Emergence of scale-free behavior in networks from limited-horizon linking and cost trade-offs

TL;DR

This paper introduces a local-growth network model on a sphere where edges form based on gain-cost criteria, leading to scale-free degree distributions under certain conditions, revealing how local rules can produce complex network behaviors.

Contribution

The study presents a novel local-growth mechanism on a spherical surface that results in scale-free networks, highlighting the emergence of power-law degree distributions from local gain-cost rules.

Findings

Power-law degree distributions emerge under specific parameter regimes.

Network properties such as connectivity and distances are analyzed.

Scale-free behavior is invariant for certain parameter scalings.

Abstract

We study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere. The growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria. Only nodes that are not too far apart on the sphere may be considered for being joined by an edge. Given two such nodes, the joining occurs only if the gain of doing it surpasses the cost. Our model is based on a multiplicative parameter lambda that regulates, in a function of node degrees, the maximum geodesic distance that is allowed between nodes for them to be considered for joining. For n nodes distributed uniformly on the sphere, and for lambda*sqrt(n) within limits that depend on cost-related parameters, we have found that our growth mechanism gives rise to power-law distributions of node degree that are invariant for constant lambda*sqrt(n). We also study connectivity- and distance-related properties of the networks.