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I just started doing things with TikZ today and I run into a problem: there is just no example code snippets for typesetting directed, weighted graphs. Can anyone supply one simple example in an answer?

I made tkz-graph and tkz-berge to help beginners to draw some graphs. tkz-berge is used for specials graphs (named graphs in graph theory)

You can use only tikz to draw graphs.
Version with tkz-graph

documentclass{article}

usepackage[utf8]{inputenc}

usepackage[usenames,dvipsnames]{xcolor}

usepackage{fullpage}

usepackage[upright]{fourier}

usepackage{tkz-graph}

usetikzlibrary{arrows}

thispagestyle{empty}

begin{document}

SetVertexNormal[Shape      = circle,

                 FillColor  = orange,

                 LineWidth  = 2pt]

SetUpEdge[lw         = 1.5pt,

           color      = black,

           labelcolor = white,

           labeltext  = red,

           labelstyle = {sloped,draw,text=blue}]

begin{center}

begin{tikzpicture}

   Vertex[x=0 ,y=0]{K}

   Vertex[x=0 ,y=2]{F}

   Vertex[x=-1,y=4]{D}

   Vertex[x=3 ,y=7]{H}

   Vertex[x=8 ,y=5]{B}

   Vertex[x=9 ,y=2]{N}

   Vertex[x=5 ,y=0]{M}

   Vertex[x=3 ,y=1]{S}

   tikzset{EdgeStyle/.append style = {bend left}}

   Edge[label = $120$](K)(F)

   Edge[label = $650$](H)(S)

   Edge[label = $780$](H)(M)

   Edge[label = $490$](D)(B)

   Edge[label = $600$](D)(M)

   Edge[label = $580$](B)(M)

   Edge[label = $600$](H)(N)

   Edge[label = $490$](F)(H)

   tikzset{EdgeStyle/.append style = {bend right}}

   Edge[label = $630$](S)(B)

   Edge[label = $210$](S)(N)

   Edge[label = $230$](S)(M)

end{tikzpicture}

end{center}



end{document} 

With arrows on edges

documentclass[11pt]{scrartcl}

usepackage{tkz-graph}





begin{document}

begin{tikzpicture}

 SetUpEdge[lw         = 1.5pt,

            color      = orange,

            labelcolor = white]

  GraphInit[vstyle=Normal] 

  SetGraphUnit{3}

  tikzset{VertexStyle/.append  style={fill}}

  Vertex{P}

  NOEA(P){B}  SOEA(P){M} NOEA(B){D}

  SOEA(B){C}  SOEA(C){L}

  tikzset{EdgeStyle/.style={->}}

  Edge[label=$3$](C)(B)

  Edge[label=$10$](D)(B)

  Edge[label=$10$](L)(M)

  Edge[label=$10$](B)(P)

  tikzset{EdgeStyle/.style={<->}}

  Edge[label=$4$](P)(M)

  Edge[label=$9$](C)(M)

  Edge[label=$4$](C)(L)

  Edge[label=$5$](C)(D)

  Edge[label=$10$](B)(M)

  tikzset{EdgeStyle/.style={<->,relative=false,in=0,out=60}}

  Edge[label=$11$](L)(D)

end{tikzpicture}

end{document}

update : Version with tikz and automata

documentclass[11pt]{scrartcl} 

PassOptionsToPackage{usenames,dvipsnames,svgnames}{xcolor}  

usepackage{tikz}

usetikzlibrary{arrows,positioning,automata}



begin{document}

begin{tikzpicture}[>=stealth',shorten >=1pt,node distance=3cm,on grid,initial/.style    ={}]

  node[state]          (P)                        {$P$};

  node[state]          (B) [above right =of P]    {$B$};

  node[state]          (M) [below right =of P]    {$M$};

  node[state]          (D) [above right =of B]    {$D$};

  node[state]          (C) [below right =of B]    {$C$};

  node[state]          (L) [below right =of C]    {$L$};

tikzset{mystyle/.style={->,double=orange}} 

tikzset{every node/.style={fill=white}} 

path (C)     edge [mystyle]    node   {$3$} (B)

      (D)     edge [mystyle]    node   {$10$} (B) 

      (L)     edge [mystyle]    node   {$10$} (M)

      (B)     edge [mystyle]    node   {$10$} (P);

tikzset{mystyle/.style={<->,double=orange}}   

path (P)     edge [mystyle]   node   {$4$} (M)

      (C)     edge [mystyle]   node   {$9$} (M) 

      (C)     edge [mystyle]   node   {$4$} (D)

      (B)     edge [mystyle]   node   {$5$} (M);

tikzset{mystyle/.style={<->,relative=false,in=0,out=60,double=orange}}

path (L)     edge [mystyle]   node   {$10$} (D); 

end{tikzpicture}

end{document}

Here is a stripped down version usking tkz-graph of Alain Matthes's code that provides reasonable looking (for CS paper) defaults.

documentclass{article}

usepackage{tkz-graph}



begin{document}



begin{figure}

centering

begin{tikzpicture}[scale=0.75,transform shape]

  Vertex[x=0,y=0]{K}

  Vertex[x=0,y=2]{F}

  Vertex[x=-1,y=4]{D}

  Vertex[x=3,y=7]{H}

  Vertex[x=8,y=5]{B}

  Vertex[x=9,y=2]{N}

  Vertex[x=5,y=0]{M}

  Vertex[x=3,y=3]{S}

  tikzstyle{LabelStyle}=[fill=white,sloped]

  tikzstyle{EdgeStyle}=

  Edge[label=$120$](K)(F)

  Edge[label=$650$](H)(S)

  Edge[label=$780$](H)(M)

  Edge[label=$490$](D)(B)

  Edge[label=$600$](D)(M)

  Edge[label=$580$](B)(M)

  Edge[label=$600$](H)(N)

  Edge[label=$490$](F)(H)

  tikzstyle{EdgeStyle}=[pre, bend right]

  Edge[label=$210$](S)(N)

  Edge[label=$230$](S)(M)

end{tikzpicture}

end{figure}



end{document}