Ellipsis positioning when drawing large graphs in Tikz

Question

I tried to use Tikz to draw large graphs with specific toologies and I am running into some problems. Since not all nodes can be represented explicitly, I am forced to use an ellipsis. I tried to define this ellipsis as a node, but the dots appear horizontally and not as shown in figure, i.e., it would be great if the ellipsis are three adjacent points on the circumference of the circle where other nodes rest.

The same problem arises even when I have a complete graph. Can someone help me out with this? Thanks!

Code so far:

\begin{tikzpicture}[->,>=stealth', shorten>=1pt, auto, node distance=2.8cm, semithick]
 \tikzstyle{peer}=[draw,circle,violet,bottom color=\lav, top color= white, text=violet, minimum width=8pt]
 \tikzstyle{superpeer}=[draw, circle, burntorange, left color=\oran, text=violet, minimum width=25pt]
  \node[superpeer] (A) {$0$};
  \node[peer]         (B)  at ($ (A) + (20:4) $) {$1$};
  \node[peer]         (C)  at ($ (A) + (-52:4) $) {$2$};
  \node[peer]         (D)   at ($ (A) + (-124:4) $){$3$};
  \node[draw=none] (F)  at ($ (A) + (-176:4) $){$\cdots$};  
  \node[peer] (E) at ($ (A) + (92:4) $) {$N$};

  \path (A) edge [color=blue, sloped] node[]{$\lambda_{01}, \mu_{01}$} (B);
  \path (A) edge [color=blue, sloped] node[] {$\lambda_{02}, \mu_{02}$} (C);
  \path (A) edge [color=blue, sloped] node[] {$\lambda_{03}, \mu_{03}$}(D);
  \path (A) edge [color=blue, sloped, below] node[] {$\lambda_{0N}, \mu_{0N}$} (E);
  \path (B) edge [color=blue, sloped] node[] {$\lambda_{10}, \mu_{10}$}(A);
  \path (C) edge [color=blue, sloped] node[] {$\lambda_{20}, \mu_{20}$}(A);
  \path (D) edge [color=blue, sloped] node[] {$\lambda_{30}, \mu_{30}$}(A);
  \path (E) edge [color=blue, sloped, above] node[] {$\lambda_{N0}, \mu_{N0}$}(A);
\end{tikzpicture}

Answer 1

This is another possible solution. Since don't have the definition of \lav color, red is used. The solution uses (A.<angle>) for the two lines drawn between nodes. The lines are -20 <center angle> +20 degrees apart. p1, p1 and p2 can be adjusted to show closeness between them

Code

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes,snakes,arrows,shadows,calc,positioning,automata}
\usepackage{xcolor}
\usetikzlibrary{shapes}
\begin{document}

\tikzset{%
peer/.style={draw,circle,violet,bottom color=red, top color= white, text=violet, minimum width=25pt},
superpeer/.style={draw, circle,  left color=orange, text=violet, minimum width=25pt},
point/.style = {fill=black,inner sep=2pt, ellipse, minimum width=10pt,align=right,rotate=60},
forward edge/.style={->, >=stealth, shorten >=0pt, thick, color=blue},
}
\begin{tikzpicture}[auto, node distance=2.8cm]

  \node[superpeer] (A) {$0$};
  \node[peer]  (B)   at ($ (A) + (18:4) $){$1$};  % 72 degree apart
  \node[peer]  (C)   at ($ (A) + (306:4)$){$2$};
  \node[peer]  (D)   at ($ (A) + (234:4)$){$3$};
  \node[point] (p1)  at ($ (A) + (172:4) $){};  
  \node[point] (p2)  at ($ (A) + (162:4) $){};            % center angle  
  \node[point] (p3)  at ($ (A) + (152:4) $){};  
  \node[peer] (E) at ($ (A) + (90:4) $) {$N$};

\path [forward edge] (A.38) edge [sloped,above=1cm] node[label=above:{$\lambda_{01}$}]{}(B.178);
\path [forward edge,] (B.218) edge [sloped,below=1cm] node[label=below:{$\mu_{01}$}]{} (A.-2);

 \path [forward edge] (A.326) edge [sloped] node[] {$\lambda_{02}$} (C.106);
 \path [forward edge] (C.146) edge [sloped] node[] {$\mu_{02}$} (A.286);

  \path [forward edge] (A.214) edge [sloped] node[label={[shift={(0ex,-1ex)}]above:{$\lambda_{03}$}}] {}(D.74);
  \path [forward edge] (D.34) edge [sloped] node[label={[shift={(0ex,2ex)}]below:{$\mu_{03}$}}] {}(A.254);

 \path [forward edge] 
(A.70) edge [sloped] node[label={[shift={(-3ex,-1ex)}]left:{$\lambda_{0N}$}}] {} (E.290) 
(E.250) edge [sloped] node[label={[shift={(-3ex,-1ex)}]right:{$\mu_{0N}$}}] {} (A.110);
\end{tikzpicture}



\end{document}

Answer 2

Remarks

I guess your question was, how to rotate the dots, such that they match the sample picture you posted. We know, that therefore it needs to be orthogonal to the path pointing there from the origin. With some simple mathematics we find

-176 + 90 = -86

Hence the node needs to be rotated by -86°.

\node[rotate=-86] (F) at ($(A) + (-176:4)$) {\mydots};

To get scaled dots, I introduced a macro \mydots. If you want to use the default Computer Modern fonts in the OT1 encoding you might want to declare

\newcommand*\mydots{{\Huge$\cdots$}}

If you load a scalable font like Latin Modern, you can specify arbitrarily large dots

\usepackage{lmodern}
\newcommand*\mydots{{\fontsize{60}{80}\selectfont$\cdots$}}

(I added an extra level of grouping, if you want to use it in another context and don't want the surroundings to be scaled as well)

Complete example

\PassOptionsToPackage{dvipsnames}{xcolor}
\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,calc}
\newcommand*\lav{Lavender}
\newcommand*\oran{Orange}
% We need a scalable font for this
\usepackage{lmodern}
\newcommand*\mydots{{\fontsize{60}{80}\selectfont$\cdots$}}
%\newcommand*\mydots{{\Huge$\cdots$}}% <- Without scalable font
\tikzset{
    peer/.style={
        draw,
        circle,
        violet,
        bottom color=\lav,
        top color= white,
        text=violet,
        minimum width=8pt
    },
    superpeer/.style={
        draw,
        circle,
        BurntOrange,
        left color=\oran,
        text=violet,
        minimum width=25pt
    }
}
\begin{document}
\begin{tikzpicture}[
        ->,
        >=stealth',
        shorten >=1pt,
        auto,
        node distance=2.8cm,
        semithick
    ]
    \node[superpeer]  (A)                       {$0$};
    \node[peer]       (B) at ($(A) + (20:4)$)   {$1$};
    \node[peer]       (C) at ($(A) + (-52:4)$)  {$2$};
    \node[peer]       (D) at ($(A) + (-124:4)$) {$3$};
    \node[rotate=-86] (F) at ($(A) + (-176:4)$) {\mydots};  
    \node[peer]       (E) at ($(A) + (92:4)$)   {$N$};

    \path (A) edge[color=blue,sloped]       node {$\lambda_{01}, \mu_{01}$} (B);
    \path (A) edge[color=blue,sloped]       node {$\lambda_{02}, \mu_{02}$} (C);
    \path (A) edge[color=blue,sloped]       node {$\lambda_{03}, \mu_{03}$} (D);
    \path (A) edge[color=blue,sloped,below] node {$\lambda_{0N}, \mu_{0N}$} (E);
    \path (B) edge[color=blue,sloped]       node {$\lambda_{10}, \mu_{10}$} (A);
    \path (C) edge[color=blue,sloped]       node {$\lambda_{20}, \mu_{20}$} (A);
    \path (D) edge[color=blue,sloped]       node {$\lambda_{30}, \mu_{30}$} (A);
    \path (E) edge[color=blue,sloped,above] node {$\lambda_{N0}, \mu_{N0}$} (A);
\end{tikzpicture}
\end{document}

Output