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\title[Tissue-specific interactomes] % (optional, use only with long paper titles)
{A Convex Optimization Approach for Identification of Human Tissue-specific Interactomes}




\author[S. Mohammadi and A. Grama] % (optional, use only with lots of authors)
{Shahin Mohammadi and Ananth Grama}

\institute[Purdue] % (optional, but mostly needed)
{
  Department of Computer Science\\
  Purdue University
}

\date[ISMB] % (optional, should be abbreviation of conference name)
{July, 2016}



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\begin{document}

\begin{frame}{Combing Density and Overlap (CoDO)}{Assumptions}

\begin{block}{An overlap is significant if ...}
\begin{enumerate}
\item It is at least as dense as the containing graphs
\item It is ``large enough"
\end{enumerate}
\end{block}

\end{frame}


\begin{frame}{Combing Density and Overlap (CoDO)}{Formal Statement}

\begin{definition}
\begin{align*}
	\pCoDO
    &= \Pr[|\Set{\hat{A}} \lintersect \Set{\hat{B}}| \geq |\Set{Z}| ~\lintersect~ \delta(\Set{\hat{A}} ~\lintersect~ \Set{\hat{B}}) \geq \delta(\Set{Z})] \nonumber\\
\end{align*}
\end{definition}
where $\Set{Z}$ is the set of vertices in the overlap subgraph and $\delta()$ measures the density of a graph, i.e. $\frac{|E|}{C(|V|, 2)}$.
\end{frame}



\begin{frame}{Combing Density and Overlap (CoDO)}{Expansion}

By conditioning on the size of the overlap, we can get an explicit formula for this 
$p$-value in terms of hypergeometric tails:
\begin{align*}
	\pCoDO 
    &= \sum_{j=|\Set{Z}|}^{\min\{|\hat{A}|, |\hat{B}|\}}
      \Pr\left[|\hat{A}\lintersect \hat{B}| = j\right]
      \cdot \Pr\left[\delta(\hat{A}\lintersect \hat{B}) \geq \delta(\Set{Z}) \big|  |\hat{A} \lintersect \hat{B}| = j \right]
\end{align*}
\end{frame}



\begin{frame}{Combing Density and Overlap (CoDO)}{Example}

\begin{figure}[!h]
\centering
\hfil%
\subfigure[\footnotesize Insignificant Overlap/Insignificant Density\label{fig:inSigOver-inSigDen}]{\includegraphics[width=.35\columnwidth]{figures/inSigOverlap-inSigDensity}}%
\hfil%
\subfigure[\footnotesize Insignificant Overlap/Significant Density\label{fig:inSigOver-SigDen}]{\includegraphics[width=.35\columnwidth]{figures/inSigOverlap-SigDensity}}%
\hfil%
\subfigure[\footnotesize Significant Overlap/Insignificant Density\label{fig:SigOver-inSigDen}]{\includegraphics[width=.35\columnwidth]{figures/SigOverlap-inSigDensity}}%
\hfil%
\subfigure[\footnotesize Significant Overlap/Significant Density\label{fig:SigOver-SigDen}]{\includegraphics[width=.35\columnwidth]{figures/SigOverlap-SigDensity}}%
\end{figure}
\end{frame}




\begin{frame}{Combing Density and Overlap (CoDO)}{Application-- Social networks}
\begin{figure}[!h]
\centering
\hfil%
\subfigure[Facebook\label{fig:FB}]{\includegraphics[width=.3\columnwidth]{figures/facebook}}%
\hfil%
\subfigure[Google+\label{fig:GPlus}]{\includegraphics[width=.3\columnwidth]{figures/gplus}}%
\hfil%
\subfigure[Twitter\label{fig:Twitter}]{\includegraphics[width=.3\columnwidth]{figures/twitter}}%
\end{figure}

\begin{definition}
\small \alert{Ego Net} is the induced subgraph among friends (alters) of a given user (ego)
\end{definition}

\end{frame}


\begin{frame}{Combing Density and Overlap (CoDO)}{Application-- Biological networks}

\begin{center}

\begin{figure}
\includegraphics[width=0.6\columnwidth]{figures/CoDO_annotated}
\end{figure}

Overlap among KEGG pathways is an indicator of \alert{pathway cross-talk}
\end{center}

\end{frame}
\end{document}


