Making a network graph more readable

I have a graph showing nodes on the map of the US. There are between 100 and 400 nodes, and all two nodes have a line connecting them.

The problem is that there are so many connections that the graph becomes virtually meaningless. To deal with this, I have removed, through some simple algorithm, about 10% of nodes each time. Still, however, the graph is not very readable. This is what it looks like with 10% removed:

The purpose of the graph is to see which connections are strongest (the brightest lines indicate strongest connections). But, as I'm sure you would agree, it's pretty hard to interpret anything.

What are your suggestions for making this graph more readable?