%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% baposter Landscape Poster
% LaTeX Template
% Version 1.0 (11/06/13)
%
% baposter Class Created by:
% Brian Amberg (baposter@brian-amberg.de)
%
% This template has been downloaded from:
% http://www.LaTeXTemplates.com
%
% License:
% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%----------------------------------------------------------------------------------------
%	PACKAGES AND OTHER DOCUMENT CONFIGURATIONS
%----------------------------------------------------------------------------------------

\documentclass[portrait,a0paper,fontscale=0.285]{baposter} % Adjust the font scale/size here

\usepackage{graphicx} % Required for including images
\graphicspath{{figures/}} % Directory in which figures are stored

\usepackage{amsmath} % For typesetting math
\usepackage{amssymb} % Adds new symbols to be used in math mode

\usepackage{booktabs} % Top and bottom rules for tables
\usepackage{enumitem} % Used to reduce itemize/enumerate spacing
\usepackage{palatino} % Use the Palatino font
\usepackage[font=small,labelfont=bf]{caption} % Required for specifying captions to tables and figures

\usepackage{multicol} % Required for multiple columns
\setlength{\columnsep}{1.5em} % Slightly increase the space between columns
\setlength{\columnseprule}{0mm} % No horizontal rule between columns
\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc}    % use 8-bit T1 fonts

\usepackage{url}            % simple URL typesetting
\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}         % colors

\usepackage{mathtools}

\usepackage{stfloats}

%\usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=black]{hyperref}       % hyperlinks
\makeatletter
\def\Hy@Warning#1{} % squelch hyperref
\makeatother




\def\dominik#1{ { \bf \color{red} #1 }\normalfont\color{black} }


\usepackage{relsize}

\def\CC{C\nolinebreak[4]\hspace{-.05em}\raisebox{.4ex}{\relsize{-2}{\textbf{++}}}}





\usepackage{calc}
\usepackage{enumitem}
\usepackage{float}









\usepackage{amsmath}
\usepackage{amsthm}


\usepackage[capitalize]{cleveref}


\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc}    % use 8-bit T1 fonts

\usepackage{url}            % simple URL typesetting
\usepackage{booktabs}       % professional-quality tables
\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}         % colors

\usepackage{mathtools}

\usepackage{stfloats}

%\usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=black]{hyperref}       % hyperlinks
\makeatletter
\def\Hy@Warning#1{} % squelch hyperref
\makeatother




\def\dominik#1{ { \bf \color{red} #1 }\normalfont\color{black} }


\usepackage{relsize}

\def\CC{C\nolinebreak[4]\hspace{-.05em}\raisebox{.4ex}{\relsize{-2}{\textbf{++}}}}





\usepackage{calc}
\usepackage{enumitem}
\usepackage{float}









\usepackage{amsmath}
\usepackage{amsthm}


\usepackage[capitalize]{cleveref}


\usepackage{bbm}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{trees, shapes}
\usetikzlibrary{shapes,decorations,arrows,calc,arrows.meta,fit,positioning}
\usetikzlibrary{shadows,shadings,shapes.symbols}
\tikzset{
	double color fill/.code 2 args={
		\pgfdeclareverticalshading[%
		tikz@axis@top,tikz@axis@middle,tikz@axis@bottom%
		]{diagonalfill}{100bp}{%
			color(0bp)=(tikz@axis@bottom);
			color(50bp)=(tikz@axis@bottom);
			color(50bp)=(tikz@axis@middle);
			color(50bp)=(tikz@axis@top);
			color(100bp)=(tikz@axis@top)
		}
		\tikzset{shade, left color=#1, right color=#2, shading=diagonalfill}
	}
}
\usepgfplotslibrary{fillbetween}
\usepackage{graphicx}
\usetikzlibrary{decorations.pathreplacing, matrix}

\usepackage[boxed, linesnumbered, noend, noline]{algorithm2e}
\SetKwInput{KwData}{Input}
\SetKwInput{KwResult}{Output}
\DeclareMathOperator*{\argmin}{arg\,min}

\newcommand{\wxstar}[1]{\vW^\star_{{#1}}}
\newcommand{\wx}[1]{\vW_{{#1}}}
\newcommand{\wxj}[2]{\vW_{{#1},{#2}}}
\newcommand{\wtx}[1]{\overline{\vW}_{{#1}}}
\newcommand{\wq}{\sum_{j=1}^{s-1} 2^{j/s-1} w_j}


\renewcommand\log{\ln}
\newcommand\disteq{\sim}
\newcommand\dist{\mathrm{dist}}

\newcommand\SPIV{{\tt SPIV}}
\newcommand\DD{{DD}}
\newcommand\SCOMP{{SCOMP}}
\newcommand\COMP{{COMP}}


\newcommand{\fB}{\mathfrak B}
\newcommand{\fT}{\mathfrak T}
\newcommand{\fZ}{\mathfrak Z}

\newcommand{\va}{\vec a}
\newcommand{\vA}{\vec A}
\newcommand{\vg}{\vec g}
\newcommand{\vh}{\vec h}
\newcommand{\vd}{\vec d}
\newcommand{\vD}{\vec D}
\newcommand{\vt}{\vec t}
\newcommand{\vw}{\vec w}
\newcommand{\vc}{\vec c}
\newcommand{\vs}{\vec s}
\newcommand{\vz}{\vec z}
\newcommand{\vX}{\vec X}
\newcommand{\vY}{\vec Y}
\newcommand{\vR}{\vec R}
\newcommand{\vH}{\vec H}
\newcommand{\vC}{\vec C}

\renewcommand{\epsilon}{\eps}


\newcommand\vI{\vec I}
\newcommand\vJ{\vec J}

\newcommand{\GG}{\mathbb G}
\newcommand{\limby}{\lim_{\beta \to \infty: \beta y =\Theta(1)}}

\newcommand\ALPHA{\vec\alpha}
\newcommand\MU{\vec\mu}
%\newcommand\vY{\vec Y}
\newcommand\vB{\vec B}
\newcommand\vr{\vec r}
\newcommand\vm{\vec m}
\newcommand\vn{\vec n}
\newcommand\NU{\vec\nu}
\newcommand\cMU{\check\MU}
\newcommand\cSIGMA{\check\SIGMA}

\newcommand\vU{\vec U}
\newcommand\hG{\hat\G}

\newcommand\GAMMA{{\vec\gamma}}
\newcommand\OMEGA{\vec\omega}
\newcommand\PSI{\vec\psi}
\newcommand\RHO{{\vec\rho}}
\newcommand\PHI{\vec\Phi}
\newcommand\VARPHI{\vec\varphi}

\newcommand\nix{\,\cdot\,}
\newcommand\vV{\vec V}
\newcommand\vW{\vec W}
\newcommand\vS{\vec S}
\newcommand\lam{\lambda}

\newcommand\reg{\G}
\newcommand\dd{{\mathrm d}}

\newcommand\GR{{\bf G}}
\newcommand\GRE{G_{\text{ER}}}
\newcommand\T{\vec T}
% \newcommand\vH{\vec H}

\renewcommand{\vec}[1]{\boldsymbol{#1}}

\newcommand\bem{\bf\em}
\newcommand\bemph[1]{{\bf\em #1}}

\newcommand\KL[2]{D_{\mathrm{KL}}\bc{{{#1}\|{#2}}}}

\newcommand\SIGMA{\vec\sigma}
\newcommand\CHI{\vec\chi}
\newcommand\TAU{\vec\tau}
\newcommand\cutm{\Delta_{\Box}}
\newcommand\Cutm{\cD_{\Box}}
\newcommand\CUTM{D_{\Box}}

\newtheorem{definition}{Definition}[section]
\newtheorem{claim}[definition]{Claim}
\newtheorem{example}[definition]{Example}
\newtheorem{remark}[definition]{Remark}
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{corollary}[definition]{Corollary}
\newtheorem{Algo}[definition]{Algorithm}
\newtheorem{fact}[definition]{Fact}
\newtheorem{hypothesis}[definition]{Hypothesis}
\newtheorem{experiment}[definition]{Experiment}
\newtheorem{conjecture}[definition]{Conjecture}

\newcommand\fG{\mathfrak{G}}
\newcommand\fS{\mathfrak{S}}
\newcommand\fK{\mathfrak{K}}
\newcommand\fD{\mathfrak{D}}
\newcommand\fM{\mathfrak{M}}
\newcommand\fQ{\mathfrak{Q}}
\newcommand\fC{\mathfrak{C}}
\newcommand\fY{\mathfrak{Y}}
\newcommand\fA{\mathfrak{A}}
\newcommand\fF{\mathfrak{F}}
\newcommand\fP{\mathfrak{P}}
\newcommand\fp{\mathfrak{a}}
\newcommand\fr{\mathfrak{r}}

\newcommand\cA{\mathcal{A}}
\newcommand\cB{\mathcal{B}}
\newcommand\cC{\mathcal{C}}
\newcommand\cD{\mathcal{D}}
\newcommand\cF{\mathcal{F}}
\newcommand\cG{\mathcal{G}}
\newcommand\cE{\mathcal{E}}
\newcommand\cU{\mathcal{U}}
\newcommand\cN{\mathcal{N}}
\newcommand\cQ{\mathcal{Q}}
\newcommand\cH{\mathcal{H}}
\newcommand\cS{\mathcal{S}}
\newcommand\cT{\mathcal{T}}
\newcommand\cI{\mathcal{I}}
\newcommand\cK{\mathcal{K}}
\newcommand\cJ{\mathcal{J}}
\newcommand\cL{\mathcal{L}}
\newcommand\cM{\mathcal{M}}
\newcommand\cO{\mathcal{O}}
\newcommand\cP{\mathcal{P}}
\newcommand\cX{\mathcal{X}}
\newcommand\cY{\mathcal{Y}}
\newcommand\cV{\mathcal{V}}
\newcommand\cW{\mathcal{W}}
\newcommand\cZ{\mathcal{Z}}
\def\cR{{\mathcal R}}
\def\cE{{\mathcal E}}
\def\bC{{\bf C}}
\def\bT{{\bf T}}
\def\bM{{\bf M}}


\newcommand\fX{\mathfrak{X}}
\newcommand\ve{\vec e}
\newcommand\vu{\vec u}
%\newcommand\vv{\vec v}
\newcommand\vx{\vec x}
\newcommand\vZ{\vec Z}
\newcommand\vM{\vec M}
\newcommand\vy{\vec y}
\newcommand\THETA{\vec\theta}

\newcommand\atom{\delta}


\newcommand\eul{\mathrm{e}}
\newcommand\eps{\varepsilon}
\newcommand\del{\delta}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\NN{\mathbb{N}}
\newcommand\ZZpos{\mathbb{Z}_{\geq0}}
\newcommand\Var{\mathrm{Var}}
\newcommand\Erw{\mathbb{E}}
\newcommand{\vecone}{\vec{1}}
\newcommand{\Vol}{\mathrm{Vol}}

\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\Po}{{\rm Po}}
\newcommand{\Bin}{{\rm Bin}}
\newcommand{\Mult}{{\rm Mult}}
\newcommand{\Be}{{\rm Be}}

\newcommand\TV[1]{\left\|{#1}\right\|_{\mathrm{TV}}}
\newcommand\tv[1]{\|{#1}\|_{\mathrm{TV}}}
\newcommand\dTV{d_{\mathrm{TV}}}

\newcommand{\bink}[2] {{\binom{#1}{#2}}}

\newcommand\bc[1]{\left({#1}\right)}
\newcommand\cbc[1]{\left\{{#1}\right\}}
\newcommand\bcfr[2]{\bc{\frac{#1}{#2}}}
\newcommand{\bck}[1]{\left\langle{#1}\right\rangle}
\newcommand\brk[1]{\left\lbrack{#1}\right\rbrack}
\newcommand\scal[2]{\bck{{#1},{#2}}}
\newcommand\norm[1]{\left\|{#1}\right\|}
\newcommand\abs[1]{\left|{#1}\right|}
\newcommand\uppergauss[1]{\left\lceil{#1}\right\rceil}
\newcommand\lowergauss[1]{\left\lfloor{#1}\right\rfloor}
\newcommand\ug[1]{\left\lceil{#1}\right\rceil}
\newcommand\RR{\mathbb{R}}
\newcommand\RRpos{\RR_{\geq0}}
\newcommand{\Whp}{W.h.p.}
\newcommand{\whp}{w.h.p.}
\newcommand{\wupp}{w.u.p.p.}
\newcommand{\tensor}{\otimes}

\newcommand{\Karonski}{Karo\'nski}
\newcommand{\Erdos}{Erd\H{o}s}
\newcommand{\Renyi}{R\'enyi}
\newcommand{\Lovasz}{Lov\'asz}
\newcommand{\Juhasz}{Juh\'asz}
\newcommand{\Bollobas}{Bollob\'as}
\newcommand{\Furedi}{F\"uredi}
\newcommand{\Komlos}{Koml\'os}
\newcommand{\Luczak}{\L uczak}
\newcommand{\Mezard}{M\'ezard}
\newcommand{\Kucera}{Ku\v{c}era}
\newcommand{\Szemeredi}{Szemer\'edi}

\newcommand\pr{\mathbb{P}} 
\renewcommand\Pr{\pr} 

\newcommand\Lem{Lemma}
\newcommand\Prop{Proposition}
\newcommand\Thm{Theorem}
\newcommand\Def{Definition}
\newcommand\Cor{Corollary}
\newcommand\Sec{Section}
\newcommand\Chap{Chapter}
\newcommand\Rem{Remark}
\newcommand{\Faadi}{Fa\`a di Bruno}
\newcommand\algstyle{\small\sffamily}
\newcommand\id{\mathrm{id}}
\newcommand{\Eig[1]}{\mathrm{Eig}^{\ast} (#1)}
\newcommand{\eig}{\mathrm{Eig}}
\newcommand{\Pomast}{\cP^2_\ast (\Omega)}
\newcommand\ZZZ{\mathbb{Z}}
\newcommand{\planted}{\pi^{\mathrm{pl}}}
\newcommand{\dcond}{d_{\mathrm{cond}}}
\newcommand{\dsat}{d_{\mathrm{sat}}}

\newcommand{\cGg}{\cG_{\Gamma}}
\newcommand{\cGd}{\cG_{\Delta}}



\newcommand{\ecut}{e_{\text{cut}}}
\newcommand{\ecutm}{{e}^*_{\text{cut}}}
\newcommand{\euncut}{e_{\text{uncut}}}
\newcommand{\euncutm}{{e}^*_{\text{uncut}}}



\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\rk}[1]{\rank(#1)}
\def\bin{{\bf Bin}}
\newcommand{\supp}[1]{{\text{supp}\left(#1\right)}}
\newcommand{\ind}[1]{\mathrm{ind}(#1)}
\def\geo{{\bf Geo}}
\def\con{{\bf Con}}


\newcommand\A{\vA}

\def\E{{\mathcal E}}
\def\G{{\vec G}}


\def\po{{\bf Po}}
\def\ex{{\mathbb E}}
\def\pr{{\mathbb P}}
\newcommand\expc[1]{{\left< #1\right>}}



\def\bfa{{\bf a}}
\def\bfb{{\bf b}}
\def\bfc{{\bf c}}
\def\bfd{{\vec d}}
\def\bfe{{\bf e}}
\def\bff{{\bf f}}
\def\bfg{{\bf g}}
\def\bfh{{\bf h}}
\def\bfi{{\bf i}}
\def\bfj{\vec j}
\def\bfk{\vk}
\def\bfl{{\bf l}}
\def\bfm{{\vec m}}
\def\bfn{{\vec n}}
\def\bfo{{\bf o}}
\def\bfp{{\bf p}}
\def\bfq{{\bf q}}
\def\bfr{{\bf r}}
\def\bfs{{\bf s}}
\def\bft{{\bf t}}
\def\bfu{{\bf u}}
\def\bfv{{\bf v}}
\def\bfw{{\bf w}}
\def\bfx{{\bf x}}
\def\bfy{{\bf y}}
\def\bfz{{\bf z}}

\def\bfA{{\bf A}}
\def\bfB{{\bf B}}
\def\bfC{{\bf C}}
\def\bfD{{\bf D}}
\def\bfE{{\bf E}}
\def\bfF{{\bf F}}
\def\bfG{{\bf G}}
\def\bfH{{\bf H}}
\def\bfI{{\bf I}}
\def\bfJ{{\bf J}}
\def\bfK{{\bf K}}
\def\bfL{{\bf L}}
\def\bfM{{\bf M}}
\def\bfN{{\bf N}}
\def\bfO{{\bf O}}
\def\bfP{{\bf P}}
\def\bfQ{{\bf Q}}
\def\bfR{{\bf R}}
\def\bfS{{\bf S}}
\def\bfT{{\bf T}}
\def\bfU{{\bf U}}
\def\bfV{{\bf V}}
\def\bfW{{\bf W}}
\def\bfX{{\bf X}}
\def\bfY{{\bf Y}}
\def\bfZ{{\bf Z}}

\def\bfgamma{{\pmb{\gamma}}}
\def\bfmu{{\pmb{\mu}}}
\def\bftheta{{\pmb{\theta}}}
\def\bfsigma{{\pmb{\sigma}}}
\def\bftau{{\pmb{\tau}}}

\usepackage{lipsum}

\def\bbF{{\mathbb F}}
\def\bbN{{\mathbb N}}
\def\bbC{{\mathbb C}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}

\def\cH{{\mathcal H}}

\newcommand{\remove}[1]{}
\newcommand\eqn[1]{(\ref{#1})}


%\newcommand\coloneqq{:=}
\newcommand{\vecGamma}{\vec{\Gamma}}

\newcommand{\one}{V_1}
\newcommand{\zero}{V_0}
\newcommand{\zerominus}{V_{0-}}
\newcommand{\oneminusminus}{V_{1--}}
\newcommand{\zeroplus}{V_{0+}}
\newcommand{\oneplus}{V_{1+}}
\newcommand\w{{\omega}}


\newcommand{\muk}{\mu_{k}}
\newcommand{\sigmai}{\SIGMA_{i}}

\newcommand{\quantnsm}{\vec\Phi_i}
\newcommand{\quantns}{\vec\Psi_i}

\newcommand\mcount{m_{\mathrm{count}}}
\newcommand\mada{m_{\mathrm{ada}}}
\newcommand\madapt{m_{\mathrm{ad}}}
\newcommand\minf{m_{\mathrm{inf}}}
\newcommand{\minfm}[1]{\minf(#1)}
\newcommand\malg{m_{\mathtt{DD}}}
\newcommand\mDD{m_{\mathrm{DD}}}
\newcommand\mSC{m_{\mathrm{SPIV}}}

\newcommand\mseed{m_{\mathrm{seed}}}
\newcommand\mbulk{m_{\mathrm{bulk}}}

\newcommand{\gk}{G}
\newcommand{\gp}{\hat G}
\newcommand{\gd}{G'}

\newcommand{\aco}[1]{\textcolor{red}{#1}}
\newcommand{\pl}[1]{\textcolor{blue}{#1}}
\newcommand{\mhk}[1]{\textcolor{blue}{#1}}
\newcommand{\geb}[1]{\textcolor{green}{#1}}
\newcommand{\op}[1]{\textcolor{violet}{#1}}
\newcommand{\manpen}[1]{\textcolor{mvertexa}{#1}}
\newcommand{\nt}[1]{\textcolor{olive}{#1}}
\newcommand{\mr}[1]{\textcolor{cyan}{#1}}
\newcommand{\lk}[1]{\textcolor{magenta}{#1}}

\newcommand{\be}{\begin{equation}}
	\newcommand{\bel}[1]{\begin{equation}\lab{#1}\ }
		\newcommand{\ee}{\end{equation}}
	\newcommand{\bea}{\begin{eqnarray}}
		\newcommand{\eea}{\end{eqnarray}}
	\newcommand{\bean}{\begin{eqnarray*}}
		\newcommand{\eean}{\end{eqnarray*}}
	
	\newcommand{\vN}{\vec N}
	
	\newcommand{\mone}{\vm_1}
	\newcommand{\mzero}{\vm_0}
	
	\graphicspath{{..}}
	\pgfplotsset{compat=1.14}
	
	
	
	\tikzstyle{node} = [shape=circle,draw=black]
	\newcommand{\colLil}{violet}
	\newcommand{\colOrange}{orange}
	\newcommand{\colGreen}{green}
	\newcommand{\colBlue}{blue}
	\tikzstyle{node} = [shape=circle,draw=black]
	\tikzstyle{dottetReplace} =[fill = violet!20!white]
	\newcommand{\betweenDist}{0}
	
	
	
	
	\newcommand{\pz}{\vec{\omega}}
	\newcommand{\cc}{\vec{\omega}_c}
	\newcommand{\activenodes}{\vec{X}^\star}
	\newcommand{\candidates}{\mathcal{C}}
	\newcommand{\distactive}[1]{t_{#1}^{\activenodes}}
	\newcommand{\lca}{\cc}
\newtheorem{theo}{Theorem}
\definecolor{lightblue}{rgb}{0.145,0.6666,1} % Defines the color used for content box headers
\definecolor{mossgreen}{rgb}{0.68, 0.87, 0.68}
\definecolor{lightgreen}{rgb}{0.56, 0.93, 0.56}

\begin{document}

\begin{poster}
{
headerborder=closed, % Adds a border around the header of content boxes
colspacing=1em, % Column spacing
bgColorOne=white, % Background color for the gradient on the left side of the poster
bgColorTwo=white, % Background color for the gradient on the right side of the poster
borderColor=mossgreen, % Border color
headerColorOne=green, % Background color for the header in the content boxes (left side)
headerColorTwo=lightgreen, % Background color for the header in the content boxes (right side)
headerFontColor=black, % Text color for the header text in the content boxes
boxColorOne=white, % Background color of the content boxes
textborder=roundedleft, % Format of the border around content boxes, can be: none, bars, coils, triangles, rectangle, rounded, roundedsmall, roundedright or faded
eyecatcher=true, % Set to false for ignoring the left logo in the title and move the title left
headerheight=.125\textheight, % Height of the header
headershape=roundedright, % Specify the rounded corner in the content box headers, can be: rectangle, small-rounded, roundedright, roundedleft or rounded
headerfont=\Large\bf\textsc, % Large, bold and sans serif font in the headers of content boxes
%textfont={\setlength{\parindent}{1.5em}}, % Uncomment for paragraph indentation
linewidth=2pt % Width of the border lines around content boxes
}
%----------------------------------------------------------------------------------------
%	TITLE SECTION 
%----------------------------------------------------------------------------------------
%
{
%	\includegraphics[height=0em]{tulogo.pdf}
} % First university/lab logo on the left
%{\vspace{5pt} \\\bf\textsc{Rumor Source Detection} \\ 
%\vspace{5pt}\Large{Lena Krieg, Olga Scheftelowitsch} \\ \vspace{1pt} \large{based on}\vspace{3pt}} % Poster title
{\bf\LARGE{Inference of a Rumor's Source in the Independent Cascade Model \\\vspace{3pt} \large{Petra Berenbrink, Max Hahn-Klimroth, Dominik Kaaser, Lena Krieg, Malin Rau }}\vspace{0.5em}} % Poster title % Author names and institution
%{\includegraphics[height=4em]{tulogo.pdf}} % Second university/lab logo on the right

%----------------------------------------------------------------------------------------
%	Introduction
%----------------------------------------------------------------------------------------



\headerbox{Introduction}{name=intro,column=0,row=0}{
	
We consider the so-called \emph{Independent Cascade Model} for rumor spreading or epidemic processes popularized by Kempe et al.\ [2003].
In this model, a small subset of nodes from a network are the source of a rumor.
In discrete time steps, each informed node "infects" each of its uninformed neighbors with probability $p$.
While many facets of this process are studied in the literature, less is known about the inference problem: given a number of infected nodes in a network, can we learn the source of the rumor?
	
\vspace{0.3em} % When there are two boxes, some whitespace may need to be added if the one on the right has more content
}






%----------------------------------------------------------------------------------------
%	OBJECTIVES
%----------------------------------------------------------------------------------------



%----------------------------------------------------------------------------------------
% Applications
%----------------------------------------------------------------------------------------

%----------------------------------------------------------------------------------------
% model
%----------------------------------------------------------------------------------------

\headerbox{$\begin{array}{l}\vspace{-25pt}\\
		\text{Independant Cascade}\\ \text{ Model}
	\end{array}$}{name=mod,column=0,row=1, below = intro, boxheaderheight=40pt }{
	We observe the following model:
	Let $G=(V,E)$ be a graph. The model works in discrete steps. Every node has exactly one of the following attributes in every step: 
	\begin{itemize}
		\itemsep0em
		\item currently active
		\item formerly active
		\item inactive
	\end{itemize}
	$I_t$ denotes the set of active nodes in step $t$. All active nodes $I_t$ have a chance to infect each of their inactive neighbors with probability $p$. All the infected neighbors of the nodes ist set $I_t$ form the set $I_{t+1}$.
	
}

\headerbox{ML-Estimator}{name=ml,column=0,row=1, below = mod }{
	We assume that we observe the network at an unknown time and can only see the set nodes ($\activenodes$) that are currently active. We also assume that the source of the rumor is a single node $I_0=\{\pz\}$. Note that $\activenodes$ can be empty. 
	
	Let
	$$\candidates= \{v\in V \mid \exists t \in \mathbb{N} :\forall w \in \activenodes : \dist(v,w)=t\} $$
	As long as $\activenodes$ is not empty, our estimator is defined as the \emph{closest candidate}:
	$$\cc=\argmin_{v\in \candidates}\{dist(v,w) \mid w\in \activenodes\}$$
	%Note that the set $\candidates$ is not empty since $\pz\in \candidates$. 
	If $|\candidates|<2$, then the estimation fails.
}


%----------------------------------------------------------------------------------------
%	RESULTS 1
%----------------------------------------------------------------------------------------
\headerbox{Visualization}{name=vis,column=1,span=2,row=0}{
	\begin{figure}[H]
		\vspace{-5pt}
		\centering
		\hspace{-30pt}
		\begin{subfigure}{0.45\textwidth}
			\centering
			%    \begin{minipage}{0.4\textwidth}
				\input{circleTree}
				%    \end{minipage}\begin{minipage}{0.6\textwidth}
				\caption{
					Visualization of a possible snapshot of the spreading process in a 4-regular tree. 
					The orange nodes are the active elements. 
					The candidate set $\candidates$ of possible rumor's sources consists of all vertices in the purple sub-tree. 
				}
				\label{fig:circltree}
				%\end{minipage}
			\end{subfigure} \hspace{10pt}
			\begin{subfigure}{0.45\textwidth}
				\centering
				%    \begin{minipage}{0.7 \textwidth}
					\input{tikz}
					%    \end{minipage}\begin{minipage}{0.3 \textwidth}
					\caption{
						Here, $\cc$ spawned three sub-trees out of which two contain active elements of $\activenodes$ (orange) and one does not contain active elements (purple). Thus, $\candidates$ consists of all vertices in the purple sub-tree rooted at $\cc$.
					}
					%    \end{minipage}
				\label{fig:Notation_Tree}
			\end{subfigure}
		\end{figure}
		
	}

\headerbox{d-regular trees}{name=dreg,column=1,span=2,row=0, below=vis}{
\begin{theo}[$d$-regular trees]\label{thm_lca_dreg}
	Let $p$ be the spreading parameer of the Independant Cascade Model and $t = \omega(1)$ the amount of steps. 
	Then, the following phase-transitions occur.
	\begin{itemize}
		\item If $(d-1)\cdot p \leq 1$, any estimator fails at weak detection with probability $1-o_t(1)$.\footnote{We denote by $o_t(1)$ a quantity that tends to zero with $t \to \infty$.}
		\item If $1 < (d-1)\cdot p = \Theta(1)$ then the closest candidate $\cc$ is the source  of the rumor $\pz$ with constant probability (weak detection). 
		Furthermore, the probability that $\dist(\cc, \pz) > k$ is at most $\exp \bc{ - \Omega(k)}$.
		\item If $(d-1)\cdot p = \omega(1)$ then closest candidate $\cc$ is the source of the rumor $\pz$  with probability $1 - o_{d}(1)$ (strong detection).
	\end{itemize}
\end{theo}
}


%----------------------------------------------------------------------------------------
%	RESULTS 2
%----------------------------------------------------------------------------------------






%----------------------------------------------------------------------------------------
%	FUTURE RESEARCH
%----------------------------------------------------------------------------------------

%\headerbox{Future Research}{name=futureresearch,column=1,span=2,aligned=references,above=bottom}{ % This block is as tall as the references block
%
%\begin{multicols}{2}
%Integer sed lectus vel mauris euismod suscipit. Praesent a est a est ultricies pellentesque. Donec tincidunt, nunc in feugiat varius, lectus lectus auctor lorem, egestas molestie risus erat ut nibh.
%
%Maecenas viverra ligula a risus blandit vel tincidunt est adipiscing. Suspendisse mollis iaculis sem, in \emph{imperdiet} orci porta vitae. Quisque id dui sed ante sollicitudin sagittis.
%\end{multicols}
%}


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

\headerbox{$\Po(\lambda)$-galton-watson trees}{name=gwatson,column=1,span=2,row=1,below=dreg}{

\begin{theo}[Galton-Watson processes]\label{thm_lca_gw}
	Let $p$ be the spreading parameer of the Independant Cascade Model and $t = \omega(1)$ the amount of steps. 
	Then, the following phase-transitions occur.
	\begin{itemize}
		\item If $\lambda p \leq 1$, any estimator fails at weak detection with probability $1-o_t(1)$.
		\item If $1 < \lambda p = \Theta(1)$, then the closest candidate $\cc$ is the source  of the rumor $\pz$ with positive probability (weak detection).
		Furthermore, the probability that $\dist(\lca, \pz) > k$ is at most $\exp \bc{ - \Omega(k)}$.
		\item If $\lambda p = \omega(1)$, then closest candidate $\cc$ is the source of the rumor $\pz$  with probability $1 - o_{\lambda}(1)$  (strong detection).
	\end{itemize}
\end{theo}	
\vspace{3pt}
}


\headerbox{Experiments}{name=exp,column=0,span=2,row=0, below=ml}{
	\vspace{-27pt}
	\begin{figure}[H]
		\hspace{7pt}
		\begin{minipage}{0.3\textwidth}
			\vspace{0pt}
			\begin{figure}[H]
				\def\ps{\small}
				%\begin{minipage}{1.8in}
				\scalebox{0.95}{
					\input{plot1}%\end{minipage}\begin{minipage}{\columnwidth-1.8in}
				}
				\caption{Success rates for Erdős-Rényi graphs, random regular graphs, and random geometric graphs.}
				\label{fig:plot-1}
				%\end{minipage}
			\end{figure}
		\end{minipage} \hspace{10pt}
		\begin{minipage}{0.3\textwidth}
			\begin{figure}[H]
				\def\ps{\small}
				%\begin{minipage}{1.8in}
				\scalebox{0.95}{\input{plot2}}
				\caption{Success rates in a random geometric graph with expected node degree $16$ after $8$, $16$, $32$ steps}
				\label{fig:plot-2}
				%\end{minipage}
			\end{figure}
		\end{minipage} \hspace{10pt}
	\begin{minipage}{0.3\textwidth}
		\begin{figure}[H]
		\def\ps{\small}
		%\begin{minipage}{1.8in}
		\scalebox{0.95}{\input{plot3}}
		\caption{Histogram of the distribution of the distances of the candidates returned by our heuristic to $\pz$ for $p = 0.45, 0.5, 0.55$.}
		\label{fig:plot3}
		%\end{minipage}
	\end{figure}
	\end{minipage}
	\end{figure}
	\vspace{-10pt}
	
	%\vspace{0.75em} % When there are two boxes, some whitespace may need to be added if the one on the right has more content
}

\headerbox{References}{name=ref,column=2,row=0, below = gwatson }{
	\renewcommand{\section}[2]{\vskip 0.05em}
	\begin{thebibliography}{2}
		\bibitem{} Petra Berenbrink, Max Hahn-Klimroth, Dominik Kaaser, Lena Krieg, Malin Rau: Inference of a Rumor's Source in the Independent Cascade Model. UAI 2023.
		\bibitem{}  David Kempe, Jon Kleinberg, Éva Tardos: Maximizing the spread of influence through a social network. SIGKDD 2003. 
		
	\end{thebibliography}
}

%\begin{multicols}{2}
%
%\tikzstyle{decision} = [diamond, draw, fill=blue!20, text width=4.5em, text badly centered, node distance=2cm, inner sep=0pt]
%\tikzstyle{block} = [rectangle, draw, fill=blue!20, text width=5em, text centered, rounded corners, minimum height=4em]
%\tikzstyle{line} = [draw, -latex']
%\tikzstyle{cloud} = [draw, ellipse, fill=red!20, node distance=3cm, minimum height=2em]
%
%\begin{tikzpicture}[node distance = 2cm, auto]
%\node [block] (init) {Initialize Model};
%\node [cloud, left of=init] (Start) {Start};
%\node [cloud, right of=init] (Start2) {Start Two};
%\node [block, below of=init] (init2) {Initialize Two};
%\node [decision, below of=init2] (End) {End};
%\path [line] (init) -- (init2);
%\path [line] (init2) -- (End);
%\path [line, dashed] (Start) -- (init);
%\path [line, dashed] (Start2) -- (init);
%\path [line, dashed] (Start2) |- (init2);
%\end{tikzpicture}
%
%%------------------------------------------------
%
%\begin{itemize}\compresslist
%\item Pellentesque eget orci eros. Fusce ultricies, tellus et pellentesque fringilla, ante massa luctus libero, quis tristique purus urna nec nibh. Phasellus fermentum rutrum elementum. Nam quis justo lectus.
%\item Vestibulum sem ante, hendrerit a gravida ac, blandit quis magna.
%\item Donec sem metus, facilisis at condimentum eget, vehicula ut massa. Morbi consequat, diam sed convallis tincidunt, arcu nunc.
%\item Nunc at convallis urna. isus ante. Pellentesque condimentum dui. Etiam sagittis purus non tellus tempor volutpat. Donec et dui non massa tristique adipiscing.
%\end{itemize}
%
%\end{multicols}






%Donec faucibus purus at tortor egestas eu fermentum dolor facilisis. Maecenas tempor dui eu neque fringilla rutrum. Mauris \emph{lobortis} nisl accumsan.
%
%\begin{center}
%\begin{tabular}{l l l}
%\toprule
%\textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\
%\midrule
%Treatment 1 & 0.0003262 & 0.562 \\
%Treatment 2 & 0.0015681 & 0.910 \\
%Treatment 3 & 0.0009271 & 0.296 \\
%\bottomrule
%\end{tabular}
%\captionof{table}{Table caption}
%\end{center}
%
%Nulla ut porttitor enim. Suspendisse venenatis dui eget eros gravida tempor. Mauris feugiat elit et augue placerat ultrices. Morbi accumsan enim nec tortor consectetur non commodo.
%
%\begin{center}
%\begin{tabular}{l l l}
%\toprule
%\textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\
%\midrule
%Treatment 1 & 0.0003262 & 0.562 \\
%Treatment 2 & 0.0015681 & 0.910 \\
%Treatment 3 & 0.0009271 & 0.296 \\
%\bottomrule
%\end{tabular}
%\captionof{table}{Table caption}
%\end{center}


%----------------------------------------------------------------------------------------
%	CONTACT INFORMATION
%----------------------------------------------------------------------------------------

%\headerbox{Contact Information}{name=contact,column=3, below = ex}{ % This block is as tall as the references block
%	\begin{description}\compresslist
%		\item[Web] www.eac.cs.tu-dortmund.de
%		\item[Email] maurice.rolvien@tu-dortmund.de
%		\item[Phone] +492317554895
%	\end{description}
%}


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

%\headerbox{References}{name=references,column=3,span=1,above=bottom, below = contact }{
	
%	\renewcommand{\section}[2]{\vskip 0.05em} % Get rid of the default "References" section title
	%\nocite{*} % Insert publications even if they are not cited in the poster
	%\small{ % Reduce the font size in this block
		%\bibliographystyle{unsrt}
		%\bibliography{sample} % Use sample.bib as the bibliography file
		%}
%	\begin{thebibliography}{2}
%		\bibitem{}  A.~Coja-Oghlan, A.~Erg\"ur, P.~Gao, S.~Hetterich, M.~Rolvien: The rank of sparse random matrices. Random Structures and Algorithms, 2022, 10.1002/rsa.21085
		%\bibitem{}  A.~Coja-Oghlan, P.~Gao, M.~Hahn-Klimroth, J.~Lee, N.~Müller, M.~Rolvien: The full rank condition for sparse random matrices. arXiv:2112.14090
		 
	%\end{thebibliography}
%}




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

\end{poster}

\end{document}