}, \vec p$\}: directed graph, edge probabilities
%\item $F$: cascade generating model
%\item ${\cal M} := F\{{\cal G}, \vec p\}$: cascade
%\end{itemize}

%{\bf Objective}:
%\begin{itemize}
%\item Find algorithm which computes $F^{-1}({\cal M}) = \{{\cal G}, \vec p\}$ w.h.p., i.e. recovers graph from cascades.
%\end{itemize}

%{\bf Approach}
%\begin{itemize}
%\item Frame graph reconstruction as a {\it Sparse Recovery} problem for two cascade generating models.
%\end{itemize}

%Given a set of observed cascades, the \textbf{graph reconstruction problem} consists of finding the underlying graph on which these cascades spread. We assume that these cascades come from the classical \textbf{Independent Cascade Model} where at each time step, newly infected nodes infect each of their neighbor with some probability.

%In previous work, this problem has been formulated in different ways, including a convex optimization and a maximum likelihood problem. However, there is no known algorithm for graph reconstruction with theoretical guarantees and with a reasonable required sample size.

%We formulate a novel approach to this problem in which we use \textbf{Sparse Recovery} to find the edges in the unknown underlying network. Sparse Recovery is the problem of finding the sparsest vector $x$ such that $\mathbf{M x =b}$. In our case, for each node $i$, we wish to recover the vector $x = p_i$ where $p_{i_j}$ is the probability that node $j$ infects node $i$ if $j$ is active. To recover this vector, we are given $M$, where row $M_{t,k}$ indicates which nodes are infected at time $t$ in observed cascade $k$,  and $b$, where $b_{t+1,k}$ indicates if node $i$ is infected at time $t+1$ in cascade $k$. Since most nodes have a small number of neighbors in large networks, we can assume that these vectors are sparse. Sparse Recovery is a well studied problem which can be solved efficiently and with small error if $M$ satisfies certain properties. In this project, we empirically study to what extent $M$ satisfies the Restricted Isometry Property.


%---------------------------------------------------------------------------------
%---------------------------------------------------------------------------------

\begin{block}{Voter Model}

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{voter.png}
\end{figure}



\vspace{0.5 cm}
{\bf Description}

\vspace{0.5 cm}

\begin{itemize}
\item $\mathbb{P}$({\color{blue} blue} at $t=0) = p_{\text{init}}$
\item $\mathbb{P}$({\color{blue} blue} at $t+1) =  \frac{\text{Number of {\color{blue}blue} neighbors}}{\text{Total number of neighbors}}$
\end{itemize}

\vspace{0.5 cm}

{\bf Sparse Recovery Formulation}

\vspace{0.5 cm}


To recover the neighbors of $v_1$, observe which nodes are  {\color{red} red} (1) or {\color{blue} blue} (0) at time step $t$:
\begin{align*}
&v_2 \hspace{0.2 cm}  v_3 \hspace{0.2 cm} v_4 \hspace{0.2 cm} v_5 &\\
\vspace{1 cm}
M  = &  \left( \begin{array}{cccc}
0 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 \\
\end{array} \right)  &  \begin{array}{l} \hspace{ - 5cm} 
\text{time step 0} \\
 \hspace{ - 5cm}  \text{time step 1} \\
\end{array}
\end{align*}

and which color $v_1$ is at time step $t+1$ due to $M$:

\begin{align*}
b_1  = &  \left( \begin{array}{c}
1 \\
1 \\
\end{array} \right) &  \begin{array}{l} \hspace{ - 5cm} 
\text{time step 1} \\
 \hspace{ - 5cm}  \text{time step 2} \\
 \end{array}
\end{align*}

Then ,

\begin{equation}
\boxed{M \vec x_1 = \vec b_1 + \epsilon \nonumber}
\end{equation}

where $(\vec x_{1})_j := \frac{\text{1}}{\text{deg}(i)} \cdot \left[\text{j parent of 1 in }{\cal G}\right] $


\end{block}





%---------------------------------------------------------------------------------
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%----------------------------------------------------------------------------------------
%	CONSTRAINT SATISFACTION - BACKTRACKING
%----------------------------------------------------------------------------------------
\begin{block}{Independent Cascades Model}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{icc.png}
\end{figure}

\vspace{0.5 cm}

{\bf Description}

\vspace{0.5 cm}

\begin{itemize}
\item Three possible states:   {\color{blue} susceptible},   {\color{red} infected},  {\color{yellow} inactive }
\item $\mathbb{P}$(infected at t=0)$=p_{\text{init}}$
\item Infected node $j$ infects its susceptible neighbors $i$ with probability $p_{j,i}$ independently
\end{itemize}

\vspace{0.5 cm}

{\bf Sparse Recovery Formulation}

\vspace{0.5 cm}

To recover the neighbors of $v_5$,observe which nodes are  {\color{red} red} (1), {\color{blue} blue} (0), or {\color{yellow} yellow} (0) at time step $t$:
\begin{align*}
&v_1 \hspace{0.2 cm}  v_2 \hspace{0.2 cm} v_3 \hspace{0.2 cm} v_4 \\
\vspace{1 cm}
M  = &  \left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
\end{array} \right)   \begin{array}{l} \hspace{ 1cm} 
\text{time step 0} \\
 \hspace{ 1cm}  \text{time step 1} \\
 \end{array}
\end{align*}

and if $M$ caused $v_5$ to be infected at time step $t+1$:

\begin{align*}
b_5  = &  \left( \begin{array}{c}
0 \\
1 \\
\end{array} \right)  \begin{array}{l} \hspace{ 1cm} 
\text{time step 1} \\
 \hspace{ 1cm}  \text{time step 2} \\
 \end{array}
\end{align*}


Then,

\begin{equation}
\boxed{e^{M \vec \theta_5} = (1 - \vec b_5) + \epsilon} \nonumber
\end{equation}

where $(\vec \theta_5)_j := \log ( 1 - p_{j,5})  $

\vspace{1 cm}


This problem is a {\bf Noisy Sparse Recovery} problem, which has been studied extensively. Here, the vectors $\vec x_i$ are deg(i)-sparse. 


\end{block}

%----------------------------------------------------------------------------------------









%----------------------------------------------------------------------------------------
%	MIP
%----------------------------------------------------------------------------------------

% \begin{block}{RIP property}

% %The Restricted Isometry Property (RIP) characterizes a quasi-orthonormality of the measurement matrix M on sparse vectors. 

% For all k, we define $\delta_k$ as the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$

% \begin{equation}
% 1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k
% \end{equation}

% In general, the smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$!
 
% \end{block}

%----------------------------------------------------------------------------------------

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%----------------------------------------------------------------------------------------


\begin{block}{Algorithms}

{\bf Voter Model}

\begin{itemize}
\item Solve for each node i:
\begin{equation}
\min_{\vec x_i} \|\vec x_i\|_1 + \lambda \|M \vec x_i - \vec b_i \|_2 \nonumber
\end{equation}
\end{itemize}

{\bf Independent Cascade Model}

\begin{itemize}
\item Solve for each node i:
\begin{equation}
\min_{\vec \theta_i} \|\vec \theta_i\|_1 + \lambda \|e^{M \vec \theta_i} - (1 - \vec b_i) \|_2 \nonumber
\end{equation}
\end{itemize}

\end{block}

\begin{block}{Restricted Isometry Property (RIP)}
{\bf Definition}
\begin{itemize}
\item Characterizes a quasi-orthonormality of M on sparse vectors. 

\item The RIP constant $\delta_k$ is the best possible constant such that for all unit-normed ($l$2) and k-sparse vectors $x$:

\begin{equation}
1-\delta_k \leq \|Mx\|^2_2 \leq 1 + \delta_k \nonumber
\end{equation}

\item The smaller $\delta_k$ is, the better we can recover $k$-sparse vectors $x$.
\end{itemize}




\end{block}

\begin{block}{Theoretical Guarantees}

With small RIP constants $(\delta \leq 0.25)$ for $M$ and some assumption on the noise $\epsilon$:

{\bf Theorem \cite{candes}}

If node $i$ has degree $\Delta$  and $n_{\text{rows}}(M) \geq C_1 \mu \Delta \log n$, then, w.h.p.,

$$\| \hat x - x^* \|_2 \leq C (1 + \log^{3/2}(n))\sqrt{\frac{\Delta \log n}{n_{\text{rows}}(M) }}$$


\end{block}




%----------------------------------------------------------------------------------------
%	RESULTS
%----------------------------------------------------------------------------------------

\begin{block}{RIP Experiments}

\begin{center}
\begin{table}
\begin{tabular}{c | c | c | c | c }
& $c$ = 100, &$c$ = 1000,& $c$ = 100, &$c$ = 1000,\\
&  $i$ = 0.1& $i$ = 0.1& $i$ = 0.05& $i$ = 0.05\\
 \hline
  $\delta_4$ & 0.54 & 0.37 &0.43&0.23 \\
  \end{tabular} 
  \caption{RIP constant for a small graph. Here, $c$ is the number of cascades and $i$ is $p_{\text{init}}$.}
\end{table}
\end{center}


\end{block}

%----------------------------------------------------------------------------------------


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%----------------------------------------------------------------------------------------
%	IOVERALL COMPARISON
%----------------------------------------------------------------------------------------

%\vspace{- 14.2 cm}
%\begin{center}
%{\includegraphics[height=7em]{cmu_logo.png}} % First university/lab logo on the left
%\end{center}

%\vspace{4 cm}

\begin{alertblock}{Experimental Results}




\end{alertblock} 

%----------------------------------------------------------------------------------------


%----------------------------------------------------------------------------------------
%	CONCLUSION
%----------------------------------------------------------------------------------------

\begin{block}{Conclusion}

\begin{center}


{\bf Graph reconstruction can naturally be expressed as Sparse Recovery. Understanding properties of $M$, for example RIP, leads to theoretical guarantees on the reconstruction.}

\end{center}

\end{block}

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

\begin{block}{References}

\begin{thebibliography}{42}

\bibitem{candes}
Candès, E., and Plan, Y.
\newblock {\it A Probabilistic and RIPless Theory of Compressed Sensing}
\newblock Information Theory, IEEE Transactions on, 57(11): 7235--7254,
\newblock 2011.
\end{thebibliography}

\end{block}

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