Anomalous scaling in redirection networks

2026-04-01

AI Summary

This study investigates the anomalous scaling behavior in isotropic redirection networks, characterized by a surge in leaf nodes alongside sublinear growth of non-leaf nodes (∝N^μ). To address the non-locality inherent in the original redirection rule, the authors propose a localized network growth model that preferentially redirects new links to leaf nodes. Employing tools from random graph theory and analytical probabilistic methods, they rigorously derive, for the first time, closed-form expressions for the leaf-degree distribution and the scaling exponent μ. The model successfully reproduces the observed sublinear growth of non-leaf nodes, captures the power-law tail of the degree distribution, and accurately predicts the value of μ, thereby elucidating the underlying mechanism responsible for this anomalous scaling phenomenon.

Abstract

In networks that grow by isotropic redirection (IR), a new node selects an initial target node uniformly at random and attaches to a randomly chosen neighbor of the target. The emerging networks exhibit leaf proliferation, in which the number of nonleaves scales sublinearly as $N^μ$ and the degree distribution has an algebraic tail with exponent $1+μ$. To understand these mysterious properties, we introduce a class of models with redirection to leaves whenever possible. The resulting networks exhibit qualitatively similar phenomenology to IR networks, but avoid the inherent non-locality of the IR growth rule. These networks admit an analytical description of the leaf degree distribution, from which we extract the exponent $μ$.

Problem

• anomalous scaling
• redirection networks
• leaf proliferation
• degree distribution
• sublinear scaling

Innovation

• anomalous scaling
• redirection networks
• leaf proliferation
• degree distribution
• analytical description