My code here does what I would like, except some node outlines are dashed where they should be solid. I realise I am probably putting the dashed in the wrong place, but I just tried my way to this.
\tikzstyle{every node}=[circle, draw, fill=black!50, inner sep=0pt, minimum width=4pt]
\begin{tikzpicture}[thick,scale=0.8,->,shorten >=2pt]
\draw (0,0) node {} -- (1,1) node {};
\draw (1,1) node {} -- (2,1) node {};
\draw (2,1) node {} -- (3,2) node {};
\draw (3,2) node {} -- (4,1) node {};
\draw (0,2) node {} -- (1,1) [dashed] node {};
\draw (1,2) node {} -- (2,1) [dashed] node {};
\draw (2,1) node {} -- (3,2) node {};
\draw (3,2) node {} -- (4,1) node {};
\draw (2,1) node {} -- (3,0) [dashed] node {};
\draw (2,1) node {} -- (2,0) [dashed] node {};
\draw (3,0) node {} -- (4,0) [dashed] node {};
\draw (3,0) node {} -- (4,-1) [dashed] node {};
\draw (1,1) node {} -- (1,0) [dashed] node {};
\end{tikzpicture}

