So, we can use use that between the centrality or

any of these centrality metrics to simplify a graph.

And so, for example, if I compute the between the centrality of edges

of my network of yeast proteins.

Edges that have low between the centrality,

have few shortest paths going through them.

And edges with high between the centrality have a lot of edges going through them.

And so a lot of shortest paths going through them.

And so if I remove edges that have few shortest paths going through them

you might think the impact would be less than if I have a lot of shortest paths

going through that edge.

So I've plotted the between the centrality of edges here, and I basically

removed the lowest between the centrality, the ones that are very dark blue and

left the highest between the centrality edges, the ones that are in red here.

And I've not removed any edge if it creates a disconnected graph.

So the result is a graph that

has as many edges as it has nodes, about 1500 edges and 1500 nodes.

So I guess something like a tree that's minimally connected,

but have retained the high between the centrality edges.

And removed the lowest between the centrality edges.

And so what's left is kind of the communications backbone,

the most often used edges when I'm finding the shortest path between any two nodes.

And that simplifies the layout.

Now I've got fewer edges in order to compute

spring distances when I use the same gem layout, force-directed layout for

this graph on the right than I'm using for this graph on the left.

The nodes spread out more because I've got fewer springs.

And the nodes are freer to move around to unique places.

And you can visually see the relationship between nodes better in this layout

than in this layout.

You're also looking at fewer edges,

so you could always add back in the edges as necessary,

to add back in those low between the centrality edges to see a more complete

view of this graph, but the nodes would still be positioned better than they are.

When you try to compute the layout using all of those edges.