\begin{tikzpicture}[x=1cm,y=1cm]
  \tikzset{
    notion/.style={rectangle,draw=black,line width=.6pt,
      minimum width=3.6cm,minimum height=1.15cm,
      inner xsep=5pt,inner ysep=5pt,align=center,font=\small},
    implication/.style={-{Stealth[length=4.5pt,width=3.5pt]},
      draw=black,line width=.7pt},
    implication label/.style={font=\small,align=center},
  }

  \node[font=\small\bfseries] at (0,4.5) {Analytic characterizations};
  \node[font=\small\bfseries] at (8.4,4.5) {Structural characterizations};
  \node[notion] (holder) at (0,3.375)
    {Weak H\"older property};
  \node[notion] (norming) at (0,1.125)
    {Weakly norming};
  \node[notion] (step) at (0,-1.125)
    {Step Sidorenko property};

  \node[notion] (reflection) at (8.4,3.375)
    {Reflection bigraph};
  \node[notion] (percolating) at (8.4,1.125)
    {Cut-percolating};
  \node[notion] (edges) at (8.4,-1.125)
    {$\Fold(H)$\\edge-transitive};
  \node[notion] (parts) at (8.4,-3.375)
    {$\Fold(H)$ transitive\\on each vertex part};

  \draw[implication]
    ($(holder.south)+(-.32,0)$)--($(norming.north)+(-.32,0)$)
    node[midway,left=5pt,implication label] {immediate};
  \draw[implication]
    ($(norming.north)+(.32,0)$)--($(holder.south)+(.32,0)$)
    node[midway,right=5pt,implication label] {\cite{Hatami}};
  \draw[implication]
    ($(norming.south)+(-.32,0)$)--($(step.north)+(-.32,0)$)
    node[midway,left=5pt,implication label] {immediate};
  \draw[implication]
    ($(step.north)+(.32,0)$)--($(norming.south)+(.32,0)$)
    node[midway,right=5pt,implication label] {\cite{DGRR}};

  % The structural equivalence is one directed cycle. Only its
  % closing edge uses the new part-transitivity lemma.
  \draw[implication]
    (reflection.south)--(percolating.north)
    node[midway,right=5pt,implication label] {\cite{CL}};
  \draw[implication]
    (percolating.south)--(edges.north)
    node[midway,right=5pt,implication label] {immediate};
  \draw[implication]
    (edges.south)--(parts.north)
    node[midway,right=5pt,implication label] {immediate};
  \draw[implication]
    (parts.east)--(11,-3.375)--(11,3.375)--(reflection.east);
  \node[implication label,rotate=90] at (11.25,0)
    {Lemma~\ref{lem:transitive-reflection}};

  \draw[implication]
    ($(percolating.west)+(0,.23)$)--($(norming.east)+(0,.23)$)
    node[midway,above=5pt,implication label] {\cite{CL}};
  \draw[implication,dashed,draw=foldgray]
    ($(norming.east)+(0,-.08)$)--($(percolating.west)+(0,-.08)$)
    node[midway,below=7pt,implication label,text=foldgray!60!black]
    {Conjecture};

  \draw[implication,draw=foldgreen]
    ($(norming.east)+(0,-.4)$)--(2.2,.725)--(2.2,-.25)
    --(6.2,-.25)--(6.2,.725)--($(percolating.west)+(0,-.4)$);
  \node[implication label,text=foldgreen,below=5pt] at (4.2,-.25)
    {Theorem~\ref{thm:designs}\\only for simple $2$-designs};
\end{tikzpicture}