Inhomogeneous substructures hidden in random networks

TL;DR

This paper investigates the structure of load-based spanning trees in Erdős-Rényi networks, revealing inhomogeneous, scale-free, and starlike topologies driven by edge betweenness centrality, despite the underlying network's homogeneity.

Contribution

It uncovers how the load-based spanning tree structure varies with edge density, showing a transition from scale-free to starlike topologies, highlighting correlations in shortest path structures.

Findings

LSTs are highly inhomogeneous compared to original networks.

The topology of LSTs changes with edge density, from scale-free to starlike.

Shortest path topology is correlated despite the network's homogeneity.

Abstract

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.