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.