Illustrating symmetric key distribution

Question

Can I draw something like this programmatically with LaTeX?

I drew these with Adobe Illustrator, but it gets pretty time-consuming.
Even with relatively few nodes. What it attempts to demonstrate is the number of "symmetric" cryptographic keys required for secure communication between n parties - for which the formula is n(n-1)/2 (I tried to render that with LaTeX/MathJax, but it won't let me for some reason).

And it's hard to get it perfect. It's always off by a small amount, as you can see here. It might not seem like much but with a lot of nodes, it can add up and skew the output, and cause problems when trying to get everything to fit properly.

I wanted to draw a big one with up up to 100 nodes, so it would be best if I could do it programmatically, rather than having to type up several hundred lines of code manually. Is that kind of thing possible with LaTeX? I heard someone say TeX is a full Turing complete programming language. If that's true, I'm not sure, but it would probably make it pretty useful for generating diagrams with this kind of data, kinda like D3.js.

As you can see, by the time you get to 10 or 11 vertices, it can be pretty unruly:

Answer 1

Yes, you can.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{arrows.meta}
\newcounter{pft}
\begin{document}
\begin{tikzpicture}[font=\sffamily,pics/cgram/.style={code={
\foreach \XX [count=\YY starting from 0] in {1,...,#1}
{\pgfmathsetmacro{\mycolor}{{\LstCols}[\YY]}
\node[circle,draw,minimum size=2.5em,fill=\mycolor] (c-#1-\XX) at 
({{\LstAngles}[#1-2]-\YY*360/#1}:1.5) {\setcounter{pft}{\XX}\Alph{pft}};}
\foreach \XX [evaluate=\XX as \Ymax using {int(\XX-1)}] in {2,...,#1}
{\foreach \YY  in {1,...,\Ymax}
{\pgfmathsetmacro{\mycolorA}{{\LstCols}[\XX-1]}
\pgfmathsetmacro{\mycolorB}{{\LstCols}[\YY-1]}
\path (c-#1-\XX) -- (c-#1-\YY) coordinate[pos=0.1] (aux0) coordinate[pos=0.9] (aux1);
\fill[black] (aux0) to[bend left=2] (aux1) to[bend left=2] (aux0);
\draw[{Stealth[fill=\mycolorB,length=7pt,inset=2pt]}-{Stealth[fill=\mycolorA,length=7pt,inset=2pt]}] (c-#1-\XX) -- (c-#1-\YY);
}}}}]
\def\LstCols{"red","orange","yellow","green!70!black","blue!70!white","purple!80!white"}
\def\LstAngles{180,150,135,128,150}
\path (-5,0) pic {cgram=2} (0,0.5) pic {cgram=3} (5,0) pic {cgram=4}
 (-3,-4) pic {cgram=5}  (3,-4) pic {cgram=6};
\end{tikzpicture}
\end{document}

Zoom in:

And yes, for large numbers N of nodes it becomes busy, simply since the number of connections goes like N (N-1)/2.

\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{arrows.meta}
\definecolor{colorA}{RGB}{202, 38, 49} 
\definecolor{colorB}{RGB}{222, 146, 60} 
\definecolor{colorC}{RGB}{240, 215, 68} 
\definecolor{colorD}{RGB}{126, 183, 86} 
\definecolor{colorE}{RGB}{98, 173, 233} 
\definecolor{colorF}{RGB}{158, 76, 150}  
\newcounter{pft}
\tikzset{pics/cgram/.style={code={
\foreach \XX [count=\YY starting from 0] in {1,...,#1}
{\pgfmathtruncatemacro{\iA}{mod(\XX-1,6)+1}
\pgfmathsetmacro{\mycolor}{{\LstCols}[\iA-1]}
\node[circle,draw,minimum size=2.5em,fill=\mycolor] (c-#1-\XX) at 
({-\YY*360/#1}:\pgfkeysvalueof{/tikz/cgram radius}) {\setcounter{pft}{\iA}\Alph{pft}};}
\foreach \XX [evaluate=\XX as \Ymax using {int(\XX-1)}] in {2,...,#1}
{\foreach \YY  in {1,...,\Ymax}
 {\pgfmathtruncatemacro{\iA}{mod(\XX-1,6)+1}
  \pgfmathtruncatemacro{\iB}{mod(\YY-1,6)+1}
  \pgfmathsetmacro{\mycolorA}{{\LstCols}[\iA-1]}
  \pgfmathsetmacro{\mycolorB}{{\LstCols}[\iB-1]}
 \draw[{Stealth[fill=\mycolorB,length=7pt,inset=2pt]}-{Stealth[fill=\mycolorA,length=7pt,inset=2pt]}] (c-#1-\XX) -- (c-#1-\YY);
}}
}},cgram radius/.initial=1.5}
\begin{document}
\foreach \Nmax in {2,4,...,40}
{\begin{tikzpicture}[font=\sffamily]
\draw (-11,-11) rectangle (11,11);
\def\LstCols{"colorA","colorB","colorC","colorD","colorE","colorF"}
\pgfmathsetmacro{\myradius}{sqrt(2.5*\Nmax)}
\path  pic[cgram radius=\myradius] {cgram=\Nmax};
\end{tikzpicture}}
\end{document}

Answer 2

So this is my construction for future references.

\documentclass[border=9,tikz,rgb]{standalone}

\usetikzlibrary{arrows.meta,decorations.pathreplacing}
\begin{document}

\tikzset{
    /pgf/arrow keys/colorsize/.style={fill=#1,length=10pt}
}
\def\N{70}
\tikzdeclarecoordinatesystem{sunflower}{ % #1 is the index of vertex
    \pgfmathsetmacro\sunindex{#1-.5}
    \pgfmathsetmacro\sunangle{mod(\sunindex*16.18034,10)*36}
    \pgfmathsetmacro\sunradius{sqrt(\sunindex)*50}
    \pgfpointpolar{\sunangle}{\sunradius}
}
\globalcolorstrue
\def\definesuncolor#1{
    \pgfmathtruncatemacro\sunindex{#1-.5}
    \pgfmathsetmacro\sunhue{mod(\sunindex*16.18034,10)*36}
    \pgfmathsetmacro\sunsaturation{sqrt(\sunindex/\N)}
    \definecolor{sun#1}{Hsb}{\sunhue,\sunsaturation,1}
}
\tikz{
    \foreach\i in{1,...,\N}{
        \definesuncolor{\i}
        \path(sunflower cs:\i)node(vertex\i)
            [circle,draw,minimum size=2cm,line width=6pt]{};
        \fill[sun\i](vertex\i)+(1pt,1pt)circle(1);
    }
    \foreach\i in{2,...,\N}{
        \foreach\j in{1,...,\numexpr\i-1}{
            \path[scale=.666/sqrt(\N)]
                [shift=(vertex\i)](sunflower cs:\j)coordinate(X-\i-\j)
                [shift=(vertex\j)](sunflower cs:\i)coordinate(Y-\i-\j);
            \draw[{Stealth[colorsize=sun\j]}-{Stealth[colorsize=sun\i]}]
                [line width=.1](X-\i-\j)--(Y-\i-\j);
        }
    }
    \foreach\i in{2,...,\N}{
        \foreach\j in{1,...,\numexpr\i-1}{
            \draw[{Stealth[colorsize=sun\j]}-{Stealth[colorsize=sun\i]}]
                [dash pattern=on0off9999](X-\i-\j)--(Y-\i-\j);
        }
    }
}

\end{document}

Some comments to whomever wants to play with this:

• sunflower is the coordinate system that controls how to place vertices. It is the same algorithm that sunflower uses to place its seeds. See wikipedia
• The color of each vertex is control by \definesuncolor#1. Currently it is defined such that the sunflower looks like the HSB wheel.
• There are two nested-for-loops at the end. The former loop draws the edge, the later loop draws the arrow tips.
• The position of arrow tip is controlled by (X-\i-\j) and (Y-\i-\j). Currently they are the relative positions of the vertices. So the arrows tips on each vertex also looks like the HSB wheel.