Minimum Spanning Trees on Random Networks

We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Because of this geometric universality, we are able to characterize the energy of MST using a scaling distribution $\left[P\right(\epsilon \left)\right]$ found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to $P\left(\epsilon \right)$. We discuss the relationship to invasion percolation, to the directed polymer in a random media, to uniform spanning trees, and also the implications for the broader issue of universality in disordered systems.