Scale-Free Networks Emerging from Weighted Random Graphs

TL;DR

This paper demonstrates that merging percolation clusters in weighted Erdős-Rényi graphs results in a scale-free supernode network with a degree distribution following a power law, revealing a natural emergence of scale-free structures.

Contribution

It introduces a novel method of constructing scale-free networks from weighted random graphs by merging clusters at the percolation threshold, linking optimization to scale-free emergence.

Findings

The supernode network is scale-free with a degree distribution exponent of 2.5.

The minimum spanning tree is composed of percolation clusters interconnected by a scale-free tree.

Optimization induces the spontaneous emergence of the percolation threshold, leading to scale-free structures.

Abstract

We study Erd\"{o}s-R\'enyi random graphs with random weights associated with each link. We generate a new ``Supernode network'' by merging all nodes connected by links having weights below the percolation threshold (percolation clusters) into a single node. We show that this network is scale-free, i.e., the degree distribution is $P(k)\sim k^{-\lambda}$ with $\lambda=2.5$. Our results imply that the minimum spanning tree (MST) in random graphs is composed of percolation clusters, which are interconnected by a set of links that create a scale-free tree with $\lambda=2.5$. We show that optimization causes the percolation threshold to emerge spontaneously, thus creating naturally a scale-free ``supernode network''. We discuss the possibility that this phenomenon is related to the evolution of several real world scale-free networks.