  We study the structure of the load-based spanning tree (LST) that carries the
maximum weight of the Erdos-Renyi (ER) random network. The weight of an edge is
given by the edge-betweenness centrality, the effective number of shortest
paths through the edge. We find that the LSTs present very inhomogeneous
structures in contrast to the homogeneous structures of the original networks.
Moreover, it turns out that the structure of the LST changes dramatically as
the edge density of an ER network increases, from scale free with a cutoff,
scale free, to a starlike topology. These would not be possible if the weights
are randomly distributed, which implies that topology of the shortest path is
correlated in spite of the homogeneous topology of the random network.
