Visualizing a graph

I have a finite but huge metric graph with say 1000 vertices. It comes say as 10000x10000 symmetric matrix filled by $0,1,\dots$ and $\infty$; 0's on the diagonal and $\infty$ is for pairs of vertices which are not connected. (Most of the vertices have degree 3.)

I need to find a way to visualize this graph. I hope it will help me to see some patterns.

Is there any software which could help? Say, I want to draw this graph in the space in a form which reflects its metric geometry.

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Have you tried with Graphviz ? Or with Sage and with the plot3d() method ? Maybe your graph is too big to be drawn correctly with these tools anyway. –  Samuele Giraudo Feb 26 '13 at 21:04
@Samuelle, no I did not, can it take the metric into account? –  ε-δ Feb 26 '13 at 21:22
Where does this graph come from? –  Per Alexandersson Feb 26 '13 at 21:28