Create dashed arrows that are adjacent to a set of arrows in Tikz graph

Building on the suggestion by Jasper Habicht and removing your own color restrictions, this is what I came up with:

begin{document}
begin{tikzpicture}
    begin{scope}[every node/.style={circle,thick,draw}]
        node (0) at (0.0,0.0) {$0$};
        node (1) at (2.0,-1.5) {$1$};
        node (2) at (3.5, 0.0) {2};
        node (3) at (3.5,-3.0) {3};
        node (4) at (6.5,0.0) {4};
        node (5) at (6.5,-3.0) {5};
        node (6) at (8.0,-1.5) {6};
        node (7) at (10,0) {7};
        coordinate (8) at (5.0,-3.6);
    end{scope}
    begin{scope}[>={Stealth},
        num/.style={text=black, fill=white,circle, inner sep=0pt,minimum size=1pt},
        every edge/.style={draw,very thick}]
        path [->] (0) edge[bend right=30] (3);
        path [->] (1) edge node {} (2);
        path [->] (1) edge node {} (3);
        path [->] (2) edge node  {} (4);
        path [->] (3) edge node {} (5); 
        path [->] (4) edge node {} (6); 
        path [->] (5) edge node {}  (6); 
        path [->] (5) edge[bend right=30] (7); 
        draw [red, thick ,dashed, ->] (0)  to[out=275,in=175] (8) to[out=0,in=270] (7) ;
        node[num] at (8) {$y_j$};
        path [blue, thick, dashed, ->] (1) edge[out=330,in=210] node[num] {{$r_j$}} (6);
        path [green, thick, dashed, ->] (1) edge[out=30,in=150] node[num] {{$t_j$}} (6);
    end{scope}
end{tikzpicture}
end{document}

By using the [in=<angle>,out=<angle>] option provided by tikz and shifting the ending point of a path as @Jasper Habicht suggested I've come up with this
notice also that I've added a midpoint: (3-5). This is used so that the red arrow does not overlap with the rest of the figure.
If you want to change the distance of the arrow from the figure, change the y coordinate in the

node (3-5) at (5, -4.0) {};

command.

If you do not like the curve of the red arrow, just play with the [in=<angle>,out=<angle>] options (also true for the blue and arrows) in the

draw [->, red, thick, dashed] (0) to [out=230,in=180] (3-5) node {color{black}$y_j$} to [out=360,in=300] (7);

command. Notice that in=180 and out=360 are used to keep the line straight as it passes through the (3-5) node.

If you wish to match the arrow tip colour to its body, remove the [black] option in >={Stealth[black]}.

Code

documentclass{article}
usepackage{tikz}
usetikzlibrary{arrows.meta}

begin{document}
begin{tikzpicture}
    begin{scope}[every node/.style={circle,thick,draw}]
        node (0) at (0.0,0.0)  {$0$};
        node (1) at (2.0,-1.5) {$1$};
        node (2) at (3.5, 0.0) {$2$};
        node (3) at (3.5,-3.0) {$3$};
        node (4) at (6.5,0.0)  {$4$};
        node (5) at (6.5,-3.0) {$5$};
        node (6) at (8.0,-1.5) {$6$};
        node (7) at (10,0) {7};
        node (3-5) at (5, -4.0) {};
    end{scope}
    begin{scope}
    [
        >={Stealth[black]},
        every node/.style={fill=white,circle, inner sep=0pt,minimum size=1pt},
        every edge/.style={draw=black,very thick}
    ]
        path [->] (0) edge[bend right=30] node {}  (3);
        path [->] (1) edge node {} (2);
        path [->] (1) edge node {} (3);
        path [->] (2) edge node {} (4);
        path [->] (3) edge node {} (5); 
        path [->] (4) edge node {} (6); 
        path [->] (5) edge node {}  (6); 
        path [->] (5) edge[bend right=30] node {} (7);

        draw [->, red, thick, dashed] (0) to [out=230,in=180] (3-5) node {color{black}$y_j$} to [out=360,in=300] (7);
        draw [->, blue, thick, dashed]  ([yshift=-2pt]1.east) to [out=-50,in=-130] node {color{black}$r_j$} ([yshift=-2pt]6.west);
        draw [->, green, thick, dashed] ([yshift=2pt]1.east) to [out=50,in=130] node {color{black}$t_j$} ([yshift=2pt]6.west);
    end{scope}
end{tikzpicture}
end{document}