Spectral Coarse-Graining and Rescaling for Preserving Structural and Dynamical Properties in Graphs

TL;DR

This paper presents a spectral coarse-graining method for graphs that preserves key structural and dynamical properties, enabling efficient analysis of large networks and revealing insights into brain activity patterns across different states.

Contribution

It introduces a novel graph renormalization technique based on spectral properties that maintains essential features while reducing complexity, applicable to large-scale brain network data.

Findings

Reveals macroscopic brain activity patterns during rest and attention tasks.

Shows dynamic reorganization of neuronal interactions across scales.

Facilitates analysis of large graphs by reducing vertices without losing critical information.

Abstract

We introduce a graph renormalization procedure based on the coarse-grained Laplacian, which generates reduced-complexity representations for characteristic scales identified through the spectral gap. This method retains both diffusion probabilities and large-scale topological structures, while reducing redundant information, facilitating the analysis of large graphs by decreasing the number of vertices. Applied to graphs derived from EEG recordings of human brain activity, our approach reveals macroscopic properties emerging from neuronal interactions, such as collective behavior in the form of coordinated neuronal activity. Additionally, it shows dynamic reorganization of brain activity across scales, with more generalized patterns during rest and more specialized and scale-invariant activity in the occipital lobe during attention-focused tasks.