We live in a world of interconnected systems. From the intricate web of the internet to the complex network of neurons in our brains, understanding how these connections work is crucial.

One key concept in understanding these complex systems is “scale invariance.” This means that the system looks similar at different levels of detail. For example, a fractal, like a snowflake, exhibits scale invariance: if you zoom in on a tiny part, you’ll see the same intricate patterns as the whole snowflake.

Similarly, in a truly scale-invariant network, the overall structure remains consistent regardless of the scale at which you observe it. 

 The Renormalization Group is a set of techniques to study the effects of such scale transformations in physical systems.However, determining if a network truly exhibits scale invariance can be challenging, especially for biological and technical networks, such as organisms’ metabolic pathways and the Internet. 

A research team from CREF, in collaboration with Sapienza Università di Roma, Roma3, Univesidad de Granada and ISC-Istituto dei sistemi complessi, has broadened the scope of renormalization to arbitrary networked systems beyond grid structures. In a recent paper, Villegas and al. developed a new framework, the “Laplacian Renormalization Group,” to investigate scale invariance in networks. This method analyzes how information flows and spreads within the network, revealing patterns that emerge at different levels.

“Imagine an oil stain spreading across a network,” explains Pablo Villegas. “If the stain maintains its characteristics regardless of its size and the size of the network itself, we are likely dealing with a scale-invariant network.”

Applying this framework to the human brain, researchers found intriguing results.

“We studied the connectome, a map of neural connections in the brain,” says Villegas. “Our findings suggest that the human brain exhibits scale-invariant properties, resembling hierarchical modular networks. This is akin to a Matrioska doll, with intricate structures nested within each other.”

These findings have significant implications for our understanding of brain function. By unraveling the principles of scale invariance in the brain, we can gain valuable insights into how this complex organ operates and how it might be affected by various conditions.

The paper “Networks with Many Structural Scales: A Renormalization Group Perspective”, has been published 5 February, 2025 on Phys. Rev. Lett. 134, 057401

DOI: https://doi.org/10.1103/PhysRevLett.134.057401

Authors:

Anna Poggialini (Sapienza, CREF), Pablo Villegas (CREF, Universidad de Granada), Miguel A. Muñoz (Universidad de Granada), Andrea Gabrielli (CREF, Roma Tre, Istituto dei Sistemi Complessi (ISC)-CNR)