Metric structural human connectomes: Localization and multifractality of eigenmodes

We explore the fundamental principles underlying the architecture of the human brain's structural connectome through the lens of spectral analysis of Laplacian and adjacency matrices. Building on the idea that the brain balances efficient information processing with minimizing wiring costs, our goal is to understand how the metric properties of the connectome relate to the presence of an inherent scale. We demonstrate that a simple generative model combining nonlinear preferential attachment with an exponential penalty for spatial distance between nodes can effectively reproduce several key features of the human connectome. These include spectral density, edge length distribution, eigenmode localization, local clustering, and topological properties. Additionally, we examine the finer spectral characteristics of human structural connectomes by evaluating the inverse participation ratios (IPR _q ) across various parts of the spectrum. Our analysis shows that the level statistics in the soft cluster region of the Laplacian spectrum (where eigenvalues are small) deviate from a purely Poisson distribution due to interactions between clusters. Furthermore, we identify localized modes with large IPR values in the continuous spectrum. Multiple fractal eigenmodes are found across different parts of the spectrum, and we evaluate their fractal dimensions. We also find a power-law behavior in the return probability-a hallmark of critical behavior-and conclude by discussing how our findings are related to previous conjectures that the brain operates in an extended critical phase that supports multifractality.