How can I draw a graph like the diagram below?

Hint:

usepackage{tikz}
usetikzlibrary{intersections}




begin{tikzpicture}

node[fill=black, draw,circle] (a1) at (-7.5,-3.5) {};
node[fill=black, draw,circle] (a2) at (-7.5,-1) {};
node[fill=black, draw,circle] (a3) at (-7.5,1) {};
node[fill=black, draw,circle] (a4) at (-7.5,3) {};

node[fill=black, draw,circle] (b1) at (7.5,-3.5) {};
node[fill=black, draw,circle] (b2) at (7.5,-1) {};
node[fill=black, draw,circle] (b3) at (7.5,1) {};
node[fill=black, draw,circle] (b4) at (7.5,3) {};


draw[name path=a1b4] (b4.center) .. controls (2.8,3.4) and (-3,-1.2) .. (a1.center);
draw[name path=a2b3] (b3.center) .. controls (2.1,0.9) and (-2.3,-5.8) .. (a2.center);
draw[name path=a3b1] (b1.center) .. controls (3.9,-3.4) and (-2.6,1.5) .. (a3.center);
draw[name path=a4b2] (b2.center) .. controls (4.1,3.3) and (-0.3,2.3) .. (a4.center);

path [name intersections={of=a4b2 and a1b4, by=ab14}];
draw (ab14) node[fill=black, draw,circle,inner sep=3pt,]  {}node[above=4pt]{$14$};
end{tikzpicture}

Use color=purple inside draw[] to colorize the path. Draw other intersection points yourself.

A variation and completing of the @C.F.G. answer (+1):

documentclass[tikz, margin=6pt]{standalone}
usetikzlibrary{intersections}

begin{document}
    begin{tikzpicture}[
arr/.style = {draw=#1, very thick},
dot/.style = {circle, fill=#1, outer sep=0pt},
dot/.default = black,
                        ]
    foreach y [count=i] in {-3.5,-1,1,3}
{                        
node[dot, label=left: i] (ai) at (-7.5,y) {};
node[dot, label=right:i'] (bi) at ( 7.5,y) {};
}
%
draw[arr=blue, name path=A] (b4) .. controls (2.8, 3.4) and (-3,-1.2) .. (a1);
draw[arr=teal, name path=B] (b3) .. controls (2.1, 0.9) and (-2.3,-5.8) .. (a2);
draw[arr=teal!50, name path=C] (b1) .. controls (3.9,-3.4) and (-2.6,1.5) .. (a3);
draw[arr=blue!50, name path=D] (b2) .. controls (4.1, 3.3) and (-0.3,2.3) .. (a4);

path [name intersections={of=A and D, by=n14}]
    node (s3) [dot=green,label=14] at (n14) {};
path [name intersections={of=A and B, by=n13}]
    node (s2) [dot=green,label=13] at (n13) {};
path [name intersections={of=A and C, by=n12}]
    node (s1)[dot=green,label=12] at (n12) {};
%
path [name intersections={of=B and C, by=n23}]
    node[dot=green,label=23] at (n23) {};
path [name intersections={of=B and D, by=n24}]
    node[dot=green,label=23] at (n23) {};
%
node [below] at (a1 -| n12) {$s_1$};
node [below] at (a1 -| n13) {$s_2$};
node [below] at (a1 -| n14) {$s_1$};
node [below] at (a1 -| n23) {$s_3$};
node [below] at (a1 -| n24) {$s_2$};
end{tikzpicture}
end{document}

An alternative in Metapost. This is wrapped up in luamplib so compile with lualatex.

documentclass[border=5mm]{standalone}
usepackage{luamplib}
begin{document}
mplibtextextlabel{enable}
begin{mplibcode}
beginfig(1);
    % define a unit for the points
    numeric u; u = 1cm;

    % define all the points, using the magic z macro and note the ' in suffixes...
    z1 = (0, 1u);  z1' = (7u, 1u);
    z2 = (0, 2u);  z2' = (7u, 2u);
    z3 = (0, 3u);  z3' = (7u, 3u);
    z4 = (0, 4u);  z4' = (7u, 4u);

    z12 = (1u, 1.4u);
    z13 = (3u, 2.5u);
    z23 = (4u, 2u);
    z14 = (5u, 3.6u);
    z24 = (6u, 2.9u);

    % draw the lines with some nice colours
    input colorbrewer
    draw z4{20,-1} .. z14 .. z24 .. z2' withcolor Blues 4 4;
    draw z3 .. z13 .. z23 ..        z1' withcolor Blues 4 2;
    draw z2 .. z12 .. z23 .. z24 .. z3' withcolor Greens 4 4;
    draw z1 .. z12 .. z13 .. z14 .. z4' withcolor Greens 4 2;

    % dot labels on left, and then the right, using "forsuffix" loops...
    forsuffixes @=1,2,3,4: dotlabel.ulft("$" & str @ & "$", z@); endfor
    forsuffixes @=1',2',3',4': dotlabel.urt("$" & str @ & "$", z@); endfor

    % finally mark all the intersections
    interim labeloffset := 6;      % allow a bit more space    
    drawoptions(withcolor 3/4 Greens 3 3);   % use dark green ink
    forsuffixes @=12, 13, 14, 23, 24: 
        % mark the intersection
        dotlabel.top("$" & str @ & "$", z@);
        % strip the two numbers from the suffix
        numeric a, b; 
        a = scantokens substring (0,1) of str @;
        b = scantokens substring (1,2) of str @;
        % add the appropriate label at the bottom
        label.bot("$s_" & decimal (b-a) & "$", (x@, y1));
    endfor    
    drawoptions();

endfig;
end{mplibcode}
end{document}

Notes

• it was easier to define the intersections where they were needed and then draw the paths through them. But you could do it the other way round, by defining the paths first and then using intersectionpoint to work out where they cross.

• I am using Metapost Colorbrewer for the colours/colors.

• Notice that you can use ' in a suffix name

• Note that str suffix turns a suffix into a string.

• ... and that then you can concatenate string literals / variable with &.

• the final loop shows an example of how to turn a string back into a number using scantokens

• ... and also the convenient way we can pick out the xpart and ypart of z variables. So for the s... labels along the bottom, x@ refers to the xpart of z@ and y1 gives us the ypart of z1.