Consistency in distances between nodes on vertices along stacked, horizontal line segments in a `TikZ` diagram

Question

In the following TikZ diagram, I have depicted an expanding graph - a sequence of simple graphs. An inconsistency in this diagram is the distances between consecutive steps in the expansion. For the first four steps, I just decreased the y-coordinate by 3 of each of the vertices. A multi-line node under a vertex of the fourth graph prevented that from happening. I just put the last graph 6 cm under the fourth graph. It is unseemly.

How do I define a length between (label_for_Vertex_b.south) and (label_for_Farey_Fraction_at_Vertex_b.north) and use that distance to position the fraction (2n+1)/2 on the last graph, and from this node, draw the last horizontal line?

\documentclass{amsart}
\usepackage{amssymb}
\usepackage{mathtools,array}

\usepackage{tikz}
\usetikzlibrary{calc,intersections}

\begin{document}

\begin{tikzpicture}[nodes={inner sep=0, font=\scriptsize,
execute at begin node={\setlength\abovedisplayskip{0.75ex}%
\setlength\belowdisplayskip{0.5ex}%
\setlength\abovedisplayshortskip{0.75ex}%
\setlength\belowdisplayshortskip{0.5ex}}},
shorten/.style={shorten >=#1,shorten <=#1}]

%A sequence of graphs is drawn, starting with the vertex with the b-label b.


%Here is the blow-up of the vertex labeled b.
\draw[fill] (-4,0) circle (1.5pt);
\node[anchor=north] (label_for_Vertex_b) at ($(-4,0) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
%
%
\draw (-4,-3) -- (-2,-3);
\draw[fill] (-4,-3) circle (1.5pt);
\draw[fill] (-2,-3) circle (1.5pt);
%
\node[anchor=north] at ($(-4,-3) +(0,-0.25)$){\textit{b}};
\node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b) at ($(-4,-3) +(0,0.25)$){$\dfrac{0}{1}$};
%
\node[anchor=north] (label_for_Vertex_b-1) at ($(-2,-3) +(0,-0.25)$){$b - 1$};
\node[anchor=south] at ($(-2,-3) +(0,0.25)$){$\dfrac{1}{1}$};
%
%
%An arrow is drawn to the next diagram.
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex \textit{b}} (label_for_Farey_Fraction_at_Vertex_b);


%Here is the blow-up of the vertex labeled b-1.
\draw (-4,-6) -- (-2,-6) -- (0,-6);
\draw[fill] (-4,-6) circle (1.5pt);
\draw[fill] (-2,-6) circle (1.5pt);
\draw[fill] (0,-6) circle (1.5pt);
%
\node[anchor=north] at ($(-4,-6) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,-6) +(0,0.25)$){$\dfrac{0}{1}$};
%
\node[anchor=north] at ($(-2,-6) +(0,-0.25)$){$b-1$};
\node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b-1) at ($(-2,-6) +(0,0.25)$){$\dfrac{1}{1}$};
%
\node[anchor=north] (label_for_Vertex_b-2) at ($(0,-6) +(0,-0.25)$){$b-2$};
\node[anchor=south] at ($(0,-6) +(0,0.25)$){$\dfrac{2}{1}$};
%
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);


%Here is the blow-up of the vertex labeled b-n.
\draw (-4,-9) -- (-2,-9) -- (0,-9) (2,-9) -- (5,-9);
\draw[fill] (-4,-9) circle (1.5pt);
\draw[fill] (-2,-9) circle (1.5pt);
\draw[fill] (0,-9) circle (1.5pt);
\node at (1,-9){$\ldots$};
\draw[fill] (2,-9) circle (1.5pt);
\draw[fill] (5,-9) circle (1.5pt);
%
\node[anchor=north] at ($(-4,-9) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,-9) +(0,0.25)$){$\dfrac{0}{1}$};
%
\node[anchor=north] at ($(-2,-9) +(0,-0.25)$){$b-1$};
\node[anchor=south] at ($(-2,-9) +(0,0.25)$){$\dfrac{1}{1}$};
%
\node[anchor=north] at ($(0,-9) +(0,-0.25)$){$b-2$};
\node[anchor=south] at ($(0,-9) +(0,0.25)$){$\dfrac{2}{1}$};
%
\node[anchor=south, green] (label_for_phantom_Farey_Fraction_at_ellipses) at ($(1,-9) +(0,0.25)$){$\dfrac{1}{1}$};
%
\node[anchor=north] at ($(2,-9) +(0,-0.25)$){\textit{b-n}};
\node[anchor=south] at ($(2,-9) +(0,0.25)$){$\dfrac{n}{1}$};
%
\node[anchor=north] at ($(5,-9) +(0,-0.25)$){$b-(n+1)$};
\node[anchor=south] at ($(5,-9) +(0,0.25)$){$\dfrac{n+1}{1}$};
%
%
%
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-2) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\more vertices} (label_for_phantom_Farey_Fraction_at_ellipses);
%
%
%A "pin" is drawn between the midpoint of last two vertices and the label of the mediants of these vertices.
\draw[-latex, dashed, line width=0.8pt, shorten <=3mm, shorten >=1mm] ($(3.5,-9) +(60:2)$) -- (3.5,-9);
\path node[anchor=south, align=center, text width={width("future vertex")}]
at ($(3.5,-9) +(60:2)$){future mediant\\for vertex\[\dfrac{2n+1}{2}\]};
%
%A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
\coordinate (label_for_Edge) at ($(3.5,-9.5) +(0,-0.75)$);
\draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] (3.5,-9) -- (label_for_Edge);
\node[anchor=north, align=center, inner sep=0, font=\scriptsize] (actual_label_for_Edge) at (label_for_Edge)
{$\begin{aligned} &\text{Present edge label of} \\[-1.5ex]
&\quad 2\bigl[(b-n)+(b-(n+1))\bigr] \\[-1.5ex]
&\qquad=2^{2}b-(2n+1)2
\end{aligned}$};



\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);


\draw[-latex, line width=0.8pt, shorten <=30pt, shorten >=7.5pt] (label_for_Edge.south) -- ($(label_for_Edge.south) +(0,-4)$);

%Here is the vertex placed at the broken edge.
\draw (-4,-15) -- (-2,-15) -- (0,-15) (2,-15) -- (5,-15);
\draw[fill] (-4,-15) circle (1.5pt);
\draw[fill] (-2,-15) circle (1.5pt);
\draw[fill] (0,-15) circle (1.5pt);
\node at (1,-15){$\ldots$};
\draw[fill] (2,-15) circle (1.5pt);
\draw[fill] ({(2+5)/2},-15) circle (1.5pt);
\draw[fill] (5,-15) circle (1.5pt);
%
\node[anchor=north] at ($(-4,-15) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,-15) +(0,0.25)$){$\dfrac{0}{1}$};
%
\node[anchor=north] at ($(-2,-15) +(0,-0.25)$){$b-1$};
\node[anchor=south] at ($(-2,-15) +(0,0.25)$){$\dfrac{1}{1}$};
%
\node[anchor=north] at ($(0,-15) +(0,-0.25)$){$b-2$};
\node[anchor=south] at ($(0,-15) +(0,0.25)$){$\dfrac{2}{1}$};
%
\node[anchor=north] at ($(2,-15) +(0,-0.25)$){\textit{b-n}};
\node[anchor=south] at ($(2,-15) +(0,0.25)$){$\dfrac{n}{1}$};
%
\node[anchor=north] at ($(5,-15) +(0,-0.25)$){$b-(n+1)$};
\node[anchor=south] at ($(5,-15) +(0,0.25)$){$\dfrac{n+1}{1}$};
%
%A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
\draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] ({(2+5)/2},-15) -- ({(2+5)/2},-16);
\node[anchor=north] at ({(2+5)/2},-16){$2^{2}b-(n+1)2$};
\node[anchor=south] (Last_Farey_Fraction_in_diagram) at ($({(2+5)/2},-15) +(0,0.25)$){$\dfrac{2n+1}{2}$};


%Extra code to display distances I want between graphs.
\draw[|<->|, green] (label_for_Farey_Fraction_at_Vertex_b) -- (label_for_Vertex_b);
\node[green, anchor=west] at ($($(label_for_Farey_Fraction_at_Vertex_b)!0.5!(label_for_Vertex_b)$) +(0.1,0)$){\textit{D}};
%
\draw[|<->|, green] ($(actual_label_for_Edge.south) +(0.5,0)$) -- ($(Last_Farey_Fraction_in_diagram.north) +(0.5,0)$);
\node[green, anchor=west] at ($($(actual_label_for_Edge.south) +(0.6,0)$)!0.5!($(Last_Farey_Fraction_in_diagram.north) +(0.6,0)$)$){\textbf{Want this to be \textit{D}}};

\end{tikzpicture}

\end{document}

Answer 1

To shift complete blocks, you can put them in scopes,

\begin{scope}[yshift=-3cm]
 ...
\end{scope}

This allows you to shift the last block 3cm below the south anchor of the edge label.

\documentclass{amsart}
\usepackage{amssymb}
\usepackage{mathtools,array}

\usepackage{tikz}
\usetikzlibrary{calc,intersections}

\begin{document}


\begin{tikzpicture}[nodes={inner sep=0, font=\scriptsize,
execute at begin node={\setlength\abovedisplayskip{0.75ex}%
\setlength\belowdisplayskip{0.5ex}%
\setlength\abovedisplayshortskip{0.75ex}%
\setlength\belowdisplayshortskip{0.5ex}}},
shorten/.style={shorten >=#1,shorten <=#1}]

%A sequence of graphs is drawn, starting with the vertex with the b-label b.


%Here is the blow-up of the vertex labeled b.
\draw[fill] (-4,0) circle (1.5pt);
\node[anchor=north] (label_for_Vertex_b) at ($(-4,0) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
%
%
\begin{scope}[yshift=-3cm]
 \draw (-4,0) -- (-2,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b) at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] (label_for_Vertex_b-1) at ($(-2,0) +(0,-0.25)$){$b - 1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 %
 %An arrow is drawn to the next diagram.
 \draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b) to[bend right=30] node[midway, left=1.5mm, align=center]
 {Blow-up of\\vertex \textit{b}} (label_for_Farey_Fraction_at_Vertex_b);
\end{scope}

\begin{scope}[yshift=-6cm]
 %Here is the blow-up of the vertex labeled b-1.
 \draw (-4,0) -- (-2,0) -- (0,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b-1) at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] (label_for_Vertex_b-2) at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
\end{scope} 
%
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);

\begin{scope}[yshift=-9cm]
%Here is the blow-up of the vertex labeled b-n.
 \draw (-4,0) -- (-2,0) -- (0,0) (2,0) -- (5,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 \node at (1,0){$\ldots$};
 \draw[fill] (2,0) circle (1.5pt);
 \draw[fill] (5,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
 %
 \node[anchor=south, green] (label_for_phantom_Farey_Fraction_at_ellipses) at ($(1,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(2,0) +(0,-0.25)$){\textit{b-n}};
 \node[anchor=south] at ($(2,0) +(0,0.25)$){$\dfrac{n}{1}$};
 %
 \node[anchor=north] at ($(5,0) +(0,-0.25)$){$b-(n+1)$};
 \node[anchor=south] at ($(5,0) +(0,0.25)$){$\dfrac{n+1}{1}$};
 %
 %
 %
 %
 \draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-2) to[bend right=30] node[midway, left=1.5mm, align=center]
 {Blow-up of\\more vertices} (label_for_phantom_Farey_Fraction_at_ellipses);
 %
 %
 %A "pin" is drawn between the midpoint of last two vertices and the label of the mediants of these vertices.
 \draw[-latex, dashed, line width=0.8pt, shorten <=3mm, shorten >=1mm] ($(3.5,0) +(60:2)$) -- (3.5,0);
 \path node[anchor=south, align=center, text width={width("future vertex")}]
 at ($(3.5,0) +(60:2)$){future mediant\\for vertex\[\dfrac{2n+1}{2}\]};
 %
 %A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
 \coordinate (label_for_Edge) at ($(3.5,-0.5) +(0,-0.75)$);
 \draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] (3.5,0) -- (label_for_Edge);
 \node[anchor=north, align=center, inner sep=0, font=\scriptsize] (edge label) at (label_for_Edge)
 {$\begin{aligned} &\text{Present edge label of} \\[-1.5ex]
 &\quad 2\bigl[(b-n)+(b-(n+1))\bigr] \\[-1.5ex]
 &\qquad=2^{2}b-(2n+1)2
 \end{aligned}$};
\end{scope}


\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);



\path ($(0,0|-edge label.south)+(0,-3)$) coordinate(auxp);
\begin{scope}[shift={(auxp)}]
 %Here is the vertex placed at the broken edge.
 \draw (-4,0) -- (-2,0) -- (0,0) (2,0) -- (5,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 \node at (1,0){$\ldots$};
 \draw[fill] (2,0) circle (1.5pt);
 \draw[fill] ({(2+5)/2},0) circle (1.5pt);
 \draw[fill] (5,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
 %
 \node[anchor=north] at ($(2,0) +(0,-0.25)$){\textit{b-n}};
 \node[anchor=south] at ($(2,0) +(0,0.25)$){$\dfrac{n}{1}$};
 %
 \node[anchor=north] at ($(5,0) +(0,-0.25)$){$b-(n+1)$};
 \node[anchor=south] at ($(5,0) +(0,0.25)$){$\dfrac{n+1}{1}$};
 %
 %A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
 \draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] ({(2+5)/2},0)
 -- ({(2+5)/2},-1);
 \node[anchor=north] at ({(2+5)/2},-1){$2^{2}b-(n+1)2$};
 \node[anchor=south] (f2n) at ($({(2+5)/2},0) +(0,0.25)$){$\dfrac{2n+1}{2}$};
\end{scope}
\draw[-latex, line width=0.8pt, shorten <=30pt, shorten >=7.5pt] 
(label_for_Edge.south) -- (f2n.north);
\end{tikzpicture}

\end{document}

As mentioned here, you would possibly benefit greatly using positioning to position the blocks relative to each other. This answer is written under the assumption that there are some reasons why you do not want to use positioning.

Here is another attempt to make you happy. The trick is to put of these layers in pics and to put the ("fake") pics in a matrix.

The vertical distance between these layers/pics is entirely fixed by column sep.

If you want to add annotations that should not alter the size of these pics, use either overlay or add them outside of the matrix. Examples for both can be found in the MWE

\documentclass{amsart}
\usepackage{amssymb}
\usepackage{mathtools,array}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}


\begin{tikzpicture}[nodes={inner sep=0, font=\scriptsize,
execute at begin node={\setlength\abovedisplayskip{0.75ex}%
\setlength\belowdisplayskip{0.5ex}%
\setlength\abovedisplayshortskip{0.75ex}%
\setlength\belowdisplayshortskip{0.5ex}}},
shorten/.style={shorten >=#1,shorten <=#1},
pics/fpic/.style={code={#1}}]

%A sequence of graphs is drawn, starting with the vertex with the b-label b.

\matrix[row sep=4.5em]{%<- this defines the difference between pics
%Here is the blow-up of the vertex labeled b.
\pic{fpic={%
\draw[fill] (-4,0) circle (1.5pt);
\node[anchor=north] (label_for_Vertex_b) at ($(-4,0) +(0,-0.25)$){\textit{b}};
\node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
}};\\
%
%
\pic{fpic={%
\draw (-4,0) -- (-2,0) coordinate(lcompare);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 %
 \node[anchor=north] (bcompare) at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b) at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] (label_for_Vertex_b-1) at ($(-2,0) +(0,-0.25)$){$b - 1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 }};\\
 %
 %
\pic{fpic={%
 %Here is the blow-up of the vertex labeled b-1.
 \draw (-4,0) -- (-2,0) -- (0,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] (label_for_Farey_Fraction_at_Vertex_b-1) at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] (label_for_Vertex_b-2) at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
}};\\
\pic{fpic={
%Here is the blow-up of the vertex labeled b-n.
 \draw (-4,0) -- (-2,0) -- (0,0) (2,0) -- (5,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 \node at (1,0){$\ldots$};
 \draw[fill] (2,0) circle (1.5pt);
 \draw[fill] (5,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
 %
 \node[anchor=south, green] (label_for_phantom_Farey_Fraction_at_ellipses) at ($(1,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(2,0) +(0,-0.25)$){\textit{b-n}};
 \node[anchor=south] at ($(2,0) +(0,0.25)$){$\dfrac{n}{1}$};
 %
 \node[anchor=north] at ($(5,0) +(0,-0.25)$){$b-(n+1)$};
 \node[anchor=south] at ($(5,0) +(0,0.25)$){$\dfrac{n+1}{1}$};
 %
 %
 %
 %
 %
 %A "pin" is drawn between the midpoint of last two vertices and the label of the mediants of these vertices.
 \draw[-latex, dashed, line width=0.8pt, shorten <=3mm, shorten >=1mm,overlay] ($(3.5,0) +(60:2)$) -- (3.5,0);
 \path node[anchor=south, align=center, text width={width("future
 vertex")},overlay]
 at ($(3.5,0) +(60:2)$){future mediant\\for vertex\[\dfrac{2n+1}{2}\]};
 %
 %A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
 \coordinate (label_for_Edge) at ($(3.5,-0.5) +(0,-0.75)$);
 \draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] (3.5,0) -- (label_for_Edge);
 \node[anchor=north, align=center, inner sep=0, font=\scriptsize] (edge label) at (label_for_Edge)
 {$\begin{aligned} &\text{Present edge label of} \\[-1.5ex]
 &\quad 2\bigl[(b-n)+(b-(n+1))\bigr] \\[-1.5ex]
 &\qquad=2^{2}b-(2n+1)2
 \end{aligned}$};
}};\\
\pic{fpic={
 %Here is the vertex placed at the broken edge.
 \draw (-4,0) -- (-2,0) -- (0,0) (2,0) -- (5,0);
 \draw[fill] (-4,0) circle (1.5pt);
 \draw[fill] (-2,0) circle (1.5pt);
 \draw[fill] (0,0) circle (1.5pt);
 \node at (1,0){$\ldots$};
 \draw[fill] (2,0) circle (1.5pt);
 \draw[fill] ({(2+5)/2},0) circle (1.5pt);
 \draw[fill] (5,0) circle (1.5pt);
 %
 \node[anchor=north] at ($(-4,0) +(0,-0.25)$){\textit{b}};
 \node[anchor=south] at ($(-4,0) +(0,0.25)$){$\dfrac{0}{1}$};
 %
 \node[anchor=north] at ($(-2,0) +(0,-0.25)$){$b-1$};
 \node[anchor=south] at ($(-2,0) +(0,0.25)$){$\dfrac{1}{1}$};
 %
 \node[anchor=north] at ($(0,0) +(0,-0.25)$){$b-2$};
 \node[anchor=south] at ($(0,0) +(0,0.25)$){$\dfrac{2}{1}$};
 %
 \node[anchor=north] at ($(2,0) +(0,-0.25)$){\textit{b-n}};
 \node[anchor=south] at ($(2,0) +(0,0.25)$){$\dfrac{n}{1}$};
 %
 \node[anchor=north] at ($(5,0) +(0,-0.25)$){$b-(n+1)$};
 \node[anchor=south] at ($(5,0) +(0,0.25)$){$\dfrac{n+1}{1}$};
 %
 %A "pin" is drawn between the midpoint of the edge between the last two vertices and its label.
 \draw[draw=gray, line width=0.8pt, shorten <=1mm, shorten >=1mm] ({(2+5)/2},0)
 -- ({(2+5)/2},-1);
 \node[anchor=north] at ({(2+5)/2},-1){$2^{2}b-(n+1)2$};
 \node[anchor=south] (f2n) at ($({(2+5)/2},0) +(0,0.25)$){$\dfrac{2n+1}{2}$};
}};\\
};
%An arrow is drawn to the next diagram.
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b) to[bend right=30] node[midway, left=1.5mm, align=center]
 {Blow-up of\\vertex \textit{b}} (label_for_Farey_Fraction_at_Vertex_b);
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-2) to[bend right=30] node[midway, left=1.5mm, align=center]
 {Blow-up of\\more vertices} (label_for_phantom_Farey_Fraction_at_ellipses);
%
\draw[-latex, line width=0.8pt, shorten=7.5pt] (label_for_Vertex_b-1) to[bend right=30] node[midway, left=1.5mm, align=center]
{Blow-up of\\vertex $b - 1$} (label_for_Farey_Fraction_at_Vertex_b-1);
%
\draw[-latex, line width=0.8pt, shorten <=30pt, shorten >=7.5pt] 
(label_for_Edge.south) -- (f2n.north);
\end{tikzpicture}
\end{document}