Geometric Criticality in Scale-Invariant Networks

TL;DR

This paper explores how structural perturbations affect the dimensionality and stability of scale-invariant networks, revealing a new phenomenon called geometric criticality that influences network behavior and phase transitions.

Contribution

It introduces the concept of geometric criticality in scale-invariant networks and analyzes how structural fixed points and basin stability relate to phase transitions.

Findings

Identification of geometric breakdown leading to fractal dimensions

Discovery of hidden LRG flows toward unstable fixed points

Connection between topology, stability, and phase transitions in networks

Abstract

Dimension in physical systems determines universal properties at criticality. Yet, the impact of structural perturbations on dimensionality remains largely unexplored. Here, we characterize the attraction basins of structural fixed points in scale-invariant networks from a renormalization group perspective, demonstrating that basin stability connects to a structural phase transition. This topology-dependent effect, which we term geometric criticality, triggers a geometric breakdown hitherto unknown, which induces non-trivial fractal dimensions and unveils hidden LRG flows toward unstable structural fixed points. Our systematic study of how networks and lattices respond to disorder paves the way for future analysis of non-ergodic behavior induced by quenched disorder.