How to draw the physical tree with balls under both the stress of strings and gravity?

Question

I want to draw the physical model of graph that has a ball for each vertex and a piece of string for each edge. In this model, if you lift a ball high enough, the other balls are pulled up along with it, affected by both the stress of strings and gravity. An illustration is shown in the following figure.

However, using tikz-qtree, I can only draw an ordinary tree which cannot visualize the strings or gravity (and is somewhat ugly).

Therefore, I want to know that

How to draw the physical tree with balls under both the stress of strings and gravity?

---

My code using tikz-qtree is as follows.

\documentclass{standalone}
\usepackage{tikz}
\usepackage{tikz-qtree}

\begin{document}

\begin{tikzpicture}
  \node [draw, circle] (r) at (0,0) {$r$};
  \node [draw, circle, blue, very thick] (s) at (2,0) {$s$};
  \node [draw, circle] (t) at (4,0) {$t$};
  \node [draw, circle] (u) at (6,0) {$u$};

  \node [draw, circle] (v) at (0,-2) {$v$};
  \node [draw, circle] (w) at (2,-2) {$w$};
  \node [draw, circle] (x) at (4,-2) {$x$};
  \node [draw, circle] (y) at (6,-2) {$y$};

  \draw (r) to (v);
  \draw (r) to (s);
  \draw (s) to (w);
  \draw (t) to (u);
  \draw (t) to (w);
  \draw (t) to (x);
  \draw (u) to (x);
  \draw (u) to (y);
  \draw (x) to (y);
  \draw (y) to (u);
\end{tikzpicture}

\begin{tikzpicture}[]
  \tikzset{level distance = 36pt, sibling distance = 25pt}
  \tikzset{every node/.style = {draw, circle}}

  \Tree [.\node[blue, very thick](s){$s$}; 
            [.$r$ $v$ ]
            [.\node(w){$w$}; 
                [.\node(t){$t$}; \node(u){$u$}; ] 
                [.\node(x){$x$}; \node(y){$y$}; ]
            ] 
        ]

  % cross edges
  \draw [dashed, thick, red] (t) to [out = -45, in = 225] (x);
  \draw [dashed, thick, red] (u) to [out = -45, in = 225] (y);
  \draw [dotted, thick, purple] (x.-110) to (u);
\end{tikzpicture}

\end{document}

Answer 1

Here's some examples with the graph drawing stuff from the latest PGF version. I'm not quite sure about the best way to define new edges but at least the method I used seems to work. It requires lualatex:

\documentclass[border=5]{standalone}
\usepackage{tikz}
\usetikzlibrary{topaths,graphs,graphdrawing,decorations}
\usegdlibrary{layered,trees}
\pgfdeclaredecoration{squiggle}{draw}{
\state{draw}[width=\pgfdecoratedpathlength]{
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathcurveto{\pgfpoint{0.25*\pgfdecoratedpathlength}{0.25*\pgfdecoratedpathlength}}%
    {\pgfpoint{0.75*\pgfdecoratedpathlength}{-0.25*\pgfdecoratedpathlength}}%
    {\pgfpoint{\pgfdecoratedpathlength}{0pt}}
}}

\tikzgraphsset{
 edge squiggle/.style={
   new --/.code n args={4}{
    \path [-, every new --/.try]
      (##1\tikzgraphleftanchor)
      edge[##3, decoration=squiggle, decorate] ##4
      (##2\tikzgraphrightanchor);}
  },
  edge loop/.style={
   new --/.code n args={4}{
    \draw [-, every new --/.try]
     (##1)
      edge [out=260, in=280, looseness=3] ##4
       (##2);}
  }
}

\begin{document}
\tikz\graph [nodes={shape=circle, fill=gray!50}, edge squiggle, 
  chain shift=(0:2cm), group shift=(270:2cm)]{
  E -- S -- A;
  D -!- C -- B;
  D -- E; D -- S;
  C -- S; B -- A; 
};

\hskip1cm

\tikz\graph [layered layout, nodes={shape=circle, fill=gray!50},
  level distance=1.5cm] {
  S -- {A -- B, C -- B, E, D};
  {[edge loop] E -- D};
  {[same layer] A, C, D, E};
};
\end{document}