定义 TikZ 函数来生成 Knodel 图

Question

我正在尝试定义一个将生成 Knodel 图的函数（如下图所示为示例）。

我不完全确定 TikZ 的代码如何使用范围进行 for 循环，以及如何使用索引计算不同的值。另外，我不确定如何更改函数中参数的不同值上的边类型（粗体、虚线等）。

这是伪代码对于 TikZ 函数我想定义：

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}[auto, scale=0.9]
\tikzstyle{vertex}=[draw, circle, inner sep=0.55mm]

[declare function={
    %
    %assuming delta and n are both positive integers
    %assuming 1 <= delta <= floor(log base 2 (n))
    %
    knodel(\n, \delta) =
        % make the vertices
        for i in [1, 2]:
            for j in [0, floor(n/2) - 1]:
                \node at (i, j) [vertex] {};

        % make the edges
        for j in [0, floor(n/2) - 1]:
            for k in [0, delta - 1]:
                \draw (1,j) to (2, (j + 2^k - 1) mod floor(n/2));
}]

\draw {knodel(14, 3)}

\end{tikzpicture}
\end{document}

以下是我读过的一篇论文中的一个例子：

Answer 1

这里定义了两种类型的代码pics：一种knodel row graph将顶点放在两行上（如 OP 图像上所示），另一种knodel circle graph将顶点放在一个圆上。

两张图片都采用两个参数：度（顶点的数量）和维度（边类型的数量）。

\documentclass[tikz,border=7pt]{standalone}
% evaluate the parameters from
% #1 = the degree (number of vertices)
% #2 = the dimension (number of edge types)
\def\knodelParameters#1#2{
  \pgfmathsetmacro\m{int(#1/2-1)}
  \pgfmathsetmacro\n{int(2*\m+2)}
  \pgfmathsetmacro\d{int(#2-1)}
}
% place the vertices in two rows y=0 and y=1
\def\knodelRowVertices#1{
  \foreach \i in {0,...,#1}
    \path
      (\i,1) node[vertex,vertex 1/.try](v1\i){} node[above]{(1,\i)}
      (\i,0) node[vertex,vertex 2/.try](v2\i){} node[below]{(2,\i)}
  ;
}
% place the vertices on a circle with radius 2 (if not scaled)
\def\knodelCircleVertices#1{
  \foreach[evaluate={\a=(\i*360)/(#1+1);\b=(\i*360+180)/(#1+1)}] \i in {0,...,#1}
    \path
      (\a:2) node[vertex,vertex 1/.try](v2\i){} (\a:2.5) node{(2,\i)}
      (\b:2) node[vertex,vertex 2/.try](v1\i){} (\b:2.5) node{(1,\i)}
  ;
}
% draw the edges with styles (if defined) dim 0, dim 1, ...
\def\knodelEdges#1#2{
  \foreach[evaluate={\t=2^\k}] \k in {0,...,#2}{
    \foreach[evaluate={\j=int(mod(\i+\t-1,\m+1))}] \i in {0,...,#1}{
      \draw[dim \k/.try] (v1\i) -- (v2\j);
    }
  }
}
\tikzset{
  % two type of pics : "knodel row graph" and "knodel circle graph"
  % ------------------
  pics/knodel row graph/.style args={#1,#2}{
    code={
      \knodelRowVertices{\m}
      \knodelEdges{\m}{\d}
    }
  },
  pics/knodel row graph/.prefix code=\knodelParameters{#1}{#2},
  % ------------------
  pics/knodel circle graph/.style args={#1,#2}{
    foreground code={
      \knodelCircleVertices{\m}
    },
    background code={
      \knodelEdges{\m}{\d}
    }
  },
  pics/knodel circle graph/.prefix code=\knodelParameters{#1}{#2},
  % ------------------
  % two default styles "mono" and "rainbow"
  % ------------------
  vertex/.style={fill,circle,inner sep=1.5pt},
  mono/.style={
    dim 0/.style={very thin},
    dim 1/.style={dotted},
    dim 2/.style=thick
  },
  rainbow/.style={
    vertex 1/.style={red!70!black},
    vertex 2/.style={blue!70!black},
    dim 0/.style={red},
    dim 1/.style={blue},
    dim 2/.style={green},
    dim 3/.style={orange},
    dim 4/.style={purple}
  }
}
\begin{document}
  \begin{tikzpicture}
    \path
      (5,4) pic[rainbow]{knodel row graph={14,3}}
      (0,4) pic[mono]{knodel row graph={8,3}}
      (2,0) pic[mono]{knodel circle graph={8,3}}
      (8,0) pic[rainbow]{knodel circle graph={20,5}}
    ;
  \end{tikzpicture}
\end{document}

Answer 2

\documentclass[tikz,border=5mm]{standalone}
\usepackage{tikz-3dplot}

\begin{document}
\tdplotsetmaincoords{70}{110}
\tikz[tdplot_main_coords]{
    \def\nhalf{20}
    \def\Delta{5}
    \def\r{4}
    \def\h{5}
    \foreach\m in{0,...,\numexpr\nhalf-1}{
        \draw(\m*360/\nhalf:\r)node(\m){$\m$}
            +(0,0,\h)node(\m'){$\m'$};
    }
    \foreach\d in{0,...,\numexpr\Delta-1}{
        \foreach\m in{0,...,\numexpr\nhalf-1}{
            \pgfmathtruncatemacro\target{mod(\m+\d,\nhalf)}
            \draw(\m)--(\target');
        }
    }
}
\end{document}