\newcommand{\apptitle}[1]{
\def\toptitlebar{
	\hrule height4pt
	\vskip .25in}

\def\bottomtitlebar{
	\vskip .25in
	\hrule height1pt
	\vskip .25in}

\thispagestyle{empty}
\hsize\textwidth
\linewidth\hsize \toptitlebar {\centering
{\large\bf SUPPLEMENTARY MATERIAL \\ #1 \par}}
\vspace{-0.1in} \bottomtitlebar
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Paper specific macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\Erdos}{Erdős} 
\newcommand{\Renyi}{Rényi }
\newcommand{\B}[1]{B^{\scriptscriptstyle (#1)}}
\newcommand{\Di}[1]{D^{\scriptscriptstyle (#1)}}
\newcommand{\Bt}[1]{\widetilde{B}^{\scriptscriptstyle (#1)}}
\newcommand{\Dt}[1]{\widetilde{D}^{\scriptscriptstyle (#1)}}
\newcommand{\wtl}[1]{\widetilde{#1}}
\newcommand{\E}[1]{E^{\scriptscriptstyle (#1)}}
\newcommand{\G}[1]{G^{\scriptscriptstyle (#1)}}
\newcommand{\Gt}[1]{\widetilde{G}^{\scriptscriptstyle (#1)}}
\newcommand{\ssigma}[1]{\sigma{^{\scriptscriptstyle (#1)}}}
\newcommand{\sssigma}[2]{\sigma{^{\scriptscriptstyle (#1)}_{#2}}}
\newcommand{\var}{\sigma^2}
\newcommand{\vart}{\widetilde{\sigma}^2}
\newcommand{\mi}{-\mathrm{i}}
\newcommand{\mj}{-\mathrm{j}}
\newcommand{\D}{\Delta}
\newcommand{\DB}{{\Delta_{B}}}
\newcommand{\DBt}{{\Delta_{\widetilde{B}}}}
\newcommand{\DO}{{\Delta_{\Omega}}}
\newcommand{\DOh}{{\widehat{\Delta}_{\Omega}}}
\newcommand{\DOt}{{\Delta_{\Omega}^*}}
\newcommand{\DtO}{{\widetilde{\Delta}_{\Omega}}}
\newcommand{\X}[1]{X^{{\scriptscriptstyle (#1)}}}
\newcommand{\bt}[1]{\beta^{{\scriptscriptstyle (#1)}}}
\newcommand{\Om}[1]{\Omega^{{\scriptscriptstyle (#1)}}}
\newcommand{\Omt}[1]{\widetilde{\Omega}^{{\scriptscriptstyle (#1)}}}
\newcommand{\Sm}[1]{\Sigma^{{\scriptscriptstyle (#1)}}}
\newcommand{\Smh}[1]{\widehat{\Sigma}^{{\scriptscriptstyle #1}}}
\newcommand{\sm}[1]{\sigma^{{\scriptscriptstyle (#1)}}}
\newcommand{\smt}[1]{\widetilde{\sigma}^{{\scriptscriptstyle (#1)}}}
\newcommand{\Par}[1]{\pi^{\scriptscriptstyle (#1)}}
\newcommand{\nPar}{\pi}   % Parent set  (no super script)
\newcommand{\Chi}[1]{\phi^{\scriptscriptstyle (#1)}}
\newcommand{\nChi}{\phi}   % Children set (no super script)
\newcommand{\ri}{\mathrm{i}}
\newcommand{\mr}[1]{\mathrm{#1}}
\newcommand{\corr}[1]{\mathrm{corr}^{\scriptscriptstyle(#1)}} % partial correlation
\newcommand{\supp}{\mathrm{supp}}
\newcommand{\Anc}{\mathcal{A}}                                % Ancestors.
\newcommand{\Anct}{\widetilde{\mathcal{A}}}                                % Ancestors tilde.
\newcommand{\lAnc}[1]{{\mathcal{A}}^{\scriptscriptstyle(#1)}} % Labelled Ancestors.
\newcommand{\computeOrder}{\textproc{ComputeOrder}}
\newcommand{\orientEdges}{\textproc{orientEdges}}
\newcommand{\prune}{\textproc{prune}}
\newcommand{\vectorize}{\mathbf{\mathrm{vec}}}
\newcommand{\perr}{p_{\mathrm{err}}}
\newcommand{\Gf}{\mathcal{G}}  % Graph family.
\newcommand{\Gfr}{\widetilde{\mathcal{G}}}  % Graph family restricted.
\newcommand{\DGf}{\mathcal{G}_{\Delta}}  % Delta graph family.
\newcommand{\DGfr}{\widetilde{\mathcal{G}}_{\Delta}}  % Delta graph family restricted.
\newcommand{\Dec}{\zeta}
\newcommand{\Pf}{\mathcal{P}}  % Distribution P
\newcommand{\Qf}{\mathcal{Q}}  % Distribution Q
\newcommand{\Xf}{\mathfrak{X}} % Tuple of X_1 and X_2
\newcommand{\bh}{\widehat{b}}
\newcommand{\betah}{\widehat{\beta}}
\newcommand{\kdmin}{K^{\mathrm{d}}_{\mathrm{min}}}
\newcommand{\komax}{K^{\mathrm{o}}_{\mathrm{max}}}
\newcommand{\kdminS}{K^{\mathrm{d,S}}_{\mathrm{min}}}
\newcommand{\komaxS}{K^{\mathrm{o,S}}_{\mathrm{max}}}
\newcommand{\Sc}{S^c}
\newcommand{\xb}{\bar{x}}
\newcommand{\eigmax}{\lambda_{\mathrm{max}}}
\newcommand{\eigmin}{\lambda_{\mathrm{min}}}
\newcommand{\skel}{\mathrm{skel}}
\newcommand{\dt}{d'}