Suppose I have a $m \times m$ sheet of paper. Now suppose I draw a line of length $L$. The center of the line must lie within the piece of paper, and the orientation of the line is randomly chosen from a uniform distribution. So now I've got a random line of length L drawn on the page. Now I do this for $N$ lines, so I've a got just $N$ randomly drawn lines on the page. I'm trying to find some features of this network, like the clustering coefficient and average shortest path length between sticks.
Does this problem sound familiar to anyone? Can I use a random graph to represent such a network?
This is not really an answer but it is way too long for a comment.
How do you switch from this construction to a graph ? There are several possibilities.
You could take a vertex for each two lines intersecting within the sheet, and there in an edge if two vertices belong to the same line. i.e. a collinearity graph of your (possible degenerate) point line geometry (I precise degenerate because you may have lines with no points on them).
Or you could put an edge between two vertices only if there are in consecutive order on the line, keeping the look of your drawing.
In summary given the following configuration of your drawing :
The resulting graph could be in the first definition
or in the second case
Also how do you deal with the end of your segment/line. If it ends inside the sheet, is the end also a vertex ?
The second cases might be more appealing if you want to keep the look of your network but it is quite limited: Given that you pick an orientation among infinite possible values, with high probability your graph will have max degree 4. Hence you only have vertices of degree 2,3 or 4 (and possibly 1 depending on the end-segment case). I think that you won't be able to use standard random graphs.
However if you go for the collinearity graph of your point-line geometry, then you might have more options.